An isosceles triangle is a triangle with at least two congruent sides, such as AB ≅ AC in triangle ABC. The Isosceles Triangle Theorem says that the angles opposite those equal sides are also congruent. This matters because it lets you find missing angle measures quickly and justify why parts of a diagram match.
It is a key tool in triangle proofs, construction problems, and coordinate geometry.
Key Facts
- If AB ≅ AC in △ABC, then ∠B ≅ ∠C.
- Converse: If ∠B ≅ ∠C in △ABC, then AB ≅ AC.
- Triangle angle sum: m∠A + m∠B + m∠C = 180°.
- If the vertex angle is x°, then each base angle is (180° - x°) / 2.
- If each base angle is y°, then the vertex angle is 180° - 2y°.
- In an isosceles triangle, the symmetry line from the vertex to the base can be an angle bisector, median, altitude, and perpendicular bisector.
Vocabulary
- Isosceles triangle
- A triangle with at least two congruent sides.
- Legs
- The congruent sides of an isosceles triangle.
- Base
- The side of an isosceles triangle that is not one of the congruent legs.
- Base angles
- The two angles adjacent to the base and opposite the congruent sides.
- Line of symmetry
- A line that divides a figure into two mirror-image halves.
Common Mistakes to Avoid
- Marking the wrong angles as congruent: the equal angles are opposite the equal sides, not necessarily the angles touching the same side.
- Assuming every triangle with two equal angles was given equal sides: use the converse theorem only when you know the angles are congruent or can prove they are congruent.
- Forgetting that all three angles must total 180°: when the vertex angle is known, subtract it from 180° before dividing by 2.
- Confusing the base with a horizontal side: the base is the side between the two base angles and opposite the vertex angle, even if the triangle is tilted.
Practice Questions
- 1 In △ABC, AB ≅ AC and m∠A = 46°. Find m∠B and m∠C.
- 2 In isosceles △DEF, DE ≅ DF. If m∠E = 3x + 12 and m∠F = 5x - 10, find x and the measure of each base angle.
- 3 Triangle ABC has ∠B ≅ ∠C, but no side lengths are shown. Explain which sides must be congruent and which theorem justifies your answer.