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:
- \( m \angle M = 63^\circ \)
- \( m \angle O = 71^\circ \)
- \( m \angle N = 46^\circ \)
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:
- \( m \angle R = 58^\circ \)
- \( m \angle T = 71^\circ \)
- \( m \angle S = 51^\circ \)
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 (\( MNO \not\sim RST \)).