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 Statements Reasons EH ≅GH and FG≅GH 1. Given 2. EH ≅ FG 2. Transitive Property

Geometry California Standard 2.0 Students write geometric proofs. GEOMETRY GAME PLAN Date 10/2/13 Wednesday Section / Topic Notes: 2.6 Proving Statements about Right Angles Lesson Goal Students will be able to write proofs with reasons about congruent angles. Content Standard(s) Geometry California Standard 2.0 Students write geometric proofs. Homework P. 112-116 (#18, 21-22, 24) Announcements Math tutoring is available every Mon-Thurs in Room 307, 3-4PM! Test next Tuesday or Wednesday or Thursday Chapter 1 Test Retakes Available Until Ch 2 test

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.

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

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

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

Let’s Practice! Given: ∠AFC and ∠BFD are right angles, ∠BFD ≅∠CFE Prove: ∠AFC ≅∠CFE Statements Reasons 1. ∠AFC and ∠ 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

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

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

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

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

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

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

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

Let’s Practice! Given: m∠1 = 24°, m∠3 = 24°, ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary Prove: ∠2≅∠4 Statements Reasons 1. m∠1 = 24°, m∠3 = 24°, ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary 1. Given 2. m∠ 1 = m∠3 2. Transitive Property of Equality 3. ∠1 ≅ ∠3 3. Definition of congruent angles 4. ∠2 ≅ ∠4 4. Congruent Complements Theorem 3 4 1 2

Let’s Practice! In a diagram, ∠1 and ∠2 are supplementary and ∠2 and ∠3 are supplementary. Prove that ∠1≅∠3. Statements Reasons 1. ∠1 and ∠2 are supplementary and ∠2 and ∠3 are supplementary 1. Given 2. m∠1 + m∠2 = m∠2 + m∠3 2. Transitive Property of Equality 3. m∠1 = m∠3 3. Subtraction Property of Equality 4. ∠1 ≅ ∠3 4. Definition of Congruent Angles

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

Geometry California Standard 2.0 Students write geometric proofs. GEOMETRY GAME PLAN Date 10/4/13 Friday Section / Topic Notes: 2.6 Proving Statements about Right Angles Lesson Goal Students will be able to write proofs with reasons about congruent angles. Content Standard(s) Geometry California Standard 2.0 Students write geometric proofs. Homework Finish classwork Announcements Test next Tuesday or Wednesday or Thursday Chapter 1 Test Retakes Available Until Ch 2 test Late Start on Wednesday 10/9/13 Back to School, Wednesday 10/9/13

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

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.

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

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

5 and 6 are a linear pair, 6 and 7 are a linear pair Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6 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