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433 Cards in this Set
- Front
- Back
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Polyhedra
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Solids with flat faces
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Prism
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has two congruent parallel faces and a set of parallel edges that connect corresponding vertices of the two parallel faces
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Bases
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two parallel congruent faces
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Lateral Edges
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parallel edges joining the vertices of the bases
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Lateral Faces
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faces of a prism that are not bases
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Lateral Surface Area
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the sum of the areas of the lateral faces
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Total Surface Area
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the sum of the prism's lateral area and the area of the two bases
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Pyramid has
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has only one base and lateral edges are not parallel, but meet at a single point called the vertex
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Regular pyramid has
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has a regular polygon as its base and has congruent lateral edges.
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Surface Area of Regular Pyramid
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½ blt +B
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Lateral Surface Area of Regular Pyramid
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½ blt
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Lateral Area of cylinder
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2𝜋rh
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Theorem 113
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The lateral surface area of a cylinder is equal to the product of the height and the circumference of the base.
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Theorem 114
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The lateral surface area of a cone is equal to one half the product of the slant height and the circumference of the base.
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Lateral area of cone
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½ Cl or 𝜋rl
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Postulate of sphere surface area
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TAsphere or 4𝜋r^2
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Volume of a solid
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number of cubic units of space contained by the solid
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Postulate of volumes of prisms and pyramids
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The volume of a right rectangular prism is equal to the product of its length, its width, and its height.
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volume of right rectangular prism
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length*width*height
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Theorem 115
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The volume of a right rectangular prism is equal to the product of the height and the area of the base.
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Formula for rectangular prism volume.
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V of rect. box=Bh
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Theorem 116
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The volume of any prism is equal to the product of the height and the area of the base.
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Theorem 117
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The volume of cylinder is equal to the product of the height and the area of the base.
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Volume of cylinder
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Bh=𝜋r2h Where B is the area of the base, h is the height, and r is the radius of the base.
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Theorem 118
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The volume of a prism or a cylinder is equal to the product of the figure's cross-sectional area and its height.
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Volume of cross section prism or cylinder
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/Ch
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Theorem 119
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The volume of a pyramid is equal to one third of the product of the height and the area of the base.
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Theorem 120
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The volume of a cone is equal to one third of the product of the height and the area of the base.
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Pyramid volume
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⅓ Bh=⅓ 𝜋r^2h
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Cross section
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Cross sections are similar shapes to the base.
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Theorem 121
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In a pyramid or a cone, the ratio of the area of a cross section to the area of the base equals the square of the ratio of the figures' respective distances from the vertex.
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Theorem 122
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The volume of sphere is equal to four thirds the product of 𝜋 and the cube of the radius.
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Volume of a sphere
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4/3𝜋r^3
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The area of a closed region-
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the number of square units of space within the boundary of the region.
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Area Postulate 1
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The area of a rectangle is equal to the product of the base and the height for that base.
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Arect=bh, where b is the length of the base and h is the height.
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Theorem 99
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The area of a square is equal to the square of a side.
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Asq=s2, where s is the length of a side.
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Area postulate 2
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Every closed region has an area.
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Area postulate 3
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If two closed figures are congruent, then their areas are equal.
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Area postulate 4
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If two closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual areas.
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Theorem 100(Parallelogram area)
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The area of a parallelogram is equal to the product of the base and the height.
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A = bh, where b is the length of the base and h is the height.
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Theorem 101(Triangle area)
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The area of a triangle is equal to one-half the product of a base and the height (or altitude) for that base.
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Atri=½ bh, where b is the length of the base and h is the altitude.
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Theorem 102(Area of trapezoid)
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The area of a trapezoid equals one-half the product of the height and the sum of the bases.
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Atrap=½ h(b1+b2), where b1 is the length of one base, b2 is the length of the other base, and h is the height.
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Theorem 103(Median of trapezoid)
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The measure of the median of a trapezoid equals the average of the measures of the bases.
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M= ½ (b1+b2), where b1 is the length of one base and b2 is the length of the other base.
