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2.6 Proving Statements about Angles. Mrs. Spitz Geometry Fall 2004. Standards/Objectives. Standard 3: Students will learn and apply geometric concepts. Objectives: Use angle congruence properties Prove properties about special pairs of angles. Theorem 2.2 Properties of Angle Congruence.

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2 6 proving statements about angles

2.6 Proving Statements about Angles

Mrs. Spitz Geometry Fall 2004

standards objectives
Standards/Objectives

Standard 3: Students will learn and apply geometric concepts.

Objectives:

  • Use angle congruence properties
  • Prove properties about special pairs of angles.
theorem 2 2 properties of angle congruence
Theorem 2.2 Properties of Angle Congruence
  • Angle congruence is reflexive, symmetric, and transitive.
  • Examples:
    • Reflexive: For any angle A, A ≅ A.
    • Symmetric: If A ≅ B, then B ≅ A
    • Transitive: If A ≅ B and B ≅ C, then A ≅ C.
ex 1 transitive property of angle congruence
Ex. 1: Transitive Property of Angle Congruence
  • Prove the Transitive Property of Congruence for angles

Given: A ≅ B, B ≅ C

Prove: A ≅ C

C

B

A

ex 1 transitive property of angle congruence5
Statement:

A ≅ B, B ≅ C

mA = mB

mB = mC

mA = mC

A ≅ C

Reason:

Given

Def. Cong. Angles

Def. Cong. Angles

Transitive property

Def. Cong. Angles

Ex. 1: Transitive Property of Angle Congruence
ex 2 using the transitive property
Ex. 2: Using the Transitive Property

Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3

Prove: m1 =40

1

4

2

3

slide7
Statement:

m3 ≅ 40, 1 ≅ 2, 2 ≅ 3

1 ≅ 3

m1 = m3

m1 = 40

Reason:

Given

Trans. Prop of Cong.

Def. Cong. Angles

Substitution

Ex. 2:
theorem 2 3
Theorem 2.3

All right angles are congruent.

Example 3: Proving Theorem 2.3

Given: 1 and 2 are right angles

Prove: 1 ≅ 2

slide9
Statement:

1 and 2 are right angles

m1 = 90, m2 = 90

m1 = m2

1 ≅ 2

Reason:

Given

Def. Right angle

Transitive property

Def. Cong. Angles

Ex. 3:
properties of special pairs of angles
Properties of Special Pairs of Angles
  • Theorem 2.4: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

If m1 +m2 = 180 AND m2 +m3 = 180, then 1 ≅ 3.

congruent complements theorem
Congruent Complements Theorem
  • Theorem 2.5: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent.

If m4 +m5 = 90 AND m5 +m6 = 90, then 4 ≅ 6.

proving theorem 2 4
Proving Theorem 2.4

Given: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4

Prove: 2 ≅ 3

3

4

1

2

slide13
Statement:

1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4

m 1 + m 2 = 180; m 3 + m 4 = 180

m 1 + m 2 =m 3 + m 4

m 1 = m 4

m 1 + m 2 =m 3 + m 1

m 2 =m 3

2 ≅ 3

Reason:

Given

Def. Supplementary angles

Transitive property of equality

Def. Congruent Angles

Substitution property

Subtraction property

Def. Congruent Angles

Ex. 4:
postulate 12 linear pair postulate
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
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
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.
vertical angles theorem
Vertical Angles Theorem
  • Vertical angles are congruent.

2

3

1

4

1≅ 3; 2≅ 4

proving theorem 2 6
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

ex 6 proving theorem 2 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

Ex. 6: Proving Theorem 2.6
your assignment
Your Assignment
  • In-Class--2.6 pp. 112-114 #1-28

Coming attractions:

  • Test Ch. 2 and Binder Check Wednesday
  • Friday—Chapter 3 Definitions on page 128 and Postulates/Theorems
  • Chapter 2 Review/Algebra Review may be completed for extra credit. SHOW ALL WORK!