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In the given triangle \( \triangle DEF \), we are tasked with finding the measure of angle \( \angle DEF \).

From the diagram, the following observations can be made:

1. \( \triangle DEF \) is an isosceles triangle because the sides \( EF \) and \( DF \) are marked as congruent.
2. The angle \( \angle EDF \) is given as \( 42^\circ \).

Since it's an isosceles triangle, the base angles \( \angle DEF \) and \( \angle EFD \) are congruent. Therefore, we can denote both \( \angle DEF \) and \( \angle EFD \) as \( x \).

Now, using the fact that the sum of the interior angles of any triangle is \( 180^\circ \), we have the following equation:

\[
x + x + 42^\circ = 180^\circ
\]

Simplifying:

\[
2x + 42^\circ = 180^\circ
\]

Subtracting \( 42^\circ \) from both sides:

\[
2x = 138^\circ
\]

Dividing by 2:

\[
x = 69^\circ
\]

Thus, \( m \angle DEF = 69^\circ \).

Would you like further clarification or details?

Here are some related questions to explore:

1. How do you classify triangles based on their angles and sides?
2. What is the exterior angle theorem in triangles?
3. How can you prove that the base angles in an isosceles triangle are congruent?
4. What is the significance of the sum of the angles in any polygon?
5. How can you solve for unknown sides in an isosceles triangle using trigonometry?

**Tip:** In any triangle, knowing the properties of special triangles, such as isosceles or equilateral triangles, can help simplify solving for unknown angles or sides.

This tutorial explains how to find the measure of an angle in a right triangle when another angle is known, using the complementary angle rule. It is suitable for middle school students learning basic geometry concepts.

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