Since triangles PQR and LMN are similar (denoted by the notation \( \triangle PQR \sim \triangle LMN \)), the corresponding angles are equal.
In triangle PQR:
- \( m\angle Q = 120^\circ \)
- \( m\angle R = 34^\circ \)
To find the measure of angle \( P \) in triangle PQR, we can use the sum of angles in a triangle, which is \( 180^\circ \):
\[ m\angle P + m\angle Q + m\angle R = 180^\circ \]
Substituting the known values:
\[ m\angle P + 120^\circ + 34^\circ = 180^\circ \]
Calculating the left side:
\[ m\angle P + 154^\circ = 180^\circ \]
Now, solve for \( m\angle P \):
\[ m\angle P = 180^\circ - 154^\circ = 26^\circ \]
Now that we know all angles in triangle PQR:
- \( m\angle P = 26^\circ \)
- \( m\angle Q = 120^\circ \)
- \( m\angle R = 34^\circ \)
Since triangles PQR and LMN are similar, the corresponding angle \( m\angle N \) in triangle LMN is equal to \( m\angle R \) in triangle PQR.
Thus,
\[ m\angle N = m\angle R = 34^\circ \]
Therefore, the measure of angle \( N \) is \( 34^\circ \).
3 answers