Algebra Midterm

Front 1

Corresponding objects

Back 1

Corresponding objects are those that appear in the same place in two similar figures.

Congruency statement:

ABC=DEF

*order matters, write the statement so that corresponding angles are in the same position (for each statement)*

Front 2

Corresponding sides

Back 2

Ex: when ABC is congruent to DEF, then the corresponding sides are:

side AC = side DF

side AB = side DE

side BC = side EF

*you can use the congruency statement ( ABC = DEF) to find this answer- this is why order matters*

*you can also use a diagram*

Front 3

Corresponding angles

Back 3

Example: ABC = DEF

angle A = angle D

angle B = angle E

angle C = angle F

*you can use the congruency statement ( ABC = DEF) to find this answer- this is why order matters*

*you can also use a diagram*

Front 4

Congruent figures

Back 4

Figures are congruent if they have exactly the same size and shape. Their corresponding sides and their corresponding angles must be congruent.

Front 5

Third angles theorem

Back 5

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Example: 123 = 456

If angle 1 and angle 4 are congruent, angle 2 and angle 5 are congruent, then angle 3 and angle 6 must be congruent.

Front 6

How to use the congruency statement to find an angle measure

Back 6

ABC = DEF

You know the measure of angle A is 70 degrees.

Angle D must also be a 70 degree angle, because they are congruent angles.

Front 7

How to use a congruency statement to solve for x

Back 7

ABC = DEF

You are given two diagrams. From the first diagram, you learn that angle A measures 60 degrees. Angle D is labeled 3x.

Because angle A = angle D, you know that 60 = 3x.

60/ 3 = 20. X = 20

Front 8

Transformations (four types)

Back 8

Translation

Reflection

Rotation

Dilation (Similar figures)

Front 9

Translations

Back 9

A translation is the same as sliding an object (up, down, left, right- on a coordinate plane). The shape that is translated is still exactly the same shape and size, but is in a different place on the coordinate plane.

Front 10

Translation equation

Back 10

Example: Translate the point (1,3) two units right, and one unit down.

A (1,3)

Two units right = +2 (x- coordinate)

One unit down = -1 (y- coordinate)

A’ (x+2, y-1)

A’ (1+2, 3-1) A’ (3, 2)

Front 11

Reflections

Back 11

A reflection is basically a flip of a shape over the line of reflection. The image is congruent to the original shape.

Front 12

Reflection equation

Back 12

Reflection over the x- axis: (x, y) ➡️ (x, -y)

Reflection over the y- axis: (x,y) ➡️ (-x, y)

Example: Reflect the point (1,2) over the x- axis.

(x, y) ➡️ (x, -y)

A (1,2) ➡️ (1, -2)

Front 13

Rotations

Back 13

A rotation turns a figure 90 degrees or 180 degrees clockwise or counterclockwise. The orientation of the figure is different, but it is still congruent to the original figure.

Front 14

Rules for rotating about the origin

Back 14

90 degrees clockwise: (y, -x)

90 degrees counterclockwise: (-y, x)

Rotated 180 degrees: (-x, -y)

A 270 degree clockwise rotation is the same as a 90 degree counterclockwise rotation. A 270 degree counterclockwise rotation is the same as a clockwise 90 degree rotation.

Example: Rotate points A(1,2), B(5,2), C(6,6), and D(2,6) 90 degrees counterclockwise. (-y,x)

A(1,2) ➡️ A’(-2,1)

B(5,2) ➡️ B’(-2, 5)

C(6,6) ➡️ C’(-6,6)

D(2,6) ➡️ D’(-6,2)

Front 15

Similar figures

Back 15

Similar figures are the same shape, but are a different size. Corresponding sides are PROPORTIONAL, which corresponding angles are CONGRUENT.

Front 16

Vertical Angles

Back 16

* Opposite sides of two intersecting lines

* Congruent

* Look for an “X” shape

Front 17

Linear Pair Angles

Back 17

* Adjacent angles that form a line

* Supplementary (add up to 180 degrees)

Front 18

Interior angles of polygons

Back 18

The angles inside a polygon

Front 19

Exterior angles of polygons

Back 19

The angles outside a polygon

Front 20

The sum of the interior angles of a triangle

Back 20

180 degrees

Front 21

Remote Interior Angles

Back 21

The exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent (interior) angles of the triangle.

Front 22

Similar figure equation

Back 22

There are two ways to solve this problem: Is triangle ABC congruent to triangle DEF?

