Angle Pair Relationships

Key Concepts

• Identify relationships between pairs of angles
• Use properties of special pairs of angles
• Describe angles found in home

Right Angles Congruence Theorem

All right angles are congruent.

Proof: Right Angles Congruence Theorem

Given: ∠1 and ∠2 are right angles.

Proof: ∠1 ≅ ∠2

Congruent Supplements Theorem

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

If ∠1 and ∠2 are supplementary and

∠3 and ∠2 are supplementary, then ∠1 ≅ ∠3.

Congruent Supplements Theorem

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

If ∠4 and ∠5 are complementary and

∠6 and ∠5 are complementary, then ∠4 ≅ ∠6.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

∠1 and ∠2 form a linear pair, so ∠ 1 and ∠ 2 are supplementary, and m∠1 + m∠2 = 180.

Vertical Angles Congruence Theorem

Vertical angles are congruent.

Let’s solve some examples!

Use right angle congruence theorem

Example 1: Write a proof.

Given: AB−⊥ BC−, DC−⊥ BC−

Proof: ∠B ≅ ∠C

Solution:

Prove a case of Congruent Supplements Theorem

Example 2: Prove that two angles supplementary to the same angle are congruent.

Given: 

∠1 and ∠2 are supplements.

∠3 and ∠2 are supplements.

Prove:

∠1 ≅ ∠3

Solution:

Prove the Vertical Angles Congruence Theorem

Example 3: Prove vertical angles are congruent.

Given: ∠5 and ∠7 are vertical angles.

Prove:

∠5 ≅ ∠7

Solution:

Standardized Test Practice

Example 4: Which equation can be used to find x? Also, solve for x.

1. 32 + (3x + 1) = 90
2. 32 + (3x + 1) = 180
3. 32 = 3x + 1
4. 3x + 1 = 212

Solution:

Because ∠ TPQ and ∠ QPR form a linear pair, the sum of their measures is 180 degrees.

The correct answer is B.

32 + (3x + 1) = 180

33 + 3x = 180

3x = 180-33 = 147

x = 49

Questions to Solve

Question 1:

Identify the pair(s) of congruent angles in the figures below. Explain how you know they are congruent.

a.

Solution:

∠MSN and ∠PSQ (Both are 50⁰)

∠NPS and ∠QSR (Both are 40⁰ as ∠NPS and ∠NSM are complementary and ∠PSQ and ∠QSR are complementary)

∠PSM and ∠PSR (Both are 90⁰)

b.

Solution:

∠GML and ∠HMJ (As they are vertical angles)

∠GMH and ∠LMJ (As they are vertical angles)

∠GMK and ∠ JMK (Both are 90⁰)

Question 2:

For the following, use the diagram below.

1. If m∠1 = 145, find m∠2, m∠3, and m∠4.
2. If m∠3 = 168, find m∠1, m∠2, and m∠4.

Solution:

a.

m∠3 = m∠1 = 145⁰ (pair of vertical angles)

m∠2 + m∠3 = 180⁰ (linear pair)

m∠2 = 180⁰-145⁰ = 35⁰

m∠1 + m∠4 = 180⁰ (linear pair)

m∠4 = 180⁰-145⁰ = 35⁰ 

b.

m∠3 = m∠1 (vertical angles)

m∠1 = 168⁰

m∠4 + m∠1 = 180⁰ (linear pair)

m∠4 = 180⁰-168⁰ = 12⁰

m∠2 = m∠4 (vertical angles)

m∠2 = 12⁰

Question 3:

Find the values of x and y.

a.

Solution:

Vertical Angles:

8x + 7 = 9x – 4

x = 11

5y = 7y – 34

2y = 34

y = 17 

b.

Solution:

Vertical angles:

4x = 6x – 26

2x = 26

x = 13

6y + 8 = 7y – 12

y = 20

Key Concepts Covered

1. Right Angles Congruence Theorem

All right angles are congruent.

2. Congruent Supplements Theorem

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

3. Congruent Complements Theorem

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

4. Linear Pair Postulate

If two angles form a linear pair, then they are supplementary

5. Vertical Angles Congruence Theorem

Vertical angles are congruent.

Exercise

• Describe the relationship between the angle measures of complementary angles, supplementary angles, vertical angles, and linear pairs.
• In the questions 2 to 5, use the given statement to name two congruent angles. Then give a reason that justifies your conclusion.
• In triangle GFE, Ray GH bisects ∠EGF.
• ∠1 is a supplement of ∠6, and ∠9 is a supplement of ∠6.
• AB is perpendicular to CD, and AB and CD intersect at E.
• ∠5 is complementary to ∠12, and ∠1 is complementary to ∠12.

Use this photo of the folding table to solve the questions 6 to 8.

• If m∠1 = x, write expressions for the other three angle measures.
• Estimate the value of x. What are the measures of the other angles?
• As the table is folded up, ∠4 gets smaller.  What happens to the other three angles? Explain your reasoning.
• Explain how the Congruent Supplements Theorem and the Transitive Property of Angle Congruence can both be used to show how angles that are supplementary to the same angle are congruent.
• Two lines intersect to form ∠1, ∠2, ∠3, and ∠4. The measure of ∠3 is three times the measure of ∠1 and m∠1 = m∠2. Find all four angle measures. Explain your reasoning.