Question

In triangle, Q, R, S, comma△QRS, start overline, R, S, end overline, \cong, start overline, Q, R, end overline
RS
≅
QR
​
and m, angle, Q, equals, 46, degrees, .m∠Q=46
∘
. Find m, angle, S, .m∠S.

Answer 1

In triangle \( \triangle QRS \), we are given that \( \overline{RS} \cong \overline{QR} \). This means that triangle \( QRS \) is an isosceles triangle with the two sides \( RS \) and \( QR \) being equal.

Since angles opposite to equal sides are also equal in an isosceles triangle, we have:

\[
m\angle R = m\angle S
\]

We are given that:

\[
m\angle Q = 46^\circ
\]

Using the fact that the sum of the angles in a triangle is \( 180^\circ \):

\[
m\angle Q + m\angle R + m\angle S = 180^\circ
\]

Substituting the known values:

\[
46^\circ + m\angle R + m\angle S = 180^\circ
\]

Since \( m\angle R = m\angle S \), we can denote \( m\angle R = m\angle S = x \). Thus, we can rewrite the equation as:

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

This simplifies to:

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

Subtracting \( 46^\circ \) from both sides:

\[
2x = 180^\circ - 46^\circ
\]
\[
2x = 134^\circ
\]

Now, dividing by 2:

\[
x = \frac{134^\circ}{2} = 67^\circ
\]

Thus, we find:

\[
m\angle R = m\angle S = 67^\circ
\]

Finally, we conclude:

\[
m\angle S = 67^\circ
\]