Question

△GHI, start overline, G, H, end overline, \cong, start overline, I, G, end overline
GH
≅
IG
and m, angle, I, equals, 36, degrees, .m∠I=36
∘
. Find m, angle, G, .m∠G.

Answer 1

To find the measure of angle \( G \) in triangle \( \triangle GHI \), we can use the information provided, including the fact that \( \overline{GH} \cong \overline{IG} \). This means that triangle \( GHI \) is isosceles with \( GH = IG \).

Since \( GH \) and \( IG \) are congruent, the angles opposite these sides are also congruent. Therefore, we have:

\[
m\angle G = m\angle H
\]

Let \( m\angle G = m\angle H = x \).

The sum of the angles in any triangle is \( 180^\circ \). Hence, we can write the equation for the angles in triangle \( GHI \):

\[
m\angle G + m\angle H + m\angle I = 180^\circ
\]

Substituting the known values, we have:

\[
x + x + 36^\circ = 180^\circ
\]
\[
2x + 36^\circ = 180^\circ
\]

Now, we can solve for \( x \):

\[
2x = 180^\circ - 36^\circ
\]
\[
2x = 144^\circ
\]
\[
x = \frac{144^\circ}{2}
\]
\[
x = 72^\circ
\]

Thus, the measures of angles \( G \) and \( H \) are both:

\[
m\angle G = 72^\circ
\]

In conclusion, the measure of angle \( G \) is:

\[
\boxed{72^\circ}
\]