Question

Triangle F G H is shown with its exterior angles. Line H G extends past point G. Line G F extends past point F. Line F H extends past point H. Angle G F H is 53 degrees. Angle F H G is 45 degrees. The exterior angle at point G is x degrees.

What is the value of x?

Answer 1

To find the value of the exterior angle at point G, we can use the properties of angles in a triangle and how exterior angles relate to the interior angles.

In triangle FGH:

1. The interior angles are:
- Angle GFH = 53 degrees (angle at vertex F)
- Angle FHG = 45 degrees (angle at vertex H)

2. To find the third angle FGH (angle at vertex G), we can use the fact that the sum of angles in a triangle is 180 degrees:

\[
\text{Angle GHF} + \text{Angle FHG} + \text{Angle GHF} = 180
\]

Substituting the known values:

\[
53 + 45 + \text{Angle GHF} = 180
\]

Simplifying:

\[
98 + \text{Angle GHF} = 180
\]

\[
\text{Angle GHF} = 180 - 98 = 82 \text{ degrees}
\]

Now, the exterior angle at point G is formed by extending line HG beyond point G. The exterior angle is equal to the sum of the two non-adjacent interior angles:

\[
x = \text{Angle GHF} + \text{Angle FHG}
\]
Substituting the found values into the equation:

\[
x = 82 + 45 = 127 \text{ degrees}
\]

Thus, the value of \( x \) (the exterior angle at point G) is:

\[
\boxed{127 \text{ degrees}}
\]