Question

A triangle with exterior angles is shown. A triangle sits on a line and forms 2 exterior angles on the left and right of the triangle of (2 h) degrees. The top interior angle of the triangle is 40 degrees.
What is the value of h?

h = 20
h = 35
h = 55
h = 70

Answer 1

To determine the value of \( h \), we need to analyze the angles associated with the triangle and its exterior angles.

1. Recall that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, we have two exterior angles each measuring \( 2h \) degrees, and the top interior angle is \( 40 \) degrees.

2. Since the triangle has an exterior angle at the left and right, we can set up the relationship:
\[
2h = x + 40
\]
\[
2h = y + 40
\]
Where \( x \) and \( y \) are the two interior angles at the base of the triangle.

3. The sum of the interior angles of a triangle is \( 180 \) degrees:
\[
x + y + 40 = 180
\]
Rearranging gives us:
\[
x + y = 140
\]

4. Now, if we substitute \( y = 140 - x \) into one of the exterior angle equations:
\[
2h = (140 - x) + 40
\]
Simplifying this:
\[
2h = 180 - x
\]
From the first expression \( 2h = x + 40 \), you can also rearrange it:
\[
x = 2h - 40
\]

5. Now we have two equations:
- From the exterior angle: \( 2h = (140 - x) + 40 \) converts to \( x = 180 - 2h \).
- And from the rearrangement: \( x = 2h - 40 \).

6. Now we can set these equal:
\[
180 - 2h = 2h - 40
\]

7. Solving for \( h \):
\[
180 + 40 = 4h
\]
\[
220 = 4h
\]
\[
h = \frac{220}{4} = 55
\]

Thus, the value of \( h \) is \( 55 \). The answer is:

h = 55