This lesson focuses on the special properties of isosceles triangles. You will learn the relationship between their sides and angles, which is a key concept for solving many geometry problems and proofs.

1. Core Concepts (Short Notes)

• Isosceles Triangle: A triangle with two sides of equal length.

• Apex: The vertex (corner) where the two equal sides meet.

• Base: The side opposite the apex.

• Base Angles: The two angles at either end of the base. These are the angles opposite the equal sides.In ΔABC, if AB = AC, then A is the apex, BC is the base, and ∠B and ∠C are the base angles.

2. Key Theorems

This lesson has one main theorem and its direct opposite (its converse).

Theorem 1: Angles Opposite Equal Sides

If two sides of a triangle are equal, then the angles opposite those sides are equal.

• In short: Equal sides → Equal opposite angles.

• Formula: In ΔABC, if AB = AC, then ∠ACB = ∠ABC.

• Use Case: If you see a triangle with tick marks indicating two equal sides, you immediately know that the two base angles are also equal.

Theorem 2: Sides Opposite Equal Angles (The Converse)

If two angles of a triangle are equal, then the sides opposite those angles are equal.

• In short: Equal angles → Equal opposite sides.

• Formula: In ΔABC, if ∠ACB = ∠ABC, then AB = AC.

• Use Case: If you calculate or are given two equal angles in a triangle, you can conclude that the triangle is isosceles.

3. Tips & Tricks for Exams

• Identify, then Apply: The first step is always to identify the isosceles triangle in the diagram. Look for the tick marks on the sides or given information like AB = AC. Once identified, immediately apply the theorem to find equal angles.

• Combine with the 180° Rule: This theorem is almost always used with the "sum of interior angles" theorem from Lesson 8.

  • Scenario 1: Given the Apex Angle. Subtract the apex angle from 180°, then divide the result by 2 to find each base angle.

    • Example: If apex ∠A = 80°, then ∠B + ∠C = 180° - 80° = 100°. So, ∠B = ∠C = 100° / 2 = 50°.

  • Scenario 2: Given a Base Angle. The other base angle is the same. Find the apex angle by subtracting both base angles from 180°.

    • Example: If base ∠B = 40°, then ∠C = 40° as well. The apex ∠A = 180° - (40° + 40°) = 100°.

• Equilateral Triangle is a Special Case: An equilateral triangle has all three sides equal. This means it's an isosceles triangle in three different ways! Applying the theorem shows that all three angles are equal. Since they must add to 180°, each angle in an equilateral triangle is 180° / 3 = 60°.

• Look for Isosceles Triangles from Radii: In circle geometry problems, a triangle formed by two radii and a chord is always an isosceles triangle because both radii are equal in length.

4. Important Points to Remember

• Reasoning is Key: When writing a proof (rider), you must state the reason for your conclusion in brackets.

  • Use (angles opposite equal sides) when you use equal sides to prove angles are equal.

  • Use (sides opposite equal angles) when you use equal angles to prove sides are equal.

• Don't Mix Up the Theorem and its Converse:

  • Sides equal ⇒ Angles equal (Main Theorem)

  • Angles equal ⇒ Sides equal (Converse)

• The property applies only to the angles opposite the equal sides. The apex angle is generally different.