angle relationships

Angle Relationships Page 76

Complimentary Angles

• Two angles whose measures add up to ______

Example: ABC and CBD are complimentary angles

Supplementary Angles

• Two angles whose measures add up to ______

Example: EFG and GFH are supplementary angles

Congruent Angles

•When two or more angles have the same measure, they are ___________________

-Equality shown with matching markings

Notice• Two angles _________need to share a vertex and a

common side to be complimentary, supplementary, or congruent

Naming Parts of ShapesPage 81

A Point

Named by using a single ____________ letter

Example: Points A, B, and C

Prime NotationWhen a shape is transformed, the new shape

is named using_________________________ Example: The new point A is labeled as A’ (read as “A ↓ prime”)

↓

Line Segment

Named by naming its ___________ and placing a __________above them.

Example: , , , , ,

Line

Lines extend ____________ in either directionNamed by naming two points on a line and

placing a bar with arrows above them

Example: , , , , ,

Angle

• Named by using an __________ symbol in front of the name of the angle’s vertex

• Example: A is the angle measuring 80°

Angle• Sometimes a single letter isn’t enough.• When more than one angle share a vertex, The angle is named

with _______ letters (using the vertex as the ____________ letter)

Example: HGI or IGH are referring to the angle measuring 10°

Angle’s Measure

• To refer to an angle’s measure, place m in ________ of the angle’s name

Example: m HGI=10° means “the measure of HGI is 10°”

Transversal• A ________ that crosses two or more lines

• Example: is a transversal

Vertical Angles• Two _____________angles formed by two

intercepting lines• Always have _____________ value (congruent)• Example: c and d are a pair of vertical

angles

Corresponding Angles• Lie at the _________ position but different points of

intersection of the transversal• Congruent IF the lines intersecting the transversal

are __________________• Example: d and m are corresponding angles

(both to the right of transversal and above the intersecting line)

Systems of Linear EquationsPage 87

Systems of Linear Equations

• Set of two or more _____________equations that are given together

• Example: y = 2x y = -3x + 5

Point of Intersection• The point that makes ________ equations true• Where the lines ________________ if graphed• Example:

Point of intersection: (1, 2)

Coincide• The graphs of the two lines lie on _________

of each other• __________________number of intersections

• Example:

NO Points of Intersection• Then the lines are ____________________• They will ________________ intersect • Example:

Equal Values Method• When the two equations have the ________variable already by itself (ex: y-form)• Set them ______________ to each other• Solve for one variable• Plug in the value for the solved variable and

____________ to find the value of the other variable.

Ex: y = 2x – 3 and y = -4x +3If y = a and y = b, then a = b

2x – 3 = -4x + 3

Solve for y

Substitution Method

Ex: 2x + 5y = 31 and 3x + y = 1

Elimination Method

Step 1—Arrange the two equations in columns (so each variable and constant are lined up)

Step 2—Multiply one equation, if necessary, so that you have opposite coefficients for one variable

(ex. 2x and -2x)Step 3—add the equations from step 2

(remember this is called elimination method because you want to get rid of one of the variables during this step)

Elimination (continued)

Ex: x + 5y = 8 x – 5y = 4

More Angle Pair RelationshipsPage 91

Alternate Interior Angles• Angles that are ________ the pair of lines and

on opposite side of the transversal• Congruent IF lines intersecting transversal are

_________________• Example: f and m are alternate interior ∠ ∠

angles

Same-Side Interior Angles• _________ side of transversal and in between

the pair of intersecting lines• Supplementary IF intersecting lines are

_____________• Example: g and m are same-side interior ∠ ∠

angles

Proof By Contradiction

Page 96

Definition• Prove a claim by thinking about what

the consequences would be if it were _____________. If the claim being false would lead to an impossibility, that shows that the claim must be ______________.

Set-UpSuppose…

Then…But this is impossible, so…

ExampleProve that the lines and are parallel by proof by contradiction.

CD�

AB

Suppose and intersect at some point E.

Then the angles in AEC add up to more than

But this is impossible, so and must be parallel.

AB�

CD�

180

AB�

CD�

Definition

• The measures of all angles in a triangle add up to _____________

Example

m A + m B + m C = 180

Tiling Example

• The three angles of a triangle form a straight edge, therefore the sum of the angles of a triangle must be _________

Multiplying BinomialsPage 104

Use each factor of the product as a dimension of a rectangle and find its area

Example (2x + 5)(3x - 1)

(2x + 5)(3x - 1) =

Conditional StatementPage 108

• Written in the form: “If …, then…”• Examples:

1) If a shape is a rhombus, then it has four sides of equal length.

2) ____ it is February 14th, ___________it is Valentine’s Day.

Areas of a Triangle, Parallelogram, and Trapezoid

Page 112

Triangle

Area = bh

12

2bhOR A

The base and height are PERPENDICULAR to each other!

Parallelogram

Area = bhThe base and height are _______________ to each other!

Trapezoid

Area =

The base and height are ________________ to each other!

1 2( )2

b b hOR A

Square RootPage 115

•If the area of a square is x, then the length of a side of the square is

•Example: x

x

x = _____________

x

Area = 16

2u

Irrational Number• __________ be expressed as where a and b are integers and

b 0 • Example: is an irrational

number because 17 is not a perfect square. 4.123…

ab

17

17

Estimating Square Roots• Estimated by comparing the number

under the square root with its _________ perfect squares (perfect squares: 4, 9, 16, 25, etc.)

• Example: = 4, then is a little bigger than 4. So, 4.1 ( “The square root of 17 is approximately 4.1”)

16 1717

Triangle Inequality• pg. 118-119

• Sets the maximum & minimum limits for the length of the third side of __________triangle.

• The length of each side must be _______ than the SUM of the lengths of the other two sides.

P

Q R

PQ < PR + QR

PR < PQ + QR

QR < PQ + PR

Example:

If two of the sides are 20 and 14, then the third side is:

Max: Less than 20 + 14 = 34Min: More than 20 – 14 = 6So, 6 < x < 34 would be the _____________ that

the third side COULD be.

20

14AB < AC + BC (x < 14 + 20)

AC < AB + BC (14 < x + 20)

BC < AB + AC (20 < x + 14)

A

B C

x

In other words: 20 – 14 < x < 20 + 14

Right Triangle Vocabulary

Page 119

Right Triangle: A triangle that contains a ___ angle

Legs: The sides that meet at the ______angle

Hypotenuse: The side opposite of the right angle (the ____________________ side)

Pythagorean TheoremPage 123

• In a RIGHT triangle:

• The Pythagorean Theorem can be used to help find the length of a missing _______ in a right triangle

• Example: