An isosceles triangle is a triangle that has (at least) two equal side lengths. If all three side lengths are equal, the triangle is also equilateral. Isosceles triangles are very helpful in determining unknown angles.
In an isosceles triangle, the two equal sides are called legs, and the remaining side is called the base. The angle opposite the base is called the vertex angle, and the point associated with that angle is called the apex. The two equal angles are called the isosceles angles.
If the triangle is also equilateral, any of the three sides can be considered the base.
Because angles opposite equal sides are themselves equal, an isosceles triangle has two equal angles (the ones opposite the two equal sides). Thus, given two equal sides and a single angle, the entire structure of the triangle can be determined. Likewise, given two equal angles and the length of any side, the structure of the triangle can be determined.
Determining the area can be done with only a few pieces of information (namely, 3):
- A triangle with base and equal side has area .
- A triangle with base and isosceles angle has area , as the length of the altitude to the base is . Alternatively, if the apex angle has measure , the area is .
- A triangle with equal side and isosceles angle has area , as the vertex angle has measure , and . Alternatively, if the vertex angle has measure , the area is .
The altitude to the base also satisfies important properties:
- The altitude to the base is the perpendicular bisector of the base.
- The altitude to the base is the angle bisector of the vertex angle.
- The altitude to the base is the line of symmetry of the triangle.
- The altitude to the base is the median from the apex to the base.
It is immediate that any -sided regular polygon can be decomposed into isosceles triangles, where each triangle contains two vertices and the center of the polygon.
A regular -gon is composed of isosceles congruent triangles. Any isosceles triangle is composed of two congruent right triangles as shown in the sketch. These right triangles are very useful in solving -gon problems. The relation given could be handy. Apart from the above-mentioned isosceles triangles, there could be many other isoceles triangles in an -gon.
More interestingly, any triangle can be decomposed into isosceles triangles, for any positive integer . The picture to the right shows a decomposition of a 13-14-15 triangle into four isosceles triangles.
Isosceles triangle has and . What is the measure of ?
Because , we know that . Additionally, the sum of the three angles in a triangle is , so , and since , we have . Thus .
In isosceles triangle , the measure of is . What are the possible measures of ?
There are three possible cases:
- Case 1: . This is easy to deal with, as then immediately.
- Case 2: . Since the angles of a triangle add up to , we know that , and so .
- Case 3: . Again, the angles of a triangle add up to , so , and so .
Therefore, the possible values of are , and .
In the above figure, and the measure of is . What is the measure of ?
From the given condition, both and are isosceles triangles.
In , , implying by the exterior angle of a triangle.
In , , implying by the angle sum of a triangle.