Lesson 5.3: Special Triangles

Learning Objectives

• Identify isosceles, equilateral, and right triangles.
• Apply angle and side relationships in special triangles.
• Use properties of 45-45-90 and 30-60-90 triangles.
• Solve geometric problems using special triangle rules.

Topics Covered

• Isosceles triangle properties
• Equilateral triangle properties
• Right triangle basics
• 45-45-90 triangle ratios
• 30-60-90 triangle ratios

Key Terms

• Isosceles Triangle: Triangle with two equal sides and two equal base angles.
• Equilateral Triangle: Triangle with all sides and angles equal (each \(60^\circ\)).
• Right Triangle: Triangle containing one \(90^\circ\) angle.
• Hypotenuse: Longest side opposite the right angle.
• Special Right Triangles: Right triangles with predictable side ratios.

Lesson Examples

Example 1: Isosceles Triangle Angles

1. An isosceles triangle has vertex angle \(40^\circ\).
2. The remaining angles are equal.
3. Sum of angles is \(180^\circ\).
4. Each base angle = \( (180-40)/2 = 70^\circ\).

Example 2: 45-45-90 Triangle

1. If each leg = 5
2. Hypotenuse = \(5\sqrt{2}\)

Example 3: 30-60-90 Triangle

1. If shortest side = 4
2. Hypotenuse = 8
3. Longer leg = \(4\sqrt{3}\)

Practice Problems

1. Find the base angles of an isosceles triangle with vertex angle \(50^\circ\).
2. A 45-45-90 triangle has leg 7. Find the hypotenuse.
3. A 30-60-90 triangle has shortest side 6. Find the other sides.
4. Explain why every equilateral triangle is also isosceles.

Tips & Common Mistakes

• Do not assume a triangle is isosceles unless sides or angles confirm it.
• Remember special right triangle ratios must match correct angle positions.
• The hypotenuse is always opposite the right angle.
• Equilateral triangles automatically have all angles \(60^\circ\).

Summary

Special triangles have predictable properties that simplify geometry problems. Recognizing them allows fast calculation of angles and side lengths without complex algebra.

Challenge Problems

1. Prove the base angles of an isosceles triangle are congruent.
2. Show why an equilateral triangle has three 60° angles.
3. Derive the \(1:\sqrt3:2\) ratio of the 30-60-90 triangle.
4. A square of side 10 is cut along its diagonal. Identify the resulting triangle and find all side lengths.