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

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

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)

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

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

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.

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

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

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

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.

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

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.

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

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

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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