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Theorem 104(Trapezoid area)
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The area of a trapezoid is the product of the median and the height.
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Atrap=Mh, where M is the length of the median and h is the height.
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Theorem 105(Kite area)
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The area of a kite equals half the product of its diagonals.
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Akite=½ d1d2, where d1 is the length of one diagonal and d2 is the length of the other diagonal.
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Theorem 106(Equilateral triangle area)
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The area of an equilateral triangle equals the product of one-fourth the square of a side and the square root of three.
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A eq. tri=s2/4 ·√3, where s is the length of a side.
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A radius of a regular polygon
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a segment joining the center to any vertex.
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An apothem of a regular polygon
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a segment joining the center to the midpoint of any side.
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Theorem 107(Regular polygon area)
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The area of a regular polygon equals one-half the product of the apothem and the perimeter.
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Areg. poly.=½ ap, where a is the length of the apothem and p is the perimeter.
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Area postulate 5
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The area of a circle is equal to the product of 𝜋 and the square of the radius.
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Acirc=𝜋r2, where r is the radius.
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A sector of a circle
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a region bounded by two radii and an arc of the circle.
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Theorem 108(Area of circle sector)
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The area of a sector of a circle is equal to the area of the circle times the fractional part of the circle determined by the sector's arc.
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A segment of a circle
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a region bounded by a chord of the circle and its corresponding arc.
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Theorem 109(Ratio of areas)
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If two figures are similar, then the ratio of their areas equals the square of the ratio of corresponding segments. (Similar-Figures Theorem)
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Where A1 and A2 are areas and s1 and s2 are measures of corresponding segments.
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Theorem 110
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A median of a triangle divides the triangle into two triangles with equal areas.
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Theorem 111(Heron's Formula)
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Where a, b, and c are the lengths of the sides of the triangle and s = semiperimeter.
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(Heron's or Hero's formula) [Sq. root(s(s-a)(s-b)(s-c))]
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Theorem 112(Bhramagupta's formula)
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Only in a quadrilateral that can be inscribed in a circle known as a cyclic quadrilateral,
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Where a, b, c, and d are the sides of the quadrilateral and s = the semiperimeter. [Square root((s-a)(s-b)(s-c)(s-d))}
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Circle
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the set of all points in a plane that are a given distance from a given point in the plane
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Center
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- the given point of a circle
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Radius
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- the given distance of a circle
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Concentric
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- two or more coplanar circles with the same center
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Congruent
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- two circles are congruent if they have the same radii
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Interior point
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- a point is inside the circle if its distance from the center is less than the radius
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Exterior
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- a point is on the exterior of a circle if its distance from the center is greater than the radii
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On the circle
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- a point is on the circle if its distance from the center is equal to the radius
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Chord
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- a segment joining any two points on a circle
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Diameter
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- a chord that passes through the center of the circle
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Area of a Circle
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- A=𝜋r²
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Circumference of a Circle
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- C=𝜋d
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Radius Chord Relationship
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- the distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord.
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Arc
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- two points on a circle and all the points on the circle needed to connect the points by a single path
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Center of an arc
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- the center of the circle of which the arc is a part
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Central angle
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- an angle whose vertex is at the center of a circle
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Minor arc
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- an arc whose points are on or between the sides of a central angle
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Major arc
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- an arc whose points are on or outside of a central angle
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Semicircle
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- an arc whose endpoints are the endpoints of a diameter
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The measure of a major arc
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- 360 minus the measure of the minor arc with the same endpoints
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Two arcs are congruent whenever
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- they have the same measure and are parts of the same circle or congruent circles
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A secant
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is a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.)
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A tangent
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is a line that intersects a circle at exactly one point. The point is called the point of tangency or point of contact.
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Chapter 10 postulate 1
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A tangent line is perpendicular to the radius drawn to the point of contact.
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Chapter 10 postulate 2
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If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle
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A tangent segment
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the part of a tangent line between the point of contact and a point outside the circle.
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A secant segment
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the part of a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle.
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The external part of a secant segment
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the part of a secant line that joins the outside point to the nearer intersection point.
