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Geometry: Lesson 2.5 – Proving Angle Relationships Geometry Oklahoma Academic Standards: G.2D.1.2 Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real world and mathematical problems using algebraic reasoning and proofs. Lesson Objectives: 1. To prove angle relationships with definitions. 2. To use the proven theorems to solve problems. Introduction: Now that we have proven segment relationships, we can start to prove some angle relationships. Vocabulary: Definition – Adjacent An adjacent angle is one that shares a side with another angle. Which angle pairs represent adjacent angles in the above diagram? ____________________________________________________________________ The opposite of adjacent is vertical. Geometry: Lesson 2.5 1 Vocabulary: Definition – Vertical A vertical angle is one is directly across from another angle. Which angle pairs represent vertical angles in the above diagram? ____________________________________________________________________ Vocabulary: Definition – Complementary A complementary angle is an angle that adds to be 90 degrees with another angle. Visually: Symbolically: _________________________ Geometry: Lesson 2.5 2 Vocabulary: Definition – Supplementary A supplementary angle is an angle that adds to be 180 degrees with another angle. Visually: Symbolically: _________________________ Vocabulary: Definition – Linear Pair A Linear Pair are a couple of angles that are both adjacent and supplementary. Visually: Symbolically: _________________________ Using these definitions, we can now begin writing some theorems about how the various kinds of angles relate to each other. Geometry: Lesson 2.5 3 Conjecture: If two angles are vertical, then they are congruent. Conjecture: If two angles are supplementary to a third angle, then the two angles are congruent. Geometry: Lesson 2.5 4 Conjecture: If an angle is complementary to itself, then the angle must measure 45 degrees. Conjecture: If two angles are complementary and supplementary, then each angle is a right angle. Geometry: Lesson 2.5 5 Conjecture: If two angles are right, then they are congruent. Conjecture: If two angles are congruent and supplementary, then each angle is a right angle. Geometry: Lesson 2.5 6 Now that we have proven some theorems about angles, let’s use them to solve some problems. Example 1: Find the measure of angle 6. Example 2: Find the value of x. Example 3: Find the measures of angles 2 and 3. Geometry: Lesson 2.5 7 Assignment 2.5 You have the choice of the following: 1. Prove or disprove the following conjectures: “If two angles are straight angles, then they must be congruent.” “If an angle is complementary to a third angle and another angle is complementary to that third angle, then the two angles are congruent.” “If an angle is vertical to a second angle and the second angle is adjacent to a third angle, then the first and third angles must also be adjacent.” 2. Complete the handout, “Geometry: Handout 2.5” and turn it in. Geometry: Lesson 2.5 8