2.2

What’s the Relationship?

Pg. 8Complementary, Supplementary, and Vertical Angles

2.2 – What's the Relationship?________________Complementary, Supplementary, and Vertical Angles

In Chapter 1, you compared shapes by looking at similarities between their parts. For example, two shapes might have sides of the same length or equal angles. In this chapter you will examine relationships between parts within a single shape or diagram. Today you will start by looking at angles to identify relationships in a diagram that make angle measures equal.

2.10 – ANGLE RELATIONSHIPSWhen you know two angles have a certain relationship, learning something about one of them tells you something about the other. Certain angle relationships come up often enough in geometry that we given them special names.

76°

90 – 76 = 14°

62°

180 – 62 = 118°

23°

157°

23°

157°

23°

157°

23°

157°CEB

and

AEC

DEB

54°

126°

54°

126°

b. Based on your observations, write a conjecture (a statement based on an educated guess that is unproven). Start with , "Vertical angles are ...“

Vertical angles are _________________.congruent

2.12 – PROVING VERTICAL ANGLES CONGRUENTThe last problem used what is called inductive reasoning to show that vertical angles are congruent. We are now going to start to use deductive reasoning to prove that all vertical angles are congruent, no matter what the angles measure. Below you are given the steps in order to prove that vertical angles are congruent. Your job is to explain why each statement is true. Match the reasons with the given statements.

A. Both add to 180°B. Straight angles add to 180°C. Subtract y from both sidesD. Straight angles add to 180°

Straight angles add to 180°

Straight angles add to 180°

Both add to 180°

Subtract from both sidesy

90°

40°

50°

40°

30°

2.14 –ANGLES RELATIONSHIPS

In the problems below, you will use geometric relationships to find angle measures. Start by finding a special relationship between some of the angles, and use that relationship to write an equation. Solve the equation for the variable, then use that variable to find the missing measurement.

Angle Relationship: __________________

PNM = ____________________________

supplementary

x + 152 = 180 x = 28

28°

28°

Angle Relationship: __________________

FGH = ____________________________

Angle bisector

4x – 5 = 3x + 2 x = 7

23°

23°

Angle Relationship: __________________

DBC = ____________________________

complementary

3x + 3 = 90 x = 29

36°

36°

Angle Relationship: _____________________ x = ________________________________ Angle Relationship: _____________________ y = ____________________________________

supplementary

8x – 20 = 180 x = 25

139°

vertical

20y + 19 = 139 y = 6

2.15 – SUMMARYDiscuss each different type of angle measurement: right, complementary, straight, supplementary, congruent, and vertical. What is their relationship? Are they equal or add to something? Draw a picture of each.

Right Adjacent Angles Complementary

     

One 90° angle

Angles that share a vertex and side

12

two angles that add to 90°

Straight Linear Pair Supplementary     

One 180° angle

two adjacent angles that add to 180°

1 2

two angles that add to 180°

1 2

Congruent Angle Bisector Vertical     

Angles with same degree

Cuts an angle in half

Opposite angles that are equal

1

2