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Tangent circles
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circles that intersect each other at exactly one point.
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Externally tangent circles
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lie outside each other.
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Internally tangent circles
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lie inside each other.
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A common tangent
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a line tangent to two circles.
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Common tangent procedure step 1
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Draw the segment joining the centers.
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Common tangent procedure step 2
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Draw the radii to the points of contact.
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Common tangent procedure step 3
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Through the center of the smaller circle, draw a line parallel to the common tangent.
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Common tangent procedure step 4
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Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle.
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Common tangent procedure step 5
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Use the Pythagorean Theorem and properties of a rectangle.
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The measure of an angle whose sides intersect a circle
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determined by the measure of its intercepted arcs. The location of the vertex of each angle is the key to remembering how to compute the measure of an angle.
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An inscribed angle
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an angle whose vertex is on a circle and whose sides are determined by two chords.
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A tangent-chord angle
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an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the point of tangency.
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A secant-secant angle
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an angle whose vertex is outside a circle and whose sides are determined by two secants.
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A secant-tangent angle
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an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.
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A tangent-tangent angle
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an angle whose vertex is outside a circle and whose sides are determined by two tangents.
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When is a polygon inscribed in a circle?
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A polygon is inscribed in a circle if all of its vertices lie on the circle.
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When is a polygon circumscribed in a circle?
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A polygon is circumscribed about a circle if each of its sides is tangent to the circle.
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The center of a circle circumscribed about a polygon
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the circumcenter of the polygon.
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The center of a circle inscribed in a polygon
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the incenter of the polygon.
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Theorem 30
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The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
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Theorem 31
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If two alternate interior angles are congruent, then lines are parallel.
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Theorem 32
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If two alternate exterior angles are congruent, then lines are parallel.
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Theorem 33
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If two corresponding angles are congruent, then lines are parallel.
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Theorem 34
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If two interior angles on the same side of the transversal are supplementary, then lines are parallel.
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Theorem 35
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If two exterior angles on the same side of the transversal are supplementary, then lines are parallel.
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Theorem 36
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If two coplanar lines are perpendicular to a third line, then they are parallel.
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Parallel postulate
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Given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.
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Theorem 37
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If lines are parallel, then alternate interior angles are congruent.
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Theorem 38
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If lines are parallel, then any pair of the angles formed are either congruent or supplementary.
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Theorem 39
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If lines are parallel, then alternate exterior angles are congruent.
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Theorem 40
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If lines are parallel, then corresponding angles are congruent.
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Theorem 41
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If lines are parallel, then each pair of interior angles on the same side of the transversal are supplementary.
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Theorem 42
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If lines are parallel, then each pair of exterior angles on the same side of the transversal are supplementary.
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Theorem 43
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In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
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Theorem 44
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If two lines are parallel to a third line, then they are parallel to each other.
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Polygons
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Plane figures
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Convex polygons
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A polygon in which each interior angle has a measure less than 180 degrees
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Diagonal
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A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon
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Quadlirateral
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Four sided polygon
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Parallelogram
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A quadrilateral in which both pairs of opposite sides are parallel. Opposite sides are parallel by definition.
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Opposite sides are congruent.
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Opposite angles are congruent.
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Diagonals bisect each other.
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Any pair of consecutive angles are supplementary.
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Rectangle
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A parallelogram in which at least one angle is a right angle. All properties of a parallelogram apply by definition.
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All angles are right angles.
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Diagonals are congruent.
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Rhombus
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A parallelogram in which at least two consecutive sides are congruent. All properties of a parallelogram apply by definition
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All properties of a kite apply.
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All sides are congruent (Rhombus is equilateral).
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Diagonals bisect the angles.
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Diagonals are perpendicular bisectors of each other.
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Diagonals divide the rhombus into four congruent right triangles.
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Kite
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A quadrilateral in which two disjoint pairs of consecutive sides are congruent. Two disjoint pairs of consecutive sides are congruent by definition.
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Diagonals are perpendicular.
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One diagonal is the perpendicular bisector of the other.
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One of the diagonals bisects a pair of opposite angles.
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One pair of opposite angles are congruent.
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Square
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A parallelogram that is both a rectangle and a rhombus.
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All properties of rectangles apply by definition.
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All properties of rhombus apply by definition.
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Diagonals form four isosceles right triangles (45-45-90).
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Trapezoid
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A quadrilateral with exactly one pair of parallel sides. Parallel sides are called bases.
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Isosceles Trapezoid
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A trapezoid in which nonparallel sides (legs) are congruent. Legs are congruent by definition.
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Bases are parallel.
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Lower base angles are congruent.
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Upper base angles are congruent.
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Diagonals are congruent.
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Any lower base angle is supplementary to any upper base angle.
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Theorem 30
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The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
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Theorem 31
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If two alternate interior angles are congruent, then lines are parallel.
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Theorem 32
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If two alternate exterior angles are congruent, then lines are parallel.
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Theorem 33
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If two corresponding angles are congruent, then lines are parallel.
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Theorem 34
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If two interior angles on the same side of the transversal are supplementary, then lines are parallel.
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Theorem 35
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If two exterior angles on the same side of the transversal are supplementary, then lines are parallel.
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Theorem 36
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If two coplanar lines are perpendicular to a third line, then they are parallel.
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Parallel postulate
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Given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.
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Theorem 37
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If lines are parallel, then alternate interior angles are congruent.
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Theorem 38
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If lines are parallel, then any pair of the angles formed are either congruent or supplementary.
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Theorem 39
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If lines are parallel, then alternate exterior angles are congruent.
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Theorem 40
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If lines are parallel, then corresponding angles are congruent.
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Theorem 41
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If lines are parallel, then each pair of interior angles on the same side of the transversal are supplementary.
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Theorem 42
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If lines are parallel, then each pair of exterior angles on the same side of the transversal are supplementary.
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Theorem 43
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In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
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Theorem 44
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If two lines are parallel to a third line, then they are parallel to each other.
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Polygons
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Plane figures
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Convex polygons
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A polygon in which each interior angle has a measure less than 180 degrees
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Diagonal
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A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon
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Quadlirateral
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Four sided polygon
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Parallelogram
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A quadrilateral in which both pairs of opposite sides are parallel.
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Finding Parallelogram method 1
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Opposite sides are parallel by definition.
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Finding Parallelogram method 2
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Opposite sides are congruent.
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Finding Parallelogram method 3
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Opposite angles are congruent.
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Parallelogram method 4
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Diagonals bisect each other.
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Parallelogram method 5
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Any pair of consecutive angles are supplementary.
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Rectangle
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A parallelogram in which at least one angle is a right angle.
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Rectangle method 1
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All properties of a parallelogram apply by definition.
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Rectangle method 2
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All angles are right angles.
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Rectangle method 3
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Diagonals are congruent.
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Rhombus
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A parallelogram in which at least two consecutive sides are congruent.
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Finding a rhombus method 1
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All properties of a parallelogram apply by definition.
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Finding a rhombus method 2
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All properties of a kite apply.
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Finding a rhombus method 3
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All sides are congruent (Rhombus is equilateral).
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Finding a rhombus method 4
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Diagonals bisect the angles.
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Finding a rhombus method 5
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Diagonals are perpendicular bisectors of each other.
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Finding a rhombus method 6
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Diagonals divide the rhombus into four congruent right triangles.
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Kite
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A quadrilateral in which two disjoint pairs of consecutive sides are congruent.
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Finding a Kite method 1
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Two disjoint pairs of consecutive sides are congruent by definition.
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Finding a Kite method 2
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Diagonals are perpendicular.
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Finding a Kite method 3
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One diagonal is the perpendicular bisector of the other.
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Finding a Kite method 4
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One of the diagonals bisects a pair of opposite angles.
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Finding a Kite method 5
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One pair of opposite angles are congruent.
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Square
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A parallelogram that is both a rectangle and a rhombus.
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Finding a Square method 1
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All properties of rectangles apply by definition.
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Finding a Square method 2
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All properties of rhombus apply by definition.
