Question

note: figure not drawn to scale
in the figure shown, line m is parallel to line n, and line l intersects both lines. which of the following statements is true?
① ( t = 188 )
② ( s = 188 )
③ ( r = 188 )
④ ( q = 188 )

Explanation:

Step1: Find the measure of \( \angle q \)

Since \( 144^{\circ}\) and \( \angle q \) are supplementary angles (they form a linear pair), we use the formula \( \angle q+144^{\circ}=180^{\circ}\).

$$ \angle q = 180^{\circ}-144^{\circ}=36^{\circ} $$

Step2: Use the property of parallel lines and corresponding angles

Because line \( m \) is parallel to line \( n \), \( \angle s\) and \( \angle q \) are corresponding angles. So \( \angle s=\angle q = 36^{\circ}\).

Step3: Use the property of vertical angles

\( \angle r\) and \( \angle s\) are vertical angles. Vertical angles are equal, so \( \angle r=\angle s = 36^{\circ}\).

Step4: Use the property of parallel lines and alternate - interior angles

\( \angle t\) and \( 144^{\circ}\) are alternate - interior angles. For parallel lines \( m\) and \( n\) cut by a transversal \( k\), alternate - interior angles are equal. So \( \angle t = 144^{\circ}\).

Answer:

\(\angle t = 144^{\circ}\) (Option ①)