Question

In triangle, O, P, Q, comma△OPQ, start overline, P, Q, end overline, \cong, start overline, O, P, end overline
PQ
​
≅
OP
and m, angle, P, equals, 121, degrees, .m∠P=121
∘
. Find m, angle, O, .m∠O

Answer 1

In triangle \( OPQ \), we are given that \( PQ \cong OP \). This implies that triangle \( OPQ \) is isosceles with sides \( PQ \) and \( OP \) being equal.

Since \( PQ \cong OP \), the angles opposite these sides are also equal. Therefore:

\[
m\angle Q = m\angle O
\]

We also know that the sum of the interior angles in any triangle is \( 180^\circ \). Thus, we can set up the following equation:

\[
m\angle P + m\angle O + m\angle Q = 180^\circ
\]

Substituting the given information into the equation, we have:

\[
121^\circ + m\angle O + m\angle O = 180^\circ
\]

This simplifies to:

\[
121^\circ + 2m\angle O = 180^\circ
\]

To find \( m\angle O \), we first isolate the term:

\[
2m\angle O = 180^\circ - 121^\circ
\]
\[
2m\angle O = 59^\circ
\]

Now, divide both sides by 2:

\[
m\angle O = \frac{59^\circ}{2} = 29.5^\circ
\]

So, the measure of angle \( O \) is:

\[
\boxed{29.5^\circ}
\]