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Published byMaud Copeland Modified over 4 years ago

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Warm Up Given: ∠ 1 ≅ ∠ 2 m ∠ 2 = 60° m ∠ 3 = 60° Prove: ∠ 1 ≅ ∠ 3 1 2 3 1 1 2 2 3 3

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Angle Congruence Theorems Students will use the angle congruence theorems and their other theorems, postulates, and definitions to construct 2-column proofs.

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What Do We Know So Far? Our definitions: congruence, midpoint, angle bisector Our postulates: segment addition, angle addition Our algebraic properties (reflexivity, symmetry, transitivity, and addition, subtraction, multiplication, division, substitution) Our segment congruence and angle congruence theorems (reflexivity, symmetry, transitivity)

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Right Angle Congruence Theorem If two angles are right angles, then they are congruent. StatementsReasons 1. ∠ A and ∠ B are right angles Given 2. m ∠ A = 90° Definition of a right angle 3. m ∠ B = 90° Definition of a right angle 4. m ∠ A = m ∠ B Substitution property of equality 5. ∠ A ≅ ∠ B Definition of congruent angles

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Linear Pair Postulate If two angles form a linear pair, then they are supplementary. A BC D Question: Why is this a postulate?

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Congruent Supplements Theorem If two angles are supplementary to the same angle (or two congruent angles) then they are congruent. If m 1 + m 2 = 180 0 and m 2 + m 3 = 180 0, then 1 3.

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1 2 3 If m 1 + m 2 = 180 0 and m 2 + m 3 = 180 0, then 1 3.

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StatementsReasons 1. ∠ 1 and ∠ 2 are a linear pair. Given 2. ∠ 1 and ∠ 2 are supplementary Linear Pair Postulate 3. m ∠ 1 + m ∠ 2 = 180 Definition of supplementary angles 4. ∠ 3 and ∠ 2 are a linear pair. Given 5. ∠ 3 and ∠ 2 are supplementary Linear Pair Postulate 6. m ∠ 3 + m ∠ 2 = 180 Definition of supplementary angles 7. m ∠ 3 = 180 - m ∠ 2 Subtraction POE 8. m ∠ 1 = 180 - m ∠ 2 Subtraction POE 9. m ∠ 1 = m ∠ 3 Substitution 10. ∠ 1 ≅ ∠ 3 Definition of Congruence PP Proof of the Congruent Supplements Theorem

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Vertical Angles Theorem: Vertical angles are congruent. (Angle A ≅ Angle B) A B

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Congruent Complements Theorem If two angles are complementary to the same angle (or two congruent angles) then the two angles are congruent. If m 4 + m 5 = 90 0 and m 5 + m 6 = 90 0, then 4 6.

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If m 4 + m 5 = 90 0 and m 5 + m 6 = 90 0, then 4 6. 6 6 5 5 4 4

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Jigsaw Activity Step 1: Each group will complete one problem from worksheet 2.6. Each member of the group will be an expert on their particular problem. Step 2: One member from each group will move to a second group, so that each of the new groups has (at least) one expert on each problem. Step 3: Each member will present his or her problem and how they solved it.

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Exit Ticket Complete the following proof on a piece of loose-leaf paper. Given: ∠ A and ∠ B are complementary. m ∠ C + m ∠ B = 90° Prove: ∠ A ≅ ∠ C

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What did we talk about? Properties of Angle Congruence 1.Reflexive 2. Symmetric 3. Transitive Right Angle Congruence Theorem Congruent Supplements Theorem Congruent Complements Theorem Linear Pair Postulate Vertical Angles Theorem

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Practice Problems 1.Find the values of x and y. 1.What conclusions can you draw about the angles in the following diagram? Justify your answer.

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