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

2. 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 (#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

3. 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.

4. 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

5. 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  Def of congruent angles

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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 = 180° Definition of supplementary angles
m 3 + m = 180°

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

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

20. 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.

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

22. 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

23. 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