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Angle Relationships

A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

Complementary angles add to 90 degrees, supplementary angles add to 180 degrees, and vertical angles formed by intersecting lines are congruent. In Geometry Foundations, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

What you'll learn

Why it matters: Architects and carpenters use angle relationships when fitting beams, trim, and corners that must meet cleanly.

Worked example

Problem. Two supplementary angles measure x and 115 degrees. Find x.

  1. Supplementary angles add to 180 degrees.
  2. Write x + 115 = 180.
  3. Subtract 115 to get x = 65.
  4. Connect the result back to Angle Relationships so the geometric relationship is explicit.

Answer: 65 degrees

Practice problems

1. An angle is complementary to 38 degrees. What is its measure?

Show solution
  1. Warm-up: First identify exactly what the question is asking: An angle is complementary to 38 degrees. What is its measure?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Complementary angles add to 90.
  4. 90 - 38 = 52.
  5. Check the result by substituting or estimating: the response should match 52 and make sense in the original problem.

Answer: 52

2. Two angles add to 180 degrees. What are they called?

Choices: Complementary · Supplementary · Vertical · Adjacent

Show solution
  1. Warm-up: First identify exactly what the question is asking: Two angles add to 180 degrees. What are they called?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Supplementary angles add to 180 degrees.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: Supplementary

3. An angle is supplementary to 104 degrees. What is its measure?

Show solution
  1. Core Practice: First identify exactly what the question is asking: An angle is supplementary to 104 degrees. What is its measure?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 180 - 104 = 76.
  4. Check the result by substituting or estimating: the response should match 76 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: 76

4. Vertical angles measure 4x and 96 degrees. Find x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Vertical angles measure 4x and 96 degrees. Find x.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Vertical angles are equal.
  4. 4x = 96.
  5. Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.

Answer: 24

5. Angles x and 2x form a straight line. Find x.

Show solution
  1. Challenge: First identify exactly what the question is asking: Angles x and 2x form a straight line. Find x.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. x + 2x = 180.
  4. 3x = 180.
  5. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

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