Side AB of triangle ABC is 6 feet, while side DE of triangle DEF is 3 feet. Side AC is 10 feet, and side DF is 5 feet.

Way 1:

Is this true? AB/AC = DE/DF

Is this true? 6/10 = 3/5 (reduce) 3/5 = 3/5

Triangle ABC ~ triangle DEF

Way 2:

*cross product

AB/DE = AC/ DF ➡️ 6/3 = 10/5

6•5= 30 and 3•10= 30 30=30☑️

Triangle ABC ~ triangle DEF

Front 23

Rotation about a specific point

*FIX EXAMPLE*

Back 23

Example: Rotate triangle ABC 90 degrees clockwise (y, -x) about point C.

A(5,6) ➡️

B(8,1) ➡️

C(2,2) ➡️ C’(2,2)

*the point that you’re rotating about stays the same*

Front 24

How to solve for x using similar figures

Back 24

Example: triangle ABC ~ triangle DEF

Side AB= 6 feet Side AC= 10 feet

Side DE= x Side DF= 5 feet

(Write a proportion)

AB/AC = DE/DF

6/10 = x/5

10/2 = 5 So 6/2= x

X = 3

Front 25

Dilations

Back 25

A dilation changes the size of the image, but the image is the same shape. To make an image larger or smaller, multiply by the scale factor. Dilated figures are similar figures (not congruent).

Front 26

Dilation equation

Back 26

Multiple each coordinate by the scale factor (k) to get the coordinates of the dilated image.

(x,y) ➡️ (kx, ky)

If k is greater than one, the image gets larger. *enlargement*

If k is greater than zero, but less than one, the image gets smaller. *reduction*

Front 27

Corresponding Angles

Back 27

* Same position

* Congruent

Front 28

Alternate Interior Angles

Back 28

* Opposite sides of the transversal

* Inside the parallel lines

* Congruent

* Look for a “Z”

Front 29

The sum of the interior angles of a triangle

Back 29

180 degrees

Front 30

Remote Interior Angles

Back 30

The exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent (interior) angles of the triangle.

Front 31

Polygon

Back 31

A closed figure made up of three or more line segments that intersect only at their endpoints.

Front 32

Regular polygon

Back 32

A polygon in which ALL sides are congruent and ALL interior angles are congruent.

Front 33

Equation: Interior Angle Measures of a Polygon

Back 33

The sum S of the interior angle measures of a polygon with n sides is:

S = (n-2) • 180

Front 34

Exterior Angle Measures of a Polygon (Sum)

Back 34

The sum of the measures of the exterior angles of a convex polygon is 360 degrees.

Front 35

Solving Multi- Step Equations

Back 35

1) Distribute to clear grouping symbols

2) Combine like terms (on the same side of the equation)

3) Undo addition/ subtraction

4) Undo multiplication/ division

5) CHECK YOUR ANSWER

Front 36

Solving Equations with Variables on Both Sides

Back 36

1. Combine like terms

2. Get all variables on one side (pick on the little guy)

3. Get all constants together on the other side

4. Do inverse operations to get variable alone (division/multiplication)

Front 37

One solution

Back 37

Only one variable works

Example:

4x = 2x + 6

-2x -2x

2x = 6

6/2 = 3

x = 3

Front 38

No solution

Back 38

No value works

Example:

x + 3 = x + 5

-x -x

3 = 5

This statement is not true, so there is no solution.

Front 39

Infinite Solutions

Back 39

There are many solutions.

Example:

x + 4 = x + 4

-x -x

4 = 4

This statement is true, so there is an infinite number of solutions.

Front 40

Absolute Value

Back 40

The distance a number is from zero on a number line

Looks like: |x|

Front 41

Solving Absolute Value Equations

Back 41

Absolute value equations can be solved just like normal equations, but they have two cases (positive and negative), where there may be one, two, or no possible answers.

1) Isolate the absolute value (simplify)

|x + 3|-1 = 5 ➡️ |x + 3| = 6

2) Solve the equation using the positive case

|x+3|= 6

x = 3

3) Solve the equation using the negative case. X

|x+3|= -6

X = -9

Front 42

Rewriting Equations and Formulas

Back 42

Use the inverse operations of an equation to simplify the equation. Then, solve for whatever variable the directions tell you to solve for using this same method (inverse operations).

(Addition ➡️ Subtraction)

(Multiplication ➡️ Division)

Front 43

Linear equations

Back 43

An equation whose graph is a straight line. The points that make up the line are solution points.

