https://file.aiquickdraw.com/mathbot/file/1729968414009h4isf1gm.jpg

Let's go through question 2 from your image, where we're asked to find the value of \( x \) in a triangle.

In question 2:

- The triangle \( ABC \) has two sides labeled as congruent (marked with lines), meaning \( AB = AC \). Thus, triangle \( ABC \) is an **isosceles triangle**.
- Angle \( \angle A \) is labeled as \( 4x \), and angle \( \angle BDC \) (at the base) is given as \( 30^\circ \).

### Solution:

1. **Isosceles Triangle Property**: Since \( AB = AC \), the base angles \( \angle ABC \) and \( \angle ACB \) are equal. Let's call each of these angles \( y \).

2. **Angle Sum in a Triangle**: The sum of all angles in a triangle is \( 180^\circ \).
   \[
   4x + y + y = 180^\circ
   \]
   Simplify to get:
   \[
   4x + 2y = 180^\circ
   \]

3. **Base Angle Information**: Since \( \angle BDC = 30^\circ \), this is an exterior angle to triangle \( ABC \), which equals the sum of the two remote interior angles (the base angles):
   \[
   y + y = 30^\circ
   \]
   So,
   \[
   2y = 30^\circ
   \]
   \[
   y = 15^\circ
   \]

4. **Substitute \( y = 15^\circ \) back**: 
   Substitute \( y = 15^\circ \) into the angle sum equation:
   \[
   4x + 2(15^\circ) = 180^\circ
   \]
   \[
   4x + 30^\circ = 180^\circ
   \]
   \[
   4x = 150^\circ
   \]
   \[
   x = 37.5^\circ
   \]

However, this answer doesn't align with the options provided. Let me know if you'd like more detailed verification or help with other questions.

In triangle ABC, where AB = AC, the angle at A is labeled as 4x, and the exterior angle BDC at the base is given as 30°. What is the value of x?

Solve for x in an isosceles triangle where AB = AC, and the exterior angle BDC is 30°. This problem uses geometry concepts like the exterior angle theorem and angle sum in triangles. Suitable for Grades 8-10.

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