Lesson 2: Some Special Triangles

Property

An isosceles triangle has at least two sides of the same length (called legs).
The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then the angles opposite those sides (the base angles) are congruent.
Additionally, the altitude drawn from the vertex angle to the base acts as a line of symmetry, bisecting both the vertex angle and the base, and creating two congruent right triangles.

Examples

• Basic Application: In $\Delta ABC$, if side $\overline{AB}$ is the same length as side $\overline{AC}$, then the angles opposite them must be equal, meaning $\angle C \cong \angle B$.
• Finding Missing Angles: If an isosceles triangle has a top vertex angle of 40°, the remaining 140° must be split equally between the two base angles. Each base angle measures $(180^\circ - 40^\circ) / 2 = 70^\circ$.
• Using the Altitude: In an isosceles triangle with legs of 10 cm and a base of 12 cm, dropping an altitude to the base cuts the base in half (6 cm). This forms a right triangle with a hypotenuse of 10 and a leg of 6, allowing you to use the Pythagorean Theorem to find the altitude's height (8 cm).

Explanation

Property

In a 45-45-90 triangle, if each leg has length $x$, then the hypotenuse has length $x\sqrt{2}$. The side ratio is:

Examples

Property

In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$, where:

• Short leg (opposite 30°) = $x$
• Long leg (opposite 60°) = $x\sqrt{3}$
• Hypotenuse (opposite 90°) = $2x$

Examples