Front 44

Steps for graphing with a table

Back 44

1) Solve for y

2) choose three x- values (-1, 0, and 1 are usually good to pick)

3) Plug in each x- value to find it’s corresponding y- value

4) Plot the ordered pair on the graph and draw the line

Front 45

HOY

Back 45

H: horizontal

O: zero

Y: y- intercept

Front 46

VUX

Back 46

V: Vertical

U: Undefined

X: x- intercept

Front 47

0/x

Back 47

Slope: zero

Front 48

x/0

Back 48

Slope: Undefined

“If the zero is under, the slope undefined.”

Front 49

Parallel lines

Back 49

* Parallel lines have the same slope

* All vertical lines are parallel (VUX)

* All horizontal lines are parallel (HOY)

* To determine if a quadrilateral is a parallelogram, check to see if the opposite sides have the same slope

Front 50

Perpendicular lines

Back 50

* Perpendicular lines always have slopes that (when multiplied) have a product of -1

* Slopes are opposite reciprocals of each other (flip one of them and change the sign)

* Vertical lines (VUX) are always perpendicular to horizontal lines (HOY)

* To determine if a quadrilateral is a rectangle, check to see if the slopes of adjacent sides have a product of -1

Front 51

Slope- intercept form

Back 51

y = mx + b

Slope: change in y/ change in x

B: y- intercept

~ this is your starting point (begin with the “b”) It’s where the line crosses the y- axis

GRAPHING USING SLOPE- INTERCEPT FORM

1) Solve the equation for y if necessary

2) Identify the slope (m) and the y- intercept (b)

3) Plot the y- intercept (0,b); always on the y- axis

4) From (0,b) rise and run with the slope to find a few more points on the line

5) Connect the points to draw the solution line. (Don’t forget the arrows)

Front 52

Graphing linear equations in Standard Form (Method 1)

Back 52

Ax+ By = C

Convert to Slope- Intercept Form

1) Solve for y

2) Identify the slope (m) and y- intercept (b)

Front 53

Graphing equations in Standard Form (Method 2)

Back 53

Find x and y intercepts

1) Find the x- intercept

~ make the y= zero in the equation and solve for x

2) Find the y- intercept

~ make the x= zero in the equation and solve for y

3) Graph the intercepts and solve the line

Front 54

Graphing Equations in Standard Form

* SHORT CUT *

Back 54

Ax + By = C

M (slope)= -A/B

Y- intercept= C/B

X- intercept= C/A

Front 55

I can write equations in Slope- Intercept Form When given a graph.

Back 55

1) Find the slope of the line (m)

2) Identify the y- intercept (b)

3) Plug them into y= mx + b to write the equation for the graph

Front 56

I can write equations in Slope- Intercept Form when given a point and slope.

Back 56

1) Identify and label an x- value, y- value, and a slope (m).

2) Plug these values into y= mx + b

~ b should be the only variable left in the equation

3) Solve for b

4) Once you find b, go back and plug in the m and b values to write the equation for the graph

Front 57

I can’t write equations in Slope- Intercept Form when given two points.

Back 57

1) Find the slope (change in y over change in x)

2) Choose ONE of the given ordered pairs

3) Identify and label values for x, y, and m

~ b should be the only variable left in the equation

4) Solve for b

5) Go back and plug in the m and b values into y= mx + b to write the equation for the graph

Front 58

I can write equations in Point- Slope when given the slope and a point.

Back 58

1) Identify x1, y1, and slope (m)

2) Substitute these values into the point- slope equation:

Y- y1= m(x- x1)

3) Simplify

Front 59

I can write Point- Slope Form equations when given two points

Back 59

1) Find the slope

2) Choose one point. Identify values to substitute for (x1, y1)

3) Plug in the values for m, y1, and x1 into the point- slope equation:

y- y1= m(x- x1)

4) Simplify

Front 60

The slope of a line

Back 60

The slope of a line measures the line’s “steepness”.

The change in y over the change in x

Front 61

Writing equations of parallel lines

Back 61

Parallel lines have slopes that are the same

1) Identify (slopes) m1 and m2

2) Use Point- slope formula (plug in values)

3) Simplify

4) Graph

5) Check work

Front 62

Writing equations of perpendicular lines

Back 62

Perpendicular lines have slopes that are opposite reciprocals

1) Identify (slopes) m1 and m2

2) Use Point- slope formula (plug in values)

3) Simplify

4) Graph

5) Check work