Bell Work 1) Solve for each variable 2) Solve for each variable 3 and 4) Transitive Property of equality Definition of Congruence Given Definition of Congruence.

Bell Work

• 1) Solve for each variable 2) Solve for each variable

• 3 and 4)

Transitive Property of equalityDefinition of CongruenceSTAB

STJKJKAB , GivenDefinition of Congruence

Outcomes

• I will be able to:

• 1) Use angle congruence properties

• 2) Prove properties about special angle relationships

Properties of Angle Congruence

• Reflexive Property

• Symmetric Property

• Transitive Property

• For any Angle A,

• If, then

• If,• then

AA

BA AB

CBBA and CA

Proof PracticeProve the Transitive Property of Congruence for angles

Given:

• Statements

1. ∠A ≅ ∠B and ∠B ≅∠C

2. m∠A = m∠B

3. m∠B = m∠C

4. m∠A = m∠C

5. ∠A ≅ ∠C

Prove:• Reasons

1. Given

2. Definition of Congruent Angles

3. Definition of Congruent Angles

4. Transitive/Substitution

5. Definition of Congruent Angles

CBBA and

CA

Use the Transitive Property

Given: m∠3 = 40°, ∠1 ≅ ∠2, ∠2 ≅ ∠3 Statements1. m∠3 = 40°, ∠1 ≅ ∠2, ∠2 ≅ ∠3 2. ∠1 ≅ ∠3 3. m∠1 = m∠3 4. m∠1 = 40°

Prove: Angle 1 = 40°

Reasons1. Given2. Transitive Property of

Congruence.

3. Def. of Congruence

4. Substitution

Angle Investigation1) Fold your paper in half. 2) Place the corner of your second piece of paper

at the vertex of right angles and trace it3) Label the four angles from left to right 1, 2, 3, 4

as shown in the picture

4) Answer the following questions on the sheet of paper.

Complementary Questions

Angle Relationship Theorems• Right Angle Congruence Theorem: All right angles are congruent• Congruent Supplements Theorem: If two angles are

supplementary to the same angle (or to congruent angles), then they are congruent.

Ex: Angle 1 is supplementary to Angle 2 and Angle 3 is supplementary to Angle 2 Therefore,• Congruent Complements Theorem: If two angles are

complementary to the same angle (or to congruent angles), then they are congruent.

Ex: Angle 1 and Angle 3 are both complementary to Angle 2. Therefore,

Angle 1 is congruent to Angle 3

Angle 1 is congruent to Angle 3

Proof PracticeProve the Congruent Complements Theorem

Statements1. Angle 1 and Angle 2 are

complements; Angle 3 and Angle 4 are complements; Angle 2 is congruent to Angle 4

2.

3.

4.

5.

6.

7.

Reasons1. Given

2. Definition of Complementary Angles

3. Transitive Property of Equality4. Def. of Congruent Angles5. Substitution Prop. of Equality6. Subtraction Prop. of Equality7. Def. of Congruent Angles

9043

9021

mm

mm

4321 mmmm

42 mm2321 mmmm

31 mm31

Angle Relationship Theorems• Linear Pair Postulate: If two angles form a

linear pair, then they are supplementary.

• Meaning: Angle 1 + Angle 2 = 180°

• Vertical Angles Theorem:• Vertical angles are congruent• Meaning: 31 and 42

Example Using Angle Relationships

• By definition of a right angle, Angle 3 = 90°.• Angle 2 and Angle 5 are vertical angles and Angle 5 =

57°, so Angle 2 = 57°.• Angle 1 and Angle 5 form a linear pair, so Angle 1 +

Angle 5 = 180°. When you substitute 57° for angle 5, and solve for Angle 1, the result is Angle 1 = 123°.

• Angle 4 and Angle 5 are complementary, so Angle 4 + Angle 5 = 90°. When you substitute 57° for Angle 5 and solve for Angle 4, the result is Angle 4 = 33°

.4 and ,3,2,1 of measures theFind

575 and angleright a is 3 diagram, In the

m

Check Point

• ON YOUR OWN• Using your knowledge of angle pair

relationships, solve for each angle in the diagram.

• Then, match the diagram in column A with the appropriate theorem/postulate in column B

Exit Quiz• Given the following information, complete the

two-column proof below