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Finding a Square method 3
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Diagonals form four isosceles right triangles (45-45-90).
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Trapezoid
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A quadrilateral with exactly one pair of parallel sides. Parallel sides are called bases.
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Isosceles Trapezoid
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A trapezoid in which nonparallel sides (legs) are congruent.
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Finding Isosceles trapezoid method 1
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Legs are congruent by definition.
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Finding Isosceles trapezoid method 2
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Bases are parallel.
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Finding Isosceles trapezoid method 3
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Lower base angles are congruent.
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Finding Isosceles trapezoid method 4
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Upper base angles are congruent.
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Finding Isosceles trapezoid method 5
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Diagonals are congruent.
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Finding Isosceles trapezoid method 6
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Any lower base angle is supplementary to any upper base angle.
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Quadrilateral is a Parallelogram method 1
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If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.
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Quadrilateral is a Parallelogram method 2
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If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Quadrilateral is a Parallelogram method 3
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If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
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Quadrilateral is a Parallelogram method 4
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If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
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Quadrilateral is a Parallelogram method 5
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If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Quadrilateral is a Rectangle method 1
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If a parallelogram contains at least one right angle, then it is a rectangle.
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Quadrilateral is a Rectangle method 2
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If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
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Quadrilateral is a Rectangle method 3
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If all four angles of a quadrilateral are right angles, then it is a rectangle.
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Quadrilateral is a Kite method 1
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If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite.
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Quadrilateral is a Kite method 2
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If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite.
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Quadrilateral is a Rhombus method 1
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If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus.
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Quadrilateral is a Rhombus method 2
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If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.
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Quadrilateral is a Rhombus method 3
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If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus.
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Quadrilateral is a square
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If a quadrilateral is both a rectangle and a rhombus, then it is a square.
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Trapezoid is isosceles 1
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If the nonparallel sides of a trapezoid are congruent, then it is isosceles.
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Trapezoid is isosceles 2
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If the lower or upper base angles of a trapezoid are congruent, then it is isosceles.
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Trapezoid is isosceles 3
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If the diagonals of a trapezoid are congruent, then it is isosceles.
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Ratio
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A quotient of two numbers
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Proportion
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An equation stating that two or more ratios are equal (a:b=c:d) or as fractions.
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In a Proportion
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first and fourth terms (a and d)are extremes, second and third terms (b and c)are means.
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Theorem 59(Means-Extremes Theorem)
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In a proportion, the product of the means is equal to the product of the extremes.
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Theorem 60(Means-Extremes Ratio Theorem)
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If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes and the other pair the means, of a proportion.
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If 5:3=7:x, what is x?
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4 1/5
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Definition of ratio and proportion
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If the means in a proportion are equal, either the mean is called the geometric mean or the mean proportional between the two extremes.
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Arithmetic Mean
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Average of numbers.
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Definition of Similarity 1
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Similar figures have the same shape but are not necessarily the same size. (Angles remain the same and ratio of sides remains consistent.)
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Dilation
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Enlargement of a figure preserving similarity
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Reduction
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Shrinking a figure but preserving similarity
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Similar polygon method 1
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The ratios of the measures of corresponding sides are equal
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Similar polygon method 2
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Corresponding angles are congruent
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Theorem 61
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The ratio of the perimeter of two similar polygons equals the ratio of any pair of corresponding sides
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Postulate 11(AAA)
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If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangles, then the triangles are similar.
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Theorem 62(AA)
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If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar.
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Theorem 63(SSS~)
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If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of the corresponding sides are equal, then the triangles are similar.
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Theorem 64(SAS~)
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If there exists a correspondence between the vertices of two triangles such that ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.
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Theorem 65(Side-Splitter Theorem)
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If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.
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Theorem 66
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If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.
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Theorem 67(Angle-bisector Theorem)
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If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.
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Foot
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The point of intersection of a line and a plane is called the foot of the line.
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Postulate 10
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Three noncollinear points determine a plane.
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Theorem 45
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A line and a point not on the line determine a plane
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Theorem 46
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Two intersecting lines determine a plane.
