Mrs. McConaughy Geometry 1

Proving Angles Congruent

During this lesson, you will: Determine and apply conjectures

about angle relationships Prove and apply theorems about

angles

Mrs. McConaughy Geometry 2

Part I: Discovering Angle Relationships

Mrs. McConaughy Geometry 3

Definitions: Special Angle Pairs

Two angles are ___________________ if their measures add up to 90.

Two angles are ___________________ if their measures add up to 180.

complementary angles

supplementary angles

Mrs. McConaughy Geometry 4

Vocabulary Review: Pairs of Angles Formed By Intersecting

LinesOpposite (non-adjacent) angles, formed

by intersecting lines, which share a common vertex and whose sides are opposite rays are called ______________.

Adjacent angles formed by intersecting lines which share a common vertex, a common side, and with one side formed by opposite rays are called ____________.

vertical angles

linear pairs

Mrs. McConaughy Geometry 5

Given the following diagram, identify all vertical angle

pairs:

∠ 1 & ∠ 3

12

34

∠ 2 & ∠ 4

Mrs. McConaughy Geometry 6

Given the following diagram, identify all linear pairs of

angles:

2

4

68

∠ 2 & ∠ 4 ∠ 4 & ∠ 6 ∠ 6 & ∠ 8 ∠ 8 & ∠ 2

Mrs. McConaughy Geometry 7

supplementary 180

Investigative Results:

If two angles are vertical angles, then

the angles are _________. (VERTICAL ANGLES CONJECTURE)

If two angles are a linear pair of angles, then the angles are ______________ (____).

(LINEAR PAIR CONJECTURE)

congruent

Mrs. McConaughy Geometry 8

If two angles are equal and supplementary, what must be

true of the two angles?

If two angles are both equal in measure and supplementary, then each angle measures ____.

(EQUAL SUPPLEMENTS CONJECTURE)

90

Mrs. McConaughy Geometry 9

Examples: Use your conjectures to find the measure of each lettered angle.

Example B

a

b c

Example A

a b

c 70 30

Mrs. McConaughy Geometry 10

Examples: Use your conjectures to a. find the value of the

variable.EXAMPLE C

(3y + 20)

(5y – 16)

EXAMPLE D

(2x – 6)

(3x + 31)

5y – 16 = 3y + 20

5y = 3y + 365y = 3y + 36

2y = 36

y = 18

Vertical Angles Are Congruent

Linear Pairs Are Supplementary

3x + 31 + 2x – 6 = 1805x + 25 = 180

5x = 155

x = 31

Mrs. McConaughy Geometry 11

Homework Assignment:

Discovering Angle Relationships WS 1-5 all, plus select problems from text.

Mrs. McConaughy Geometry 12

Part 2: Proving and Applying Theorems

About Angles

Mrs. McConaughy Geometry 13

Congruent Supplements TheoremIf two angles are supplements of congruent angles, then the two angles are congruent.

Mrs. McConaughy Geometry 14

Given: ∠A supp ∠B; ∠C supp ∠D; ∠B ∠C Prove: ∠A ∠DSTATEMENT REASON

1. 1.

2. 2.

3.

4. 4.

5. 5.

6. 6.

7. 7.

∠A supp ∠B; ∠C supp ∠D

Given.

Def. of supp. ’s∠m A + m B = 180; m C + m D = 180∠ ∠ ∠ ∠

. Substitution Prop. of =

∠B ∠C Given.

m ∠B = m ∠C

Subtraction Prop. of =∠A ∠C

Def. of

m ∠A + m ∠B = m ∠C + m ∠D

m ∠A = m ∠ D

Def. of

Mrs. McConaughy Geometry 15

Vocabulary: Corollary

A _________ of a theorem is a theorem whose proof contains only a few additional statements in addition to the original proof.

EXAMPLE:

If two angles are supplements of the same angle, then the two angles are congruent.

corollary

Mrs. McConaughy Geometry 16

Congruent Complements TheoremIf two angles are complements of congruent angles, then the two angles are congruent.

COROLLARY: If two angles are complements of the same angle, then the two angles are congruent.

Mrs. McConaughy Geometry 17

Given: ∠A comp . ∠B; ∠C comp. ∠D; ∠B ∠C Prove: ∠A ∠D

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6

7. 7.

∠A comp ∠B; ∠C comp ∠D

Given.

m ∠A + m ∠B = 90; m ∠C + m ∠D = 90

Def. of supp. ’s∠

m ∠A + m ∠B = m ∠C + m ∠D

Substitution Prop. of =

∠B ∠C Given.

m ∠B = m ∠C Def. of m ∠A = m ∠ D (-) Prop. of =

∠A ∠C Def. of

Mrs. McConaughy Geometry 18

Vertical Angles TheoremVertical angles are congruent.

Given: ∠ 1 and ∠ 3 are

vertical angles 1

3 2

Prove: ∠ 1 ∠ 3

Mrs. McConaughy Geometry 19

Given: ∠ 1 and ∠ 3 are vertical anglesProve: ∠ 1 ∠ 3

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

∠ 1 and ∠ 3 are vertical angles

Given.

∠ 1 and ∠2 are a linear pair

Def. of linear pair

∠ 2 and ∠3 are a linear pair

Def. of linear pair∠1 supp ∠2; ∠3 supp ∠2

Linear pairs are supp.

∠ 1 ∠ 3 Supp. of same ∠

Mrs. McConaughy Geometry 20

Theorem

All right angles are congruent.

Mrs. McConaughy Geometry 21

Final Checks for Understanding

In the following exercises, ∠ 1 and ∠ 3 are a linear pair, ∠ 1 and ∠ 4 are a linear pair, and ∠ 1 and ∠ 2 are vertical angles. Is the statement true?

a. ∠ 1 ∠ 3 b. ∠ 1 ∠ 2 b. c. ∠ 1 ∠ 4 d. ∠ 3 ∠ 2 c. e. ∠ 3 ∠ 4 f. m∠ 2 + m ∠ 3 = 180

Mrs. McConaughy Geometry 22

Homework Assignment

Pages 100-101: 10-18 all. 32-35 all. Prove: 19 & 35 all.