Angle Relationships and Equations

By the end of this lesson, you will be able to:

• Define supplementary, complementary, vertical, and adjacent angles
• Write and solve equations using angle relationships
• Apply multiple angle relationships to solve multi-step problems

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Define supplementary, complementary, vertical, and adjacent angles
2. State the relationship for each type — sum = 180°, sum = 90°, or equal
3. Write an equation to represent an angle relationship in a figure
4. Solve the equation to find the unknown angle measure
5. Verify by checking the angle relationship holds in the answer
6. Apply multiple relationships in one figure — multi-step problems

One Angle Can Unlock a Whole Figure

A single angle measure of 72° is given at an intersection.

Question: How many of the four angles can we find — without measuring?

Think: What do you already know about angles on a straight line?

Supplementary Angles Share a Straight Line

• Two angles are supplementary if they sum to 180°
• They form a straight line when placed together

Complementary Angles Share a Right Angle Corner

• Two angles are complementary if they sum to 90°
• They form a right angle corner when placed together
• Memory device: Complementary → Corner (right angle)
• Memory device: Supplementary → Straight line

Side-by-Side: The Two Angle-Pair Formulas

The same four steps for every problem:

1. Name the relationship
2. Write the equation
3. Solve for , then find the angle
4. Verify the relationship holds

Supplementary Example: Find the Unknown Angle

Two supplementary angles measure and . Find .

Step 1 — Relationship: Supplementary → sum = 180°

Step 2 — Solve:

Step 3 — Verify: ✓

Complementary Example: Expression for the Angle

Angle , complement . Find the angle.

Angle: — Verify: ✓

x = 18 is a step; 42° is the angle measure.

Check In: Name Each Angle Type

Supplementary or Complementary?

1. Two angles sum to 90° → _______________
2. Two angles form a straight line → _______________
3. One angle is 140°, partner makes a straight line → partner = ___°
4. One angle is 35°, partner makes a right-angle corner → partner = ___°

Practice 1: Find the Supplement

Find the unknown angle.

A straight line is divided into two angles. One angle measures 127°. Find the other angle.

1. Name the relationship: _______________
2. Write the equation: _______________
3. Solve: _______________
4. Verify: _______________

Practice 2: Find the Complement

Find the complement.

An angle measures 37°. Find its complement.

1. Name the relationship: _______________
2. Write the equation: _______________
3. Solve: _______________
4. Verify: _______________

Practice 3: Write and Solve an Equation

An angle measures . It is supplementary to a angle. Find and the angle.

1. Name the relationship: _______________
2. Write the equation: _______________
3. Solve for : _______________
4. Find the angle measure: _______________
5. Verify: _______________

Practice 4: Expression and Complement

An angle measures . Its complement measures . Find and the angle.

1. Write the equation: _______________
2. Solve for : _______________
3. Find the angle measure: _______________
4. Verify: _______________

From One Pair to Four Angles at Once

We've found unknown angles where two angles share a line or corner.

Now: Two full lines cross — creating four angles at once.

Two new relationships: vertical angles (equal) and adjacent angles (supplementary)

Four Angles at an Intersection

• Adjacent angles share a side: ∠1 & ∠2, ∠2 & ∠3, ∠3 & ∠4, ∠4 & ∠1
• Vertical angles are opposite: ∠1 & ∠3, ∠2 & ∠4

Vertical Angles Are Always Equal

Vertical Angles Theorem: Vertical angles are congruent — they have equal measures.

Why? At any intersection:

Simple Vertical Angles: Find All Four

One angle = 72°. Find all four angles.

Check: ✓ — two equal pairs (72, 108, 72, 108)

Algebraic Vertical Angles: Find x and All Angles

Vertical angles: and . Find and all four angles.

Angle: — check: ✓

All four: 58°, 122°, 58°, 122° — x = 12 is the step; 58° is the answer

Check-In: Vertical and Adjacent Pairs

At an intersection, ∠1 = 110°. Answer all three questions.

1. Which angle equals ∠1?
2. Which angles are supplementary to ∠1?
3. What are the measures of all four angles?

Practice Figure 1: Find x and All Angles

Two lines intersect. One angle is and its adjacent angle is .

Find and all four angles.

(Hint: adjacent angles at an intersection are supplementary)

Write the equation → Solve → Find all four angles → Verify

Practice Figure 2: Identify Pairs and Solve

At an intersection, ∠2 = (3x + 5)° and ∠4 = (5x − 19)°.

1. Are ∠2 and ∠4 vertical or adjacent?
2. Write the equation based on that relationship
3. Solve for and find all four angles

Practice Figure 3: Multi-Expression Intersection

Three expressions are given at an intersection:

• ∠1 =
• ∠2 =
• ∠1 and ∠2 are adjacent angles

Find , all four angle measures, and verify.

When One Step Is Not Enough

So far: one relationship per problem. Real figures need two or more in sequence.

Multi-step strategy:

• Work one relationship at a time
• Label each found angle before the next step
• Verify against all stated relationships at the end

Chaining Two Relationships Step by Step

∠A and ∠B supplementary. ∠B and ∠C complementary. ∠A = 140°.

Step 1:

Step 2:

Verify: ✓ and ✓

Three Rays Creating Three Angles

Step 1: (vertical to given 100° angle)

Step 2:

Check-In: Plan the Multi-Step Solution

∠p = 75°, ∠q supplementary to ∠p, ∠r complementary to ∠q.

1. Which relationship do you use first?
2. Which angle do you find first?
3. Which relationship do you use second?
4. Write the equation for ∠r.

Practice: Three Angles on One Line

A straight line has two rays creating three angles.

• ∠1 = , ∠2 = , ∠3 =

All three angles lie on a straight line.

Find , all three angle measures, and verify the sum = 180°.

Practice: Applying Multi-Step at an Intersection

Two lines intersect. ∠a = 50° (given).

• ∠b is vertical to ∠a
• ∠c is supplementary to ∠a
• ∠d is vertical to ∠c

Find all four angles. Show each step with the relationship used.

Bringing All Four Relationships Together

You now have four tools:

In mixed problems: read the figure → identify the relationship → write the equation

Mixed Problem 1: Real-World Angle

Two boards meet at a roof peak, forming a 50° angle on one side.

What is the angle on the other side? Write the equation, solve, and verify.

Mixed Problems 2–3: Complementary and Vertical

Problem 2: Angle , complement . Find both angles.

Problem 3: One angle , vertical angle . Find and both angle measures.

For each: identify relationship → equation → solve → angle → verify.

Mixed Practice: Two Multi-Step Problems

Problem 4: ∠A = and ∠B = are supplementary. ∠C is complementary to ∠B. Find all three angles.

Problem 5: Two lines intersect. ∠1 = . Find ∠2 (adjacent), ∠3 (vertical to ∠1), ∠4 (vertical to ∠2). State the relationship used for each.

Key Takeaways and Common Warnings

Watch out:

• C for Corner (90°), S for Straight (180°)
• Vertical angles are equal, not supplementary
• Find the angle measure — not just
• Always verify the relationship holds

Preview of the Next Lesson

Coming up: 7.G.B.6 — Surface Area and Volume

We'll apply equation-solving to three-dimensional figures.

Before next class:

• Complete the workbook practice set
• Show all five steps: relationship → equation → solve → angle measure → verify