https://file.aiquickdraw.com/mathbot/file/17266882077350eio0nse.jpg

The image shows a triangle \( PQR \) with the following details:
- Angle \( \angle P = 40^\circ \)
- Angle \( \angle R = 2x^\circ \)
- \( PR = PQ \), indicating that triangle \( PQR \) is isosceles.

Since \( PR = PQ \), the angles opposite these sides must be equal, so:
\[
\angle QPR = \angle QRP = 2x^\circ
\]

The sum of the angles in any triangle is \( 180^\circ \). Therefore, for triangle \( PQR \):
\[
40^\circ + 2x^\circ + 2x^\circ = 180^\circ
\]
Simplifying:
\[
40^\circ + 4x^\circ = 180^\circ
\]
\[
4x^\circ = 180^\circ - 40^\circ
\]
\[
4x^\circ = 140^\circ
\]
\[
x = \frac{140^\circ}{4} = 35^\circ
\]

Thus, the value of \( x \) is \( 35^\circ \).

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:
1. What would change if \( PR \neq PQ \)?
2. How would you calculate the angles in a non-isosceles triangle?
3. How does the exterior angle theorem apply in this situation?
4. What are the properties of isosceles triangles regarding angle congruence?
5. Can you generalize this approach to any isosceles triangle problem?

**Tip:** In isosceles triangles, knowing the properties of congruent angles can simplify many geometry problems!

Find the value of x in triangle PQR where angle P = 40°, angle R = 2x°, and triangle PQR is isosceles with PR = PQ.

Find the value of x in an isosceles triangle where angle P is 40° and angle R is 2x°. This solution uses triangle angle sum property and the isosceles triangle theorem to solve the equation. Ideal for students in Grades 8-10.

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