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Lesson 4-2

Triangle & Quadrilateral Angle Sums

Discover the angle sum properties of triangles and quadrilaterals. Learn geometric proofs through hands-on angle investigation activities.

Learning Scenario: Angle Investigation Lab

Scenario: You're a geometry detective! You need to investigate a mystery: "What's the total of all angles inside different shapes?" You have triangles, squares, rectangles, and other quadrilaterals. Measure each angle and add them up. What pattern do you discover?

This investigation will reveal one of geometry's most important secrets: the angle sum properties of triangles and quadrilaterals!

Triangle Angle Sum

The sum of all three interior angles in any triangle always equals 180°.

Formula:

∠A + ∠B + ∠C = 180°

This works for ALL triangles: equilateral, isosceles, scalene, right, acute, obtuse!

Quadrilateral Angle Sum

The sum of all four interior angles in any quadrilateral always equals 360°.

Formula:

∠A + ∠B + ∠C + ∠D = 360°

This works for ALL quadrilaterals: squares, rectangles, parallelograms, trapezoids!

Step-by-Step Learning
1

Triangle Investigation

Measure all three angles of different triangles and add them together. You'll discover they always equal 180°.

Example: Right triangle: 90° + 45° + 45° = 180° ✓

2

Quadrilateral Investigation

Measure all four angles of different quadrilaterals and add them together. You'll discover they always equal 360°.

Example: Rectangle: 90° + 90° + 90° + 90° = 360° ✓

3

Using the Properties

Use these properties to find missing angles. If you know some angles, you can calculate the unknown ones.

Example: Triangle with angles 60° and 80°: Missing angle = 180° - 60° - 80° = 40°

Interactive Activities

Activity 1: Triangle Angle Hunt

Find the missing angles in these triangles:

  • • Triangle 1: 60°, 70°, ?
  • • Triangle 2: 90°, 30°, ?
  • • Triangle 3: 45°, 45°, ?
  • • Triangle 4: 100°, 40°, ?
  • • Triangle 5: 80°, 50°, ?

Activity 2: Quadrilateral Challenge

Find the missing angles in these quadrilaterals:

  • • Quad 1: 90°, 90°, 90°, ?
  • • Quad 2: 120°, 60°, 120°, ?
  • • Quad 3: 80°, 100°, 80°, ?
  • • Quad 4: 70°, 110°, 70°, ?
  • • Quad 5: 60°, 120°, 60°, ?
Practice Problems

Problem Set 1: Triangle Angles

1. Triangle has angles 50° and 60°. Find the third angle.

2. Right triangle has angles 90° and 30°. Find the third angle.

3. Triangle has angles 40° and 80°. Find the third angle.

4. Triangle has angles 70° and 50°. Find the third angle.

Problem Set 2: Quadrilateral Angles

5. Quadrilateral has angles 90°, 90°, 90°. Find the fourth angle.

6. Quadrilateral has angles 100°, 80°, 100°. Find the fourth angle.

7. Quadrilateral has angles 60°, 120°, 60°. Find the fourth angle.

8. Quadrilateral has angles 70°, 110°, 70°. Find the fourth angle.

Problem Set 3: Real-World Applications

9. A triangular garden has angles 60° and 70°. What's the third angle?

10. A rectangular table has three 90° corners. What's the fourth angle?

11. A triangular roof has angles 45° and 45°. What's the third angle?

12. A kite has angles 120°, 60°, 120°. What's the fourth angle?

Common Mistakes to Avoid

❌ Confusing triangle and quadrilateral angle sums

Wrong: Thinking all shapes have the same angle sum

Correct: Triangles = 180°, Quadrilaterals = 360°

❌ Forgetting to subtract from the total

Wrong: Adding known angles instead of subtracting from total

Correct: Missing angle = Total - Known angles

Key Takeaways

Triangle interior angles always sum to 180°

Quadrilateral interior angles always sum to 360°

Use these properties to find missing angles

These rules work for ALL triangles and quadrilaterals