Question

find the measure of ∠rtn. m∠rtn = \boxed{}^\circ (63 - 6x)^\circ (55 - 2x)^\circ

Explanation:

Step1: Identify parallel lines and transversal

Lines \( RQ \) and \( PO \) are parallel (marked by red arrows), and \( NM \) is a transversal. So, \( \angle RTS \) and \( \angle RTN \) are alternate interior angles? Wait, no, actually, the angles \( (63 - 6x)^\circ \) and \( (55 - 2x)^\circ \) should be equal because they are alternate interior angles (since \( RQ \parallel PO \) and \( NM \) is transversal). Wait, no, maybe corresponding angles? Wait, looking at the diagram, \( RQ \) and \( PO \) are parallel, and \( NM \) intersects them at \( T \) and \( S \). So \( \angle RTS \) (wait, no, the angles at \( T \) and \( S \)): Wait, the angle \( \angle RTN \) is \( (63 - 6x)^\circ \) and the angle \( \angle PST \) or something? Wait, actually, since \( RQ \parallel PO \), the alternate interior angles should be equal. So \( 63 - 6x = 55 - 2x \)? Wait, no, maybe I got the angles wrong. Wait, let's set up the equation. Since the lines are parallel, the angles \( (63 - 6x)^\circ \) and \( (55 - 2x)^\circ \) are equal (alternate interior angles). So:

\( 63 - 6x = 55 - 2x \)

Step2: Solve for \( x \)

Subtract \( 55 \) from both sides:

\( 63 - 55 - 6x = -2x \)

\( 8 - 6x = -2x \)

Add \( 6x \) to both sides:

\( 8 = 4x \)

Divide both sides by \( 4 \):

\( x = 2 \)

Step3: Find \( m\angle RTN \)

Substitute \( x = 2 \) into \( 63 - 6x \):

\( 63 - 6(2) = 63 - 12 = 51 \)

Answer:

\( 51 \)