Given that triangle ABC is similar to DEF, the measure of D is 102 degrees, the measure of A is 102 degrees, and the measure of B is 63+4, find the value of X.

Since \(\triangle ABC \sim \triangle DEF\), their corresponding angles are equal.

Given:
- \(\angle D = 102^\circ\)
- \(\angle A = 102^\circ\)
- \(\angle B = 63 + 4 = 67^\circ\)

Using the triangle sum theorem:

\[
\angle A + \angle B + \angle C = 180^\circ
\]

Substituting values:

\[
102^\circ + 67^\circ + \angle C = 180^\circ
\]

\[
\angle C = 180^\circ - 169^\circ = 11^\circ
\]

Since \(\triangle ABC \sim \triangle DEF\), corresponding angles are equal:

\[
\angle C = \angle F = 11^\circ
\]

\[
\angle B = \angle E = 67^\circ
\]

\[
\angle A = \angle D = 102^\circ
\]

Thus, \(X = 11\).

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### Related Questions:
1. How do you prove that two triangles are similar?
2. What is the triangle sum theorem and how is it used?
3. How do corresponding angles help in solving similarity problems?
4. What are other methods to solve for missing angles in a triangle?
5. How does similarity affect side length ratios?

**Tip:** Always check if angles sum to 180° in a triangle to verify correctness!

Learn how to determine the missing angle X in similar triangles ABC and DEF using the Triangle Sum Theorem. This problem involves applying the angle sum property of a triangle and is suitable for students in Grades 7-9.

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