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Warm Up

Warm Up. Write a two column proof for the following information. Given: EH GH and FG GH Prove: FG EH. F. G. E. X. H. We will be continuing our quest to understand geometric proofs.

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Warm Up

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  1. Warm Up • Write a two column proof for the following information. • Given: EH GH and FG GH • Prove: FG EH F G E X H

  2. We will be continuing our quest to understand geometric proofs. • Today, the proofs will focus on right angle congruence and the congruence of supplements and complements.

  3. 1 and 2 are right angles GIVEN 1 2 PROVE THEOREM THEOREM 2.3Right Angle Congruence Theorem All right angles are congruent. You can prove Theorem 2.3 as shown.

  4. Proving Theorem 2.3 1 2 3 4 1 and 2 are right angles GIVEN Statements Reasons 1 2 PROVE 1 and 2 are right anglesGiven m 1 = 90°, m 2 = 90°Definition of right angles m 1 = m 2 Transitive property of equality 1  2Defof congruent angles

  5. D C A B Let’s Practice! • Given: ∠DAB and ∠ ABC are right angles; ∠ABC ∠BCD • Prove: ∠DAB ∠BCD Statements Reasons 1. ∠ DAB, ∠ ABC are right angles 1. Given 2. ∠ DAB ∠ ABC 2. Right angles are congruent 3. ∠ ABC ∠ BCD 3. Given 4. ∠ DAB ∠ BCD 4. Transitive Property of Congruence

  6. B C D Let’s Practice! A E F • Given: ∠AFC and ∠BFD are right angles, ∠BFD ∠CFE • Prove: ∠AFC ∠CFE Statements Reasons 1. ∠AFCand ∠ BFD are right angles 1. Given 2. ∠ AFC ∠BFD 2. Right angles are congruent 3. ∠BFD ∠CFE 3. Given 4. ∠AFC ∠CFE 4. Transitive Property of Congruence

  7. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2 1 3

  8. 1 2 3 1 and 3 If m 1 + m 2 = 180° m 2 + m 3 = 180° 1  3 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. then

  9. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 5 6 4

  10. 5 6 4 6 4 and If m 4 + m 5 = 90° m 5 + m 6 = 90° 4  6 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. then

  11. Proving Theorem 2.4 1 2 Statements Reasons 1 and 2 are supplements Given 3 and 4 are supplements 1 4 m 1 + m 2 = 180° Definition of supplementary angles m 3 + m 4 = 180° 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE

  12. Proving Theorem 2.4 3 5 4 m 1 + m 2 = Transitive property of equality m 3 + m 1 m 3 + m 4 m 1 = m 4 Definition of congruent angles m 1 + m 2 = Substitution property of equality 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

  13. Proving Theorem 2.4 6 7 m 2 = m 3 Subtraction property of equality 2 3Definition of congruent angles 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

  14. Let’s Practice! • Given: m∠1 = 24°, m∠3 = 24°, ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary • Prove: ∠2∠4 3 4 1 2

  15. Let’s Practice! • In a diagram, ∠1 and ∠2 are supplementary and ∠2 and ∠3 are supplementary. • Prove that ∠1∠3.

  16. W Warm Up 10/4/13 GIVEN: X, Y, and Z are collinear, XY = YZ, YW = YZ PROVE: Y is the midpoint of XZ X Y Z Statements: Reasons: • X, Y, and Z are collinear • XY = YW • YW = YZ • 2) XY = YZ • 3) XT ≅ YZ • 4) Y is the midpoint of XZ 1) Given 2) Transitive Property 3) Definition of Congruent Segments 4) Definition of Midpoint

  17. Postulate 12: Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2 m 1 + m 2 = 180

  18. Example 5: Using Linear Pairs In the diagram, m8 = m5 and m5 = 125. Explain how to show m7 = 55 7 8 5 6 Solution: • Using the transitive property of equality m8 = 125. • The diagram shows that m 7 + m 8 = 180. • Substitute 125 for m 8 to show m 7 = 55.

  19. Vertical Angles Theorem • Vertical angles are congruent. 2 3 1 4 1≅ 3; 2≅ 4

  20. Proving Theorem 2.6 Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6

  21. Statement: 5 and 6 are a linear pair, 6 and 7 are a linear pair 5 and 6 are supplementary, 6 and 7 are supplementary 5 ≅ 7 Reason: Given Linear Pair Postulate Congruent Supplements Theorem Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6

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