Question

A triangle has a bottom left angle of (x + 15) degrees and a bottom right angle of (x degrees). 1 parallel and horizontal line sits at the top point of the triangle. Another parallel and horizontal line sits at the base of the triangle. The exterior angle to the bottom right angle of the triangle is (4 x minus 20) degrees.
What is the value of x?

x = 32
x = 36
x = 37
x = 40

Answer 1

To solve for \( x \) in the triangle, we need to use the properties of angles in a triangle and the relationship between the interior angles and exterior angles.

1. **Identify the angles:**
- The bottom left angle is \( (x + 15) \) degrees.
- The bottom right angle is \( x \) degrees.
- The top angle of the triangle can be found using the fact that the sum of all angles in a triangle is \( 180 \) degrees.

2. **Set up the equation for the triangle's angles:**
\[
(x + 15) + x + \text{top angle} = 180
\]

The top angle can be denoted as \( y \):
\[
(x + 15) + x + y = 180
\]
\[
2x + 15 + y = 180
\]
\[
y = 180 - 2x - 15
\]
\[
y = 165 - 2x
\]

3. **Use the exterior angle relationship:**
The exterior angle corresponding to the bottom right angle (\( x \) degrees) is given by \( (4x - 20) \) degrees. The exterior angle is also equal to the sum of the two opposite interior angles.
\[
4x - 20 = (x + 15) + y
\]
Substitute \( y \):
\[
4x - 20 = (x + 15) + (165 - 2x)
\]
Simplify the right side:
\[
4x - 20 = x + 15 + 165 - 2x
\]
\[
4x - 20 = -x + 180
\]

4. **Solve for \( x \):**
Rearranging gives:
\[
4x + x = 180 + 20
\]
\[
5x = 200
\]
\[
x = 40
\]

Thus, the value of \( x \) is \( \boxed{40} \).