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The triangle in the image is an isosceles triangle because it has two sides of equal length, both labeled 5. The angle \( \angle Q \) is given as \( 68^\circ \). We are tasked with finding the measure of \( \angle R \).

### Solution:
1. **Isosceles Triangle Property:**
   Since the triangle is isosceles, the base angles (i.e., \( \angle P \) and \( \angle R \)) are equal. Hence, \( \angle P = \angle R \).

2. **Sum of Interior Angles in a Triangle:**
   The sum of the angles in any triangle is \( 180^\circ \). Thus, we can set up the following equation:
   \[
   \angle P + \angle Q + \angle R = 180^\circ
   \]
   Substituting the given value of \( \angle Q = 68^\circ \) and noting that \( \angle P = \angle R \), we have:
   \[
   \angle R + 68^\circ + \angle R = 180^\circ
   \]
   \[
   2\angle R + 68^\circ = 180^\circ
   \]
   \[
   2\angle R = 180^\circ - 68^\circ
   \]
   \[
   2\angle R = 112^\circ
   \]
   \[
   \angle R = \frac{112^\circ}{2} = 56^\circ
   \]

Therefore, the measure of \( \angle R \) is **56°**, which does not match any of the given answer options. This may indicate a possible mistake in the question options, but the correct calculation yields \( \angle R = 56^\circ \).

Would you like further details or to ask questions? Here are five related questions:

1. How do we prove a triangle is isosceles based on side lengths?
2. What other methods could verify angle measures in an isosceles triangle?
3. Why is the sum of angles in any triangle always 180 degrees?
4. Can we apply the exterior angle theorem to solve this problem?
5. How would the process differ if this were a scalene triangle?

**Tip:** Always check for symmetry in triangles; equal sides mean equal opposite angles!

What is the measure of ∠R in the isosceles triangle with sides of length 5 and an angle ∠Q = 68°?

This geometry problem focuses on finding the measure of angle ∠R in an isosceles triangle with two sides of equal length (5 units) and angle ∠Q = 68°. This problem applies the isosceles triangle theorem and the angle sum property of triangles. Suitable for students in Grades 8-10.

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