Question

In triangle, K, L, M, comma△KLM, start overline, K, L, end overline, \cong, start overline, M, K, end overline
KL
≅
MK
and m, angle, M, equals, 22, degrees, .m∠M=22
∘
. Find m, angle, K, .m∠K

Answer 1

Given that triangle \( KLM \) has the sides \( \overline{KL} \cong \overline{MK} \) and \( \angle M = 22^\circ \), we can deduce that triangle \( KLM \) is isosceles with \( KL = MK \).

In isosceles triangles, the angles opposite the equal sides are also equal. Therefore, we can set \( m\angle K = m\angle L \).

Let:

- \( m\angle K = m\angle L = x \)

According to the triangle angle sum property, the sum of the angles in any triangle is \( 180^\circ \):

\[
m\angle K + m\angle L + m\angle M = 180^\circ
\]

Substituting for \( m\angle K \), \( m\angle L \), and \( m\angle M \):

\[
x + x + 22^\circ = 180^\circ
\]

This simplifies to:

\[
2x + 22^\circ = 180^\circ
\]

Now, isolating \( 2x \):

\[
2x = 180^\circ - 22^\circ
\]

\[
2x = 158^\circ
\]

Next, divide by 2 to find \( x \):

\[
x = \frac{158^\circ}{2} = 79^\circ
\]

Thus, we find:

\[
m\angle K = 79^\circ
\]

So, the measure of angle \( K \) is \( \boxed{79^\circ} \).