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Understanding Complementary and Supplementary Angles: Definitions, Properties, and Proofs

This lesson focuses on complementary and supplementary angles, essential concepts in geometry. Students will learn how to define and identify these angles through examples. Complementary angles sum to 90° while supplementary angles sum to 180°. The session will include proofs to reinforce understanding and application of these concepts. Students will work collaboratively in pairs to complete proof exercises and present their findings, enhancing their problem-solving skills in geometry.

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Understanding Complementary and Supplementary Angles: Definitions, Properties, and Proofs

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  1. Warm Up • Given: Angle MOR = (3x + 7)° • Angle ROP = (4x – 1)° • MO OP • Which angle is larger, angle MOR or angle ROP? M R O P

  2. Complementary and Supplementary Angles I CAN… • Define and identify complementary and supplementary angles • Write proofs involving complementary and supplementary angles

  3. Complementary Angles • Two angles whose sum is 90 • A is complementary to B • JOC is the complement of COB B 38 52 A J C O B

  4. Supplementary Angles • Two angles whose sum is 180 • A is supplementary to B • HAM is the supplement of HAB B 64 116 A H M A B

  5. Definitions: If-then Form • If two angles form a right angle, then they are complementary. • If two angles form a straight angle, then they are supplementary.

  6. Example 1 • Given: TVK is a right  • Prove: 1 is comp. to 2 1 2 Statement Reason T X V K

  7. Example 2 • Given: Diagram as shown • Prove: 1 is supp. to 2 1 2 Statement Reason A B C

  8. Example 3 • The measure of one of two complementary angles is three greater than twice the measure of the other. Find the measure of each.

  9. Example 4 • The measure of the supplement of an angle is 60 less than 3 times the measure of the complement of the angle. Find the measure of the complement.

  10. Groups of 2 • Work with your partner to complete the proof I give you. • Write a paragraph proof. • Be prepared to present your proof to the class!!

  11. 1 3 • Given: 1 is supp. to 2 3 is supp. to 2 • Prove: 1  3 Write a paragraph proof: 2

  12. 1 3 • Given: 1 is comp. to 2 3 is comp. to 2 • Prove: 1  3 Write a paragraph proof: 2

  13. A B D C • Given: A is supp. to B C is supp. to D B  D • Prove: A  C Write a paragraph proof:

  14. A B D C • Given: A is comp. to B C is comp. to D B  D • Prove: A  C Write a paragraph proof:

  15. Theorem • If two angles are supplementary to the same angle, then they are congruent. • If 1 is supp. to 2 and 3 is supp. to 2, then 1  3. 1 3 2

  16. A Theorem B D C • If two angles are supplementary to congruent angles, then they are congruent. • If A is supp. to B, C is supp. to D, and B  D. then A  C.

  17. Theorem 1 3 • If two angles are complementary to the same angle, then they are congruent. • If 1 is comp. to 2 and 3 is comp. to 2, then 1  3. 2

  18. A Theorem B D C • If two angles are complementary to congruent angles, then they are congruent. • If A is comp. to B, C is comp. to D, and B  D, then A  C.

  19. Homework • p. 69 # 7 – 13, 16, 17, 21 • Read Sample Problems on pages 77 & 78 • p. 79 #1 - 9, 11, 14

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