Warm Up•Write a two column proof for the following

information.

•Given: EH GH and FG GH•Prove: FG EH

Statements Reasons

1. EH GH and FGGH 1. Given

2. EH FG 2. Transitive Property

F

E

G

HX

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 ThursdayChapter 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.

THEOREM

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

Proving Theorem 2.3

Statements Reasons

1

2

3

4

m 1 = 90°, m 2 = 90° Definition of right anglesm 1 = m 2 Transitive property of equality1 2 Def of congruent angles

GIVEN 1 and 2 are right angles

PROVE 1 2

1 and 2 are right angles Given

Let’s Practice!•Given: ∠DAB and ∠ ABC are right

angles; ∠ABC ∠BCD•Prove: ∠DAB ∠BCDStatemen

tsReasons

1. ∠ DAB, ∠ ABC are right angles

2. ∠ DAB ∠ ABC

3. ∠ ABC ∠ BCD

4. ∠ DAB ∠ BCD

1. Given

2. Right angles are congruent3. Given

4. Transitive Property of Congruence

A

D C

B

Let’s Practice!•Given: ∠AFC and ∠BFD are right angles, ∠BFD ∠CFE•Prove: ∠AFC ∠CFE

Statements Reasons

1. ∠AFC and ∠ BFD are right angles

2. ∠ AFC ∠BFD

3. ∠BFD ∠CFE

4. ∠AFC ∠CFE

1. Given

2. Right angles are congruent

3. Given

4. Transitive Property of Congruence

A

B CD

EF

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.

1 233

If m 1 + m 2 = 180°

m 2 + m 3 = 180°

and1

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.

45

6

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.

4

If m 4 + m 5 = 90°

m 5 + m 6 = 90°

and

then

4 6

566

4

Proving Theorem 2.4

Statements Reasons

1

2

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

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°

Proving Theorem 2.4

Statements Reasons

3

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

4

5 m 1 + m 2 = Substitution property of equalitym 3 + m 1

m 1 + m 2 = Transitive property of equalitym 3 + m 4

m 1 = m 4 Definition of congruent angles

Proving Theorem 2.4

Statements Reasons

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

6

7

m 2 = m 3 Subtraction property of equality

2 3 Definition of congruent angles

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

complementary•Prove: ∠2∠4

4 3

2

1

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

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

Warm Up 10/4/13

GIVEN: X, Y, and Z are collinear, XY = YZ, YW = YZ

PROVE: Y is the midpoint of XZ

Statements: Reasons:

1) 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

W

ZX Y

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 ThursdayChapter 1 Test Retakes Available Until Ch 2 testLate Start on Wednesday 10/9/13Back 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, m8 = m5 and m5 = 125.Explain how to show m7 = 55

5 6 7 8

• Using the transitive property of equality m8 = 125.

• The diagram shows that m 7 + m 8 = 180. • Substitute 125 for m 8 to show m 7 = 55.

Solution:

Vertical Angles Theorem

•Vertical angles are congruent.

1

23

4

1 ≅ 3; 2 ≅ 4

Proving Theorem 2.6Given: 5 and 6 are a linear pair, 6

and 7 are a linear pairProve: 5 7

56

7

Statement:

1. 5 and 6 are a linear pair, 6 and 7 are a linear pair

2. 5 and 6 are supplementary, 6

and 7 are supplementary

3. 5 ≅ 7

Reason:

1. Given

2. Linear Pair Postulate

3. Congruent Supplements Theorem

Given: 5 and 6 are a linear pair, 6 and 7 are a linear pairProve: 5 7

56

7