## 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*.

- 32 + (3x + 1) = 90
- 32 + (3x + 1) = 180
- 32 = 3x + 1
- 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 50^{0})

∠NPS and ∠QSR (Both are 40^{0} as ∠NPS and ∠NSM are complementary and ∠PSQ and ∠QSR are complementary)

∠PSM and ∠PSR (Both are 90^{0})

**b.**

**Solution:**

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

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

∠GMK and ∠ JMK (Both are 90^{0})

**Question 2:**

For the following, use the diagram below.

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

**Solution:**

**a.**

**m∠3 **= m∠1 = 145^{0} (pair of vertical angles)

m∠2 + m∠3 = 180^{0} (linear pair)

**m∠2 **= 180^{0}-145^{0} = 35^{0}

m∠1 + m∠4 = 180^{0} (linear pair)

**m∠4 **= 180^{0}-145^{0} = 35^{0}

**b.**

m∠3 = m∠1 (vertical angles)

**m∠1** = 168^{0}

m∠4 + m∠1 = 180^{0} (linear pair)

**m∠4** = 180^{0}-168^{0} = 12^{0}

m∠2 = m∠4 (vertical angles)

**m∠2** = 12^{0}

**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.

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