2.8 Proving angle relationships cont. ink.notebook

2.8 Proving angle relationships cont. ink.notebook

1

September 20, 2017

page 81

2.8 cont.

page 82

page 83 page 84 Lesson Objectives Standards Lesson Notes

Press the tabs to view details.

2.8 Proving Angle Relationships

Cont.


2

September 20, 2017

Lesson Objectives


Lesson NotesStandards

After this lesson, you should be able to successfully write proofs involving complementary and supplementary angles, congruent angles, and right angles.

Lesson Objectives Standards


Lesson Notes

G.CO.9 Prove theorems about lines and angles.

The REFLEXIVE Property of Congruence, SYMMETRIC Property of Congruence, and TRANSITIVE Property of Congruence all hold true for angles. The following theorems also hold true for angles.

Theorem 2.10: Right Angle Congruence Theorem:

All right angles are _________________.

page 55


3

September 20, 2017

STATEMENTS REASONS

1. ∠1 and ∠2 are right angles 1. GIVEN

2. m∠1 = 90; m∠2=90 2. DEF. OF RT. ANGLE

3. m∠1 = m∠ 2 3. SUBSTITUTION PROP.

4. ∠1 ∠ 2 4. DEF OF CONGRUENT ANGLES

Given: ∠1 and ∠2 are right angles

Prove: ∠1 ≅ ∠2

pull rectangles away to reveal answers

Supplement Theorem: (Linear Pair Theorem)If two angles form a linear pair, then they are _____________________.

∠1 and ∠2 form a linear pair, so ∠1 and ∠2 are

supplementary and m∠1 + m∠2 = ___________.


4

September 20, 2017

Theorem 2.6: Congruent Supplements Theorem:If two angles are supplementary to the

same angle (or to congruent angles),

then they are _____________ .

If ∠1 and ∠2 are supplementary and ∠3 and ∠2 are supplementary, then _____________.

Theorem 2.7: Congruent Complements Theorem:If two angles are complementary to

the same angle (or to congruent angles),

then they are _____________.

If ∠4 and ∠5 are complementary and ∠6 and ∠5 are complementary, then _____________.


5

September 20, 2017

Theorem 2.8: Vertical Angles Congruence Theorem:

Vertical Angles are _____________________.

2.9: Perpendicular lines intersect to form ____ right angles.

2.11: Perpendicular lines form congruent ________ angles.

2.12: If two angles are ______________ and

________________ , then they are right angles.

2.13: If two congruent angles form a linear pair, then they are ___________ angles.

Other Right Angle Theorems:


6

September 20, 2017

STATEMENTS REASONS

1. GIVEN

2. m∠1 = 90; m∠2=90 2. DEF OF PERPENDICULAR LINES

3. ∠1 ∠ 2 3. ALL RT. ANGLES CONGRUENT

STATEMENTS REASONS

1. ∠1 and ∠2 are supplements ∠1 and ∠4 are supplements

1. GIVEN

2. ∠2 ≅ ∠4 2. Congruent Supp. THM

3. m∠2 = m∠ 4 3. DEF. OF CONGRUENT ANGLES

4. m∠2 = 45 4. GIVEN

5. m∠4 = 45 5. Substitution


7

September 20, 2017

STATEMENTS REASONS

1. ∠4 is a right angle 1. GIVEN

2. m∠4 = 90 2. DEF. OF RT. ANGLE

3. ∠2 ∠ 4 3. VERTICAL ANGLES CONGRUENT THM

4. m∠2 = m∠4 4. DEF. OF CONGRUENT ANGLES

5. m∠2 =90 5. SUBSTITUTION

6. m∠2 + m∠4 = 180 6. ADDITION

7. ∠2 AND ∠4 are Supp. 7. Def of SUPP ANGLE

If m⁄1 + m⁄2 = 90°then ⁄1 is complementary to ⁄2 1

2

If m⁄1 + m⁄2 = m⁄3 + m⁄2then m⁄1 = m⁄31 2

3

A

BC

E G If AC º EG

13

55°

55°

If m⁄1 = 55° and m⁄3 = 55°

then m⁄1 = m⁄3

If m⁄1 = m⁄2then ⁄1 § ⁄2

then m⁄3 = m⁄4If ⁄3 § ⁄4

If ⁄1 and ⁄3 are vertical anglesthen ⁄1 § ⁄31 2

34

then ⁄ABG is a right angle

Complementary

Subtraction

Perpendicular Lines

Substitution

Congruent Angles

Congruent Angles

are Congruent

Definition of

Angles

Propertyof Equality

Definition of

Definition of

Definition of

Property

Theorem:Vertical Angles

MN + NO = MO

A

BE

D

A

B

C

DA

C

B

M

NO

A

B

C

1 2

35

4

m⁄ABD + m⁄DBC = m⁄ABC

If ⁄3 § ⁄4 and ⁄4 § ⁄5

If ⁄ABC § ⁄CBDthen BC bisects ⁄ABD

If m⁄ABE is a right angle then m⁄ABE = 90°

If ⁄1 is supplementary to ⁄2 then m⁄1 + m⁄2 = 180°

then ⁄3 § ⁄5

If B is the midpoint of ACthen AB § BC

Segment Addition

Angle Bisector

Angle Addition

Right Angle

Angles

Transitive

Midpoint

Definition of

Property

Definition of

Postulate

Postulate

Definition of

Definition of

Supplementary


8

September 20, 2017

On the worksheet Practice

1. a.) If m∠4 = 63°, find m∠1 and m∠2.

b.) If m∠3 = 121°, find m∠1, m∠2, and m∠4.

2. Write and solve an equation to find x. Use x to find m∠AEB.


9

September 20, 2017

Prove: ∠3 § ∠4

Example 6:Given: ∠1 and ∠4 form a linear pair and m∠1 + m∠3 = 180

3.

4.

g.

12

3

5. Given: ⁄3 and ⁄2 are complementary m⁄1 + m⁄2 = 90°

Prove: ⁄1 § ⁄3

6. Given: AC = BD Prove: AB = CD


10

September 20, 2017

E O

B

F

R

T

S V

U1

2

Given: m⁄1 = m⁄2Prove: m⁄RSU = m⁄TSV

8. Match the correct reason for each step in the proof

Answers:

1a) m¤1 = 117, m¤2 = 63 1b) m¤1 = 121, m¤2 = 59, m¤4 = 59

3. ¤1 and ¤4 form a linear pair – given, ¤1 and ¤4 are supp – supp thm, m¤1 + m¤3 = 180 – given, ¤1 and ¤3 are supp – supp thm, ¤3 § ¤4 – § supp thm

5. given, def of comp ¤s, given, substitution, subtraction, substitution

7. given, def of bisect, segment add post, substitution, addition

Book WorkPg. 154 157 #6, 9, 10, 12, 35


11

September 20, 2017


12

September 20, 2017

Documents

2.8 Proving angle relationships cont. ink.notebook