Question

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Explanation:

Step1: Find \( m\angle2 \)

Given \( m\angle PQR = 106^\circ \) and \( m\angle1 = 49^\circ \), \( m\angle2 = m\angle PQR - m\angle1 = 106^\circ - 49^\circ = 57^\circ \)

Step2: Find \( m\angle3 \)

In triangle \( QRS \) (or relevant triangle), using angle - sum property. Wait, maybe vertical angles or triangle angle - sum. Wait, looking at the triangle with angles \( 35^\circ \), \( 49^\circ \), and \( m\angle3 \)? Wait, no, actually, in triangle \( QRS \), if we consider the angles, but maybe \( m\angle3 \): Let's see, in triangle with angle \( 35^\circ \) and \( 49^\circ \), the third angle (but maybe \( m\angle3 \) is equal to \( 49^\circ \)? Wait, no, maybe vertical angles or isosceles? Wait, maybe \( m\angle3 = 49^\circ \) (alternate interior or vertical? Wait, maybe I made a mistake. Wait, let's re - examine. Given \( m\angle1 = 49^\circ \), maybe \( \angle3=\angle1 = 49^\circ \) (vertical angles or alternate interior).

Step3: Find \( m\angle4 \)

In triangle, sum of angles is \( 180^\circ \). If we have a triangle with angles \( 35^\circ \), \( 49^\circ \), then \( m\angle4=180^\circ-(35^\circ + 49^\circ)=180 - 84 = 96^\circ \)? Wait, no, maybe another approach. Wait, the angle marked \( 106^\circ \) and \( m\angle4 \): maybe \( m\angle4 = 106^\circ \) (vertical angles).

Step4: Find \( m\angle5 \)

\( m\angle5 = 35^\circ \) (vertical angles or alternate interior, since \( m\angle6 = 35^\circ \))

Step5: Find \( m\angle2 \) (already did, \( 57^\circ \)), \( m\angle3 = 49^\circ \), \( m\angle4 = 106^\circ \), \( m\angle5 = 35^\circ \)

Wait, let's correct:

• \( m\angle1 = 49^\circ \) (given)
• \( m\angle2 \): Since \( m\angle PQR=106^\circ \) and \( m\angle1 = 49^\circ \), \( m\angle2=106 - 49=57^\circ \)
• \( m\angle3 \): In triangle, if we consider the triangle with angle \( 35^\circ \) and \( m\angle2 = 57^\circ \)? No, wait, maybe \( \angle3=\angle1 = 49^\circ \) (vertical angles)
• \( m\angle4 \): \( \angle4 \) and \( \angle PQR \) are vertical angles? So \( m\angle4 = 106^\circ \)
• \( m\angle5 \): \( \angle5=\angle6 = 35^\circ \) (vertical angles)

Answer:

\( m\angle2 = 57^\circ \), \( m\angle3 = 49^\circ \), \( m\angle4 = 106^\circ \), \( m\angle5 = 35^\circ \)

(Note: The original problem's diagram might have some symmetry or vertical angle relationships. The above is based on typical angle - relationship problems in geometry, like vertical angles, triangle angle - sum, and linear pairs.)