Question

1. m∠2 = 97° m∠6 = 83° m∠3 = ____ m∠5 = __ m∠10 = __ m∠7 = __ m∠9 = __ m∠16 = ____

Explanation:

Step1: Find \( m\angle3 \)

\( \angle2 \) and \( \angle3 \) are supplementary (linear pair), so \( m\angle3 = 180^\circ - m\angle2 \).
\( m\angle3 = 180^\circ - 97^\circ = 83^\circ \).

Step2: Find \( m\angle5 \)

\( \angle2 \) and \( \angle5 \) are same - side interior angles? Wait, no, \( \angle2 \) and \( \angle5 \): Wait, \( \angle2 \) and \( \angle5 \): Wait, actually, \( \angle6 = 83^\circ \), and \( \angle5 \) and \( \angle6 \) are supplementary (linear pair), so \( m\angle5 = 180^\circ - m\angle6 \).
\( m\angle5 = 180^\circ - 83^\circ = 97^\circ \). Also, \( \angle2 \) and \( \angle5 \): Wait, alternatively, \( \angle2 \) and \( \angle5 \) are same - side interior? No, better to use linear pair for \( \angle5 \) and \( \angle6 \).

Step3: Find \( m\angle10 \)

\( \angle2 \) and \( \angle10 \) are corresponding angles (since lines \( t\) and \( s\) are cut by transversal \( m\)), so \( m\angle10 = m\angle2 = 97^\circ \).

Step4: Find \( m\angle7 \)

\( \angle6 \) and \( \angle7 \) are supplementary (linear pair), so \( m\angle7 = 180^\circ - m\angle6 \). Wait, no, \( \angle6 \) and \( \angle7 \) are adjacent angles forming a linear pair? Wait, \( \angle5 \) and \( \angle6 \) are linear pair, \( \angle6 \) and \( \angle7 \): Wait, \( \angle6 \) and \( \angle7 \) are vertical? No, \( \angle6 \) and \( \angle8 \) are vertical, \( \angle5 \) and \( \angle7 \) are alternate interior? Wait, no, \( \angle6 = 83^\circ \), \( \angle7 \) and \( \angle6 \) are supplementary? Wait, no, \( \angle5 + \angle6 = 180^\circ \), \( \angle5 \) and \( \angle7 \) are alternate interior angles (if lines \( t\) and \( s\) are parallel? Wait, since \( \angle2 + \angle6 = 97^\circ+83^\circ = 180^\circ \), so lines \( t\) and \( s\) are parallel (same - side interior angles supplementary). So \( \angle6 \) and \( \angle10 \): No, \( \angle7 \) and \( \angle6 \): Wait, \( \angle6 = 83^\circ \), \( \angle7 \) and \( \angle6 \) are alternate interior angles? Wait, no, \( \angle5 \) and \( \angle7 \) are alternate interior, \( \angle5 = 97^\circ \), so \( \angle7 = 97^\circ \)? Wait, no, let's correct. Since \( t\parallel s\) (because \( \angle2 + \angle6 = 180^\circ \), same - side interior angles), then \( \angle6 \) and \( \angle10 \): No, \( \angle7 \) and \( \angle6 \): Wait, \( \angle6 = 83^\circ \), \( \angle7 \) is supplementary to \( \angle6 \)? No, \( \angle6 \) and \( \angle7 \) are adjacent, forming a linear pair? Wait, the diagram: \( \angle5\), \( \angle6\) on line \( t\), \( \angle7\), \( \angle8\) below. So \( \angle6 \) and \( \angle7 \) are same - side interior? No, \( \angle5 \) and \( \angle7 \) are alternate interior. Since \( t\parallel s\), \( \angle5 = \angle7\) (alternate interior angles). We found \( \angle5 = 97^\circ \), so \( m\angle7 = 97^\circ \)? Wait, no, \( \angle6 = 83^\circ \), \( \angle5 = 97^\circ \), \( \angle7 \) and \( \angle6 \): Wait, \( \angle7 \) and \( \angle6 \) are vertical? No, \( \angle6 \) and \( \angle8 \) are vertical. \( \angle7 \) and \( \angle5 \) are alternate interior, so \( m\angle7 = m\angle5 = 97^\circ \).

Step5: Find \( m\angle9 \)

\( \angle3 \) and \( \angle9 \) are corresponding angles (lines \( t\) and \( s\) cut by transversal \( m\)), so \( m\angle9 = m\angle3 = 83^\circ \).

Step6: Find \( m\angle16 \)

First, \( \angle6 = 83^\circ \), \( \angle14 \) and \( \angle6 \) are corresponding angles (lines \( t\) and \( s\) cut by transversal \( n\)), so \( m\angle14 = m\angle6 = 83^\circ \). Then \( \angle14 \) and \( \angle16 \) are vertical angles, so \( m\angle16 = m\angle14 = 83^\circ \). Also, since \( \angle6 = 83^\circ \), \( \angle16 \) is equal to \( \angle6 \) (corresponding and vertical).

Wait, let's re - check:

- \( m\angle3 \): \( \angle2 + \angle3 = 180^\circ\) (linear pair), so \( 97^\circ+\angle3 = 180^\circ\), \( \angle3 = 83^\circ\).
- \( m\angle5 \): \( \angle5+\angle6 = 180^\circ\) (linear pair), \( \angle5 = 180 - 83=97^\circ\).
- \( m\angle10 \): \( \angle2\) and \( \angle10\) are corresponding angles (transversal \( m\), parallel lines \( t\) and \( s\) (since \( \angle2+\angle6 = 180^\circ\))), so \( \angle10=\angle2 = 97^\circ\).
- \( m\angle7 \): \( \angle5\) and \( \angle7\) are alternate interior angles (transversal \( n\), parallel lines \( t\) and \( s\)), so \( \angle7=\angle5 = 97^\circ\).
- \( m\angle9 \): \( \angle3\) and \( \angle9\) are corresponding angles (transversal \( m\), parallel lines \( t\) and \( s\)), so \( \angle9=\angle3 = 83^\circ\).
- \( m\angle16 \): \( \angle6\) and \( \angle14\) are corresponding angles (transversal \( n\), parallel lines \( t\) and \( s\)), so \( \angle14 = \angle6 = 83^\circ\). \( \angle14\) and \( \angle16\) are vertical angles, so \( \angle16=\angle14 = 83^\circ\).

Answer:

\( m\angle3 = 83^\circ \), \( m\angle5 = 97^\circ \), \( m\angle10 = 97^\circ \), \( m\angle7 = 97^\circ \), \( m\angle9 = 83^\circ \), \( m\angle16 = 83^\circ \)