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Chapter 2 Lesson 5

Chapter 2 Lesson 5. Objective: To identify angle pairs and prove & apply theorems. Vertical Angles. Two angles whose sides form two pairs of opposite rays. 1. 3. 4. 2. Adjacent Angles. Two coplanar angles with a common side, a common vertex, and no common interior points. 1. 2. 1. 2.

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Chapter 2 Lesson 5

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  1. Chapter 2 Lesson 5 Objective: To identify angle pairs and prove & apply theorems.

  2. Vertical Angles Two angles whose sides form two pairs of opposite rays 1 3 4 2

  3. Adjacent Angles Two coplanar angles with a common side, a common vertex, and no common interior points 1 2 1 2

  4. A B Complementary Angles Two angles whose measures have sum 90 Each angles is called the complement of the other. 50° 1 40° 2

  5. Supplementary Angles Two angles whose measures have sum 180 Each angle if called the supplement of the other. 105° 75° 2 1

  6. Example 1: In the diagram identify pairs of numbered angles that are related as follows: a. complementary 2 and 3 b. supplementary 2 4 and 5; 3 and 4 1 3 5 c. vertical 4 3 and 5

  7. 2 3 1 4 Example 2: Name all pairs of angles in the diagram that are: a. vertical 1 and 3; 2 and 4 b. supplementary 2 and 3; 3 and 4 4 and 1 c. complementary none

  8. Example 3: What can you conclude from the information in the diagram? 3 2 4 5 1 • 1 2, by the markings • 2 and 3, are adjacent angles • 4 and 5, are adjacent supplementary angles • m 4 + m 5 = 180 by the Angle Addition Postulate • 1 and 4, are vertical angles

  9. Example 4: T P W Q V Can you make each conclusion from the information in the diagram? Explain. a. TW WV Yes; the congruent segments are marked b. PW WQ No; there are no markings c. TV PQ No; there are no markings d. TV bisects PQ No; there are no markings e. W is the midpoint of TV Yes; the congruent segments are marked

  10. Theorem:the statement that you prove true Theorem 2-1:Vertical angles are congruent. 2 4 3 1 1 2 and 3 4 Paragraph proof: written as sentences in a paragraph

  11. Example 5: Write a paragraph proof. Given: 1 and 2 are vertical angles 1 3 2 Prove: 1 2 Paragraph Proof: By the Angle Addition Postulate, m 1 + m 3 = 180 and m 2 + m 3 = 180. By substitution, m 1 + m 3 = m 2 + m 3. Subtract m 3 from each side. You get m 1 = m 2.

  12. Example 6: Find the value of x. 4x° (3x+35)° 4x = 3x + 35 X = 35 Vertical angles are congruent Subtract 3x from each side

  13. Theorem 2-2:Congruent Supplements Theorem If two angles are supplements of the same angle (or of congruent angles), then two angles are congruent Example 7: Given: 1 and 2 are supplementary. 3 and 2 are supplementary. 2 1 Prove: 1 3 Proof: By the definition of supplementary angles, m 1 + m 2 = 180 and m 3 + m 2 = 180. By substitution, m 1 + m 2 = m 3 + m 2. Subtract m 2 from each side. You get m 1 = m 3. 3

  14. Theorem 2-3:Congruent Complements TheoremIf two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2-4: All right angles are congruent. Theorem 2-5: If two angles are congruent and supplementary, then each is a right angle.

  15. Homework Page 100-103 #1-54 skip 37;38 (Due Monday)

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