Conjectures that lead to Theorems 2.5

Definition Vertical angles are the opposite angles formed by two intersecting lines. 1 and 3 are vertical angles 2 and 4 are vertical angles

Vertical Angles Theorem If two angles form a pair of vertical angles, then they are congruent.

Prove the vertical angle theorem 3 1 2 Given: 1 and 2 are vertical angles Prove: 1 is congruent to 2 4 Statement Reason 1. 1 and 2 are vertical angles 1. Given 2. 1 + 3 = 180° , 2 + 3 = 180° 2. Linear Pair Property 3. 1 + 3 = 2 + 3 3. Substitution Property of Equality 4. 1 = 2 4. Subtraction Property of Equality 5. 1 is congruent to 2 5. Definition of Congruent

Identify the vertical angles in the figure. 1. 1 and _____ 2. 2 and _____ 3. 3 and _____ 4. 4 and _____ 5. 5 and _____ 6. 6 and _____

Find the value of x. 1. 2. 3. 4. 130° 5x° 25° x° 125 ° x° 40° 1. 2. 3. 4. 130° 5x° 25° x° 125 ° x° 40° (x – 10)°

Find the value of x.

Find the value of x.

What is the measure of the angle? 5y – 50 What type of angles are these? 4y – 10 5y – 50 = 4y – 10 y = 40 Plug y back into our angle equations and we get

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example: 3 2 4 1 ADJACENT 5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example: 3 2 4 1 VERTICAL 5

Example: ADJACENT, COMPLEMENTARY Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example: 3 2 4 1 ADJACENT, COMPLEMENTARY 5

Example: ADJACENT, SUPPLEMENTARY, LINEAR PAIR Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example: 3 2 4 ADJACENT, SUPPLEMENTARY, LINEAR PAIR 1 5

Find x, y, and z. Example: x = 129, y = 51, z = 129

L P A T O Example: Find x. x = 8

L P A T O Example: Find Since we have already found the value of x, all we need to do now is to plug it in for LAT. 155

Example: Find the value of x.

Answer the questions for each figure NO 1b. Are 1 and 5 a linear pair? YES 1a. Are 1 and 2 a linear pair? 2b. Are 1 and 2 a linear pair? NO 2a. Are 1 and 3 vertical angles? NO 3b. Are 1 and 4 vertical angles? YES 3a. Are 1 and 4 a linear pair? YES NO 4b. Are 3 and 5 vertical angles? NO 4a. Are 2 and 4 vertical angles?

Congruent Supplements Theorem If 2 angles are supplementary to the same angle, then they are congruent. If 1 & 2 are supplementary, and 2 & 3 are supplementary, then 1  3. 1 2 3

Congruent Complements Theorem If 2 angles are complementary to the same angle, then they are congruent. 1 2 3 If 1 & 2 are complementary, and 2 & 3 are complementary, then 1  3.

Right Angle Congruence Theorem All right angles are congruent. 90 90

Definitions Inductive Reasoning: The process of forming conjectures based on observations or experiences. Deductive Reasoning: The process of drawing conclusions by using logical reasoning in an argument.

Find the measure of each angle. 1 2 3 4 5 6 B G V F A C E 8 60

Assignment Geometry: 2.5B and 2.5C Section 10 - 20