2.2 What’s the Relationship? Pg. 8 Complementary, 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 RELATIONSHIPS When 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°

and

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 CONGRUENT The 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 sides D. Straight angles add to 180 ° Straight angles add to 180° Both add to 180° Subtract from both sides

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 = 180 x = 28 28°

Angle Relationship: __________________ FGH = ____________________________ Angle bisector 4x – 5 = 3x + 2 x = 7 23°

Angle Relationship: __________________ DBC = ____________________________ complementary 3x + 3 = 90 x = 29 36°

Angle Relationship: _____________________ x = ________________________________ Angle Relationship: _____________________ y = ____________________________________ supplementary 8x – 20 = 180 x = ° vertical 20y + 19 = 139 y = 6

2.15 – SUMMARY Discuss 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 1 2 two angles that add to 90°

StraightLinear PairSupplementary One 180° angle two adjacent angles that add to 180° 1 2 two angles that add to 180° 1 2

CongruentAngle BisectorVertical Angles with same degree Cuts an angle in half Opposite angles that are equal 1 2