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Mathematics › 🔄 Grade 7 › Angle Relationships: Complementary, Supplementary, Vertical & Adjacent
🔄 Grade 7 · Lesson 8 of 14

Angle Relationships: Complementary, Supplementary, Vertical & Adjacent

Two crossing lines hide four angles — and two surprising equalities.

Grade 7Middle School
Angle Relationships: Complementary, Supplementary, Vertical & Adjacent — illustration
💡
The big idea: When two lines cross, they create four angles with predictable relationships: adjacent angles share a side and add up to 180°; vertical angles sit opposite each other and are always equal. Two specific sums to remember: complementary pairs add to 90°, supplementary pairs add to 180°.
🎯 By the end, you'll be able to
  • Identify complementary and supplementary angle pairs from their angle measures
  • Name vertical angles and adjacent angles in a figure of two intersecting lines
  • Explain why vertical angles must be equal using supplementary angle reasoning
  • Solve for an unknown angle using angle-pair relationships
📎 You should already know
  • Measuring angles with a protractor (Grade 4 geometry)
  • What a straight angle (180°) is

When two lines cross — what do you get?

Draw two straight lines crossing each other. You immediately see four angles arranged around the crossing point. These angles are not independent — they are locked together by two powerful rules that connect their measures. Understanding those rules lets you find any missing angle in a diagram with almost no work.

🔑 Four angle relationships

Complementary angles: Two angles whose measures add to 90°. Neither has to be in the crossing-lines figure — any two angles that sum to 90° are complementary.

Supplementary angles: Two angles whose measures add to 180°. A straight line creates two supplementary adjacent angles.

Adjacent angles: In the crossing-lines figure, two angles that share a side. They are supplementary (they line up along one of the straight lines).

Vertical angles: Two angles that sit across from each other at the crossing point. They are always equal.

🎮 Angle Pair Explorer LIVE
Drag the crossing point or one arm to change the angles. All four regions update live. Each pair is shown with both a distinct color AND a distinct hatch pattern so they are distinguishable regardless of display settings. The supplementary sums are always 180°.

Why are vertical angles equal? Here's the proof.

Call the four angles a, b, c, d going around. Angles a and c are vertical; b and d are vertical.

  1. a and b are adjacent (on a straight line): a + b = 180°
  2. b and c are adjacent: b + c = 180°
  3. Both equal 180°, so: a + b = b + c
  4. Subtract b from both sides: a = c.

No special measuring needed — it follows purely from supplementary angles adding to 180°.

⚠️ Vertical angles are NOT always 90°!

A very common misconception: students assume vertical angles must be right angles because of the word “vertical.” They do not have to be 90° — they just have to be equal to each other. If one angle at the crossing is 65°, its vertical partner is also 65° (and the other two are each 115°). Only if the lines are perpendicular do any of the four angles equal 90°.

📝 Worked example: Two lines cross. One angle is 72°. Find the measures of the other three angles.
  1. Call the known angle a = 72°.
  2. Vertical angle c = a = 72°.
  3. Adjacent angle b is supplementary to a: b = 180° − 72° = 108°.
  4. Vertical angle d = b = 108°.
✓ The four angles are 72°, 108°, 72°, 108°. They add to 360°, which is correct (a full turn around the crossing point).
📝 Worked example: Two adjacent angles are complementary. One measures (3x + 5)°. The other measures (2x)°. Find x and both angles.
  1. Complementary means they add to 90°: (3x + 5) + 2x = 90.
  2. Combine like terms: 5x + 5 = 90.
  3. Subtract 5: 5x = 85.
  4. Divide by 5: x = 17.
  5. First angle: 3(17) + 5 = 51 + 5 = 56°.
  6. Second angle: 2(17) = 34°.
  7. Check: 56° + 34° = 90°. ✓
✓ x = 17; angles are <strong>56°</strong> and <strong>34°</strong>.

Check your understanding

1. Angles A and B are supplementary. Angle A = 113°. What is angle B?
Supplementary means the pair adds to 180°. B = 180° − 113° = 67°.
2. Two lines cross. One angle is 40°. What is the measure of its vertical angle?
Vertical angles are always equal. The vertical angle is also 40°.
3. Angles P and Q are complementary. P = 28°. What is Q?
Complementary means the pair adds to 90°. Q = 90° − 28° = 62°.
4. Two crossing lines form angles of x° and (x + 40)°. These two angles are adjacent (supplementary). What is x?
x + (x + 40) = 180 → 2x + 40 = 180 → 2x = 140 → x = 70.
5. Why are vertical angles always equal?
Each vertical angle (say a and c) is supplementary to the same adjacent angle b. So a + b = 180 and c + b = 180, giving a = c.
✅ Key takeaways
  • Complementary angles add to 90°; supplementary angles add to 180°.
  • When two lines cross, adjacent angles are supplementary (on a straight line).
  • Vertical angles (opposite each other at the crossing) are always equal.
  • Vertical angles are not necessarily 90° — they equal each other, not necessarily a right angle.
  • Use the proof: both vertical angles are supplementary to the same adjacent angle, so they must be equal.