Question

In triangle, I, J, K, comma△IJK, start overline, J, K, end overline, \cong, start overline, I, J, end overline
JK
≅
IJ
and m, angle, I, equals, 23, degrees, .m∠I=23
∘
. Find m, angle, J, .m∠J.

Answer 1

In triangle \( \triangle IJK \), we are given that \( \overline{JK} \cong \overline{IJ} \). This means that the triangle is isosceles, with the sides \( IJ \) and \( JK \) being equal.

Since \( \overline{JK} \cong \overline{IJ} \), the angles opposite those sides are also equal. Therefore, we have:

\[
m \angle J = m \angle K
\]

Given that \( m \angle I = 23^\circ \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \):

\[
m \angle I + m \angle J + m \angle K = 180^\circ
\]

Substituting in the known values, we get:

\[
23^\circ + m \angle J + m \angle J = 180^\circ
\]

This simplifies to:

\[
23^\circ + 2m \angle J = 180^\circ
\]

Now, we can isolate \( m \angle J \):

\[
2m \angle J = 180^\circ - 23^\circ
\]
\[
2m \angle J = 157^\circ
\]
\[
m \angle J = \frac{157^\circ}{2}
\]
\[
m \angle J = 78.5^\circ
\]

Thus, the measure of angle \( J \) is:

\[
\boxed{78.5^\circ}
\]