Question

Find the measures of the angles of an isosceles triangle such that, when an angle bisector is drawn, two more isosceles triangles are formed.

Short answer

The angle measures are 60°, 30°, and 30°.

Step-by-step solution

Understanding the Problem

We need to find the measures of the angles in an isosceles triangle. When an angle bisector is drawn from the vertex angle to the opposite side, it splits the triangle into two smaller isosceles triangles.

Essential Properties and Variables Introduction

Let the original isosceles triangle be triangle ABC, where \( AB = AC \). Let \( \angle BAC = \theta \) be the vertex angle, and \( \angle ABC = \angle ACB = \alpha \) be the base angles. The interior angles of the triangle sum to 180°, so \( \theta + 2\alpha = 180° \).

Evaluate Conditions for Forming Two Isosceles Triangles

Draw the angle bisector of \( \angle BAC \), which meets side BC at point D. For the two new triangles \( \triangle ABD \) and \( \triangle ACD \) to be isosceles, we need \( AB = BD \) and \( AC = CD \). This implies that \( \angle ABD = \angle BAD = \frac{\theta}{2} \) and \( \angle ACD = \angle CAD = \frac{\theta}{2} \).

Make Use of Isosceles Triangle Properties

For \( \triangle ABD \), having \( \angle ABD = \angle BAD \), the measures of remaining angle \( \angle ADB = \alpha \). Since \( \angle ACB = \angle ACD + \angle DCB = \frac{\theta}{2} + \alpha \), it also leads to the condition \( \frac{\theta}{2} = \alpha \).

Solve the Angle Equations

Substituting \( \alpha = \frac{\theta}{2} \) into the angle sum equation for \( \triangle ABC \):\[ \theta + 2 \left(\frac{\theta}{2}\right) = 180° \] \[ \theta + \theta = 180° \] \[ 2\theta = 180° \] \[ \theta = 60° \]Then, the base angles \( \alpha \) are \( \frac{60°}{2} = 30° \).

Verification of Angles Forming Isosceles Triangles

Check the conditions: \( \triangle ABD \) and \( \triangle ACD \) are isosceles as they have two sides and two angles equal (\( \angle ADB = 30° \) and base angles in both small triangles are 30°), confirming consistency with original measures.

Key concepts

Angle Bisector

An angle bisector is a line or ray that divides an angle into two equal smaller angles. In the context of an isosceles triangle like triangle ABC, when the angle bisector is drawn from the vertex angle (\( \angle BAC \)) to the base (BC), it creates two segments: AD and BC, where D is the point on BC. This action is crucial for forming two smaller triangles \( \triangle ABD \) and \( \triangle ACD \), which will also be isosceles if certain conditions are met.
The angle bisector ensures that the vertex angle \( \theta \) is split into two equal angles \( \frac{\theta}{2} \). This is a key property that can help us solve angle-related problems in triangles. By balancing the angles on either side of the bisector, the geometry naturally forms conditions that help us analyze and further understand the triangle's properties.

Triangle Angle Sum

In any triangle, the sum of the three interior angles is always equal to 180 degrees. This is an essential rule known as the Triangle Angle Sum Theorem. It is a foundational concept when working with triangle-related problems. In the case of our original isosceles triangle ABC, we use this property extensively to determine unknown angles.
For triangle ABC, where \( AB = AC \), we let \( \angle BAC = \theta \) and the base angles \( \angle ABC = \angle ACB = \alpha \). Applying the Triangle Angle Sum gives us the equation:

• \( \theta + 2\alpha = 180° \)

By solving the equation with given conditions in the problem, one can deduce the measure of angles in the isosceles triangle. This theorem is vital for verifying that any computed angles fit well into the geometrical structure set out by the triangle.

Isosceles Triangle Properties

Isosceles triangles possess unique characteristics that distinguish them from other triangles. The primary property is that they have at least two sides of equal length, resulting in two angles of equal measure opposite those sides. These properties are essential when constructing solutions for problems involving isosceles triangles.
For instance, in our original problem, triangle ABC is isosceles because \( AB = AC \). This makes the base angles \( \angle ABC \) and \( \angle ACB \) equal. When an angle bisector is drawn from \( \angle BAC \), it not only splits the angle \( \theta \) but also forms two smaller isosceles triangles \( \triangle ABD \) and \( \triangle ACD \). This occurs because the bisector ensures that \( \angle ABD = \angle BAD \) and \( \angle ACD = \angle CAD \), thus making \( BD = AB \) and \( CD = AC \), preserving the isosceles property.
Understanding these properties helps one grasp how and why smaller isosceles triangles form within a larger one, providing a clear path to solve related geometric problems.