Download Lesson 2.7 Notes - Dr. Dorena Rode

Geometry
Lesson 2.7 NOTES
Prove Angle Pair Relationships
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Theorem 2.3 Right Angle Congruence Theorem
All right angles are congruent.
PROOF
Given ∠1 and ∠2 are right angles
Prove ∠1 ≅ ∠2
Statements
1. ∠1 and ∠2 are right angles
2. m∠1 = 90o , ∠2 = 90o
3. m∠1 = m∠2
4. ∠1 ≅ ∠2
Reasons
1. Given
2. Definition of right angle
3. Transitive property of equality
4. Definition of congruent angles
Example 1 Write a proof.
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅ , 𝐷𝐶
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅
Given: 𝐴𝐵
Prove: ∠𝐵 ≅ ∠𝐶
Statements
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅ , 𝐷𝐶
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅
1. 𝐴𝐵
2. ∠𝐵 and ∠𝐶 are right angles
3. ∠𝐵 ≅ ∠𝐶
2
1
D
C
A
Reasons
1. Given
2. Definition of perpendicular lines
3. Right angle congruence theorem
B
Theorem 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle then they are congruent.
Theorem 2.5 Congruent Complements Theorem
If two angles are complementary to the same angle then they are congruent.
Example 2 Prove Theorem 2.4: Two angles are supplementary to the same angle are congruent.
Given: ∠1 and ∠2 are supplements
∠3 and ∠2 are supplements
1
Prove: ∠1 ≅ ∠3
Statements
1. ∠1 and ∠2 are supplements
∠3 and ∠2 are supplements
2. m∠1 + m∠2 =180o
m∠3 + m∠2 =180o
3. m∠1 + m∠2 = m∠3 + m∠2
4. m∠1 = m∠3
5. ∠1 ≅ ∠3
Reasons
1. Given
2. Definition of supplementary angles
3. Transitive Property of Equality
4. Subtraction Property of Equality
5. Definition of congruent angles
2
3
Guided Practice On Worksheet 2.7 – problems 1+2
When two lines intersect, pairs of vertical angles and linear pairs are formed (lesson 1.5).
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
1
2
∠1 and ∠2 form a linear pair, so ∠1 and ∠2 are supplementary and m∠1 + m∠2 =180o
Theorem 2.6 Vertical Angles Congruence Theorem
Vertical angles are congruent.
∠1 ≅ ∠3
∠2 ≅ ∠4
Example 3
Prove the Vertical Angles Congruence Theorem
Given: ∠5 and ∠7 are vertical angles
Prove: ∠5 ≅ ∠7
Statements
1. ∠5 and ∠7 are vertical angles
2. ∠5 and ∠6 are a linear pair
∠6 and ∠7 are a linear pair
3. ∠5 and ∠6 are supplementary
∠6 and ∠7 are supplementary
4. ∠5 ≅ ∠7
2
1
4
3
7
5
6
Reasons
1. Given
2. Definition of a linear pair, as shown in the diagram.
3. Linear Pair Postulate
4. Congruent Supplements Theorem
Guided Practice on Worksheet 2.7 – problems 3+4