Question

MNO has m<M=63 degrees and m<O=71 degrees. RST has m<R=58 degrees and m<T=71 degrees. applying the proof of the AA similarity theorem, is MNO~RST?

Answer 1

To determine if triangles MNO and RST are similar using the Angle-Angle (AA) similarity theorem, we need to check if two angles in triangle MNO are equal to two angles in triangle RST.

Given:
- \( m \angle M = 63^\circ \)
- \( m \angle O = 71^\circ \)

First, we can find the third angle in triangle MNO. The sum of the angles in a triangle is always \( 180^\circ \).

Calculating \( m \angle N \):
\[
m \angle N = 180^\circ - m \angle M - m \angle O = 180^\circ - 63^\circ - 71^\circ = 46^\circ
\]

So, the angles in triangle MNO are:
1. \( m \angle M = 63^\circ \)
2. \( m \angle O = 71^\circ \)
3. \( m \angle N = 46^\circ \)

Now for triangle RST:
- \( m \angle R = 58^\circ \)
- \( m \angle T = 71^\circ \)

Next, we calculate the third angle in triangle RST:
\[
m \angle S = 180^\circ - m \angle R - m \angle T = 180^\circ - 58^\circ - 71^\circ = 51^\circ
\]

So, the angles in triangle RST are:
1. \( m \angle R = 58^\circ \)
2. \( m \angle T = 71^\circ \)
3. \( m \angle S = 51^\circ \)

Now we can match the angles:
- \( m \angle O = 71^\circ \) corresponds to \( m \angle T = 71^\circ \).
- \( m \angle M = 63^\circ \) does not correspond to any angle in triangle RST.

We have:
- Triangle MNO has angles \( 63^\circ, 71^\circ, \) and \( 46^\circ \).
- Triangle RST has angles \( 58^\circ, 71^\circ, \) and \( 51^\circ \).

Since they do not have two angles that are the same, we cannot conclude that triangle MNO is similar to triangle RST based on the AA similarity theorem.

**Conclusion:** Triangles MNO and RST are not similar (\( MNO \not\sim RST \)).