Question

Triangle OMN is congruent to triangle RPQ by the HL Theorem. if m angle O = 57 degrees, then what is the measure of angle P?

Answer 1

In congruent triangles, corresponding angles are equal. Given that triangle OMN is congruent to triangle RPQ by the HL (Hypotenuse-Leg) Theorem, we know that:

- Angle O in triangle OMN corresponds to angle R in triangle RPQ
- Angle M in triangle OMN corresponds to angle P in triangle RPQ
- Angle N in triangle OMN corresponds to angle Q in triangle RPQ

Since you mentioned that \( m \angle O = 57^\circ \), and because the triangles are congruent, we have:

\[
m \angle R = m \angle O = 57^\circ
\]

Now, to find \( m \angle P \), we need to realize that in a right triangle, the angles sum up to 90 degrees, since one angle is the right angle (90 degrees). Therefore, if we know one angle in triangle RPQ is \( 57^\circ \), we can express the measure of angle P as follows:

Let \( m \angle P + m \angle R + m \angle Q = 180^\circ \), but since it is a right triangle, we have:

\[
m \angle P + 57^\circ + 90^\circ = 180^\circ
\]

This simplifies to:

\[
m \angle P + 147^\circ = 180^\circ
\]

Subtract \( 147^\circ \) from both sides:

\[
m \angle P = 180^\circ - 147^\circ = 33^\circ
\]

Thus, the measure of angle P is:

\[
\boxed{33^\circ}
\]