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Theorem 47
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Two parallel lines determine a plane.
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Postulate 11
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If two planes intersect, their intersection is exactly one line.
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Definition of line and plane perpendicularity
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A line is perpendicular to a plane if it is perpendicular to every one of the lines in the plane that pass through its foot.
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Theorem 48
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If a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane.
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Definition of parallel lines 1
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A line and a plane are parallel if they do not intersect.
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Definition of parallel lines 2
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Two planes are parallel if they do not intersect.
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Definition of parallel lines 3
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Two lines are skew if they are not coplanar.
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Theorem 49
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If a plane intersects two parallel planes, the lines of intersection are parallel.
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Parallelism of Lines and planes 1
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If two planes are perpendicular to the same line, they are parallel to each other.
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Parallelism of Lines and planes 2
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If a line is perpendicular to one of the two parallel plane, it is perpendicular to the other plane as well.
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Parallelism of Lines and planes 3
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If two planes are parallel to the same plane, they are parallel to each other.
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Parallelism of Lines and planes 4
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If two lines are perpendicular to the same plane, they are parallel to each other.
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Parallelism of Lines and planes 5
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If a plane is perpendicular to one of two parallel lines, it is perpendicular to the other line as well.
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Theorem 50
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The sum of the measures of the three angles of a triangle is 180°.
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Definition of triangle application
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An exterior angle of a polygon is an angle that is adjacent to and supplementary to an interior angle of the polygon.
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Theorem 51
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The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
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Theorem 52(Midline theorem)
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A segment joining the midpoints of two midpoints of a triangle is parallel to the third side, and its length is one-half the length of the third side.
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Theorem 53(No-choice Theorem)
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If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent.
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Theorem 54(AAS)
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If there exists a correspondence between the vertices of two triangles such that two angles and a non included side of one are congruent to the corresponding parts of the other, then the triangles are congruent.
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Triangle
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Polygon with 3 sides
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Quadrilateral
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Polygon with 4 sides
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Pentagon
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5 sided polygon
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Hexagon
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6 sided polygon
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Heptagon
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7 sided polygon
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Octagon
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8 sided polygon
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nonagon
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9 sided polygon
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decagon
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10 sided polygon
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dodecagon
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12 sided polygon
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pentadecagon
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15 sided polygon
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Theorem 55
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The sum S₁ of the measures of the angles of a polygon with n sides is given by the formula
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S₁=(n-2)180
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Finding interior angle of an angle polygon
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S₁=(n-2)180
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Theorem 56
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If one exterior angle is taken at each vertex, the sum Sₑ of the measures of the exterior angles of a polygon is given by the formula Sₑ=360
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Sum of exterior angles formula
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Sₑ=360
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Theorem 57
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The number of d diagonals that can be drawn in a polygon of n sides is given by the formula
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d=n(n-3)/2
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number of diagonals formula
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d= (n^2-3n)/2 or n(n-3)/2
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Regular polygon
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A polygon that is both equilateral and equiangular.
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Theorem 58
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The measure E of each exterior angle of an equiangular polygon of n sides is given by the formula
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E=360/n
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Finding exterior angle formula
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E=360/n
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1^2
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1
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2^2
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4
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3^2
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9
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4^2
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16
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5^2
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25
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6^2
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36
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7^2
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49
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8^2
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64
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9^2
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81
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10^2
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100
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11^2
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121
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12^2
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144
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13^2
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169
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14^2
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196
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15^2
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225
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16^2
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256
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17^2
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289
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18^2
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324
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19^2
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361
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20^2
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400
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21^2
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441
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22^2
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484
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23^2
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529
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24^2
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576
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25^2
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625
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26^2
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676
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27^2
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729
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28^2
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784
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29^2
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841
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30^2
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900
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hypotenuse of right triangle with lengths 3 and 4
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5
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hypotenuse of right triangle with lengths 5 and 12
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13
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hypotenuse of right triangle with lengths 7 and 24
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25
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hypotenuse of right triangle with lengths 8 and 15
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17
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hypotenuse of right triangle with lengths 9 and 40
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41
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