Understanding the Isosceles Triangle Problem

The question asks us to find the measure of angle ∠BAC in a triangle ABC. We are given that two sides, AB and AC, are equal in length, and one angle, ∠ABC, measures 72°.

A triangle with two sides of equal length is called an isosceles triangle. A key property of isosceles triangles is that the angles opposite the equal sides are also equal.

Applying Isosceles Triangle Properties

In triangle ABC, we are given that AB = AC.

• Side AB is opposite to angle ∠ACB.
• Side AC is opposite to angle ∠ABC.

Since AB = AC, the angles opposite these sides must be equal. Therefore, ∠ACB = ∠ABC.

We are given that ∠ABC = 72°. So, ∠ACB must also be 72°.

Let's summarize the known angles:

Calculating Angle BAC using Angle Sum Property

The sum of the interior angles in any triangle is always 180°. For triangle ABC, this means:

$\text{∠BAC} + \text{∠ABC} + \text{∠ACB} = 180°$

Now, substitute the known values for ∠ABC and ∠ACB:

$\text{∠BAC} + 72° + 72° = 180°$

Combine the angles:

$\text{∠BAC} + 144° = 180°$

To find ∠BAC, subtract 144° from 180°:

$\text{∠BAC} = 180° - 144°$

$\text{∠BAC} = 36°$

Conclusion: Finding the Angle BAC

Based on the properties of isosceles triangles and the angle sum property of triangles, the measure of angle ∠BAC is 36°.

Revision Table: Key Concepts for Triangle Angles

Additional Information: Types of Triangles and Angles

Triangles can be classified by their sides and angles.

Classification by Sides:

• Equilateral Triangle: All three sides are equal. All three angles are equal ($60°$ each).
• Isosceles Triangle: Two sides are equal. The angles opposite the equal sides are equal.
• Scalene Triangle: All three sides are different lengths. All three angles are different measures.

Classification by Angles:

• Acute Triangle: All three angles are acute (less than $90°$).
• Right Triangle: One angle is a right angle ($90°$).
• Obtuse Triangle: One angle is obtuse (greater than $90°$).

Understanding these classifications helps in solving geometry problems involving triangles.