Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle 1 = 99^circ ). find ( mangle 6 ) and ( mangle 8 ).
( mangle 6 = square^circ )
( mangle 8 = square^circ )

Explanation:

Step1: Find \( m\angle6 \)

\(\angle1\) and \(\angle5\) are corresponding angles (since two parallel lines cut by a transversal, corresponding angles are equal), so \( m\angle5 = m\angle1 = 99^\circ \). \(\angle5\) and \(\angle6\) are supplementary (they form a linear pair), so \( m\angle6 = 180^\circ - m\angle5 \).
\[
m\angle6 = 180^\circ - 99^\circ = 81^\circ
\]

Step2: Find \( m\angle8 \)

\(\angle1\) and \(\angle8\) are alternate exterior angles (two parallel lines cut by a transversal, alternate exterior angles are equal), or we can also use the fact that \(\angle6\) and \(\angle8\) are vertical angles (vertical angles are equal). Since \( m\angle6 = 81^\circ \), then \( m\angle8 = m\angle6 = 81^\circ \), or using alternate exterior angles: \( m\angle8 = m\angle1\) (wait, no, alternate exterior angles: \(\angle1\) and \(\angle8\)? Wait, actually \(\angle1\) and \(\angle8\): let's correct. \(\angle1\) and \(\angle5\) are corresponding, \(\angle5\) and \(\angle8\) are vertical? No, \(\angle5\) and \(\angle7\) are vertical, \(\angle6\) and \(\angle8\) are vertical. Wait, \(\angle1\) and \(\angle3\) are vertical, \(\angle3\) and \(\angle7\) are corresponding, so \(\angle1 = \angle7\), and \(\angle7\) and \(\angle8\) are supplementary? Wait, no, let's re - examine. The two parallel lines: the upper line and lower line, transversal. \(\angle1\) and \(\angle5\) are corresponding (both above the parallel lines, same side of transversal), so \( m\angle5 = 99^\circ \). \(\angle5\) and \(\angle6\) are linear pair, so \( m\angle6 = 81^\circ \). \(\angle6\) and \(\angle8\) are vertical angles, so \( m\angle8 = m\angle6 = 81^\circ \). Also, \(\angle1\) and \(\angle2\) are supplementary, \(\angle2\) and \(\angle6\) are corresponding? Wait, maybe easier: \(\angle1\) and \(\angle8\): alternate exterior angles? Wait, \(\angle1\) is top - right, \(\angle8\) is bottom - left. So alternate exterior angles: yes, because the two parallel lines, transversal, so alternate exterior angles are equal? Wait, no, alternate exterior angles: \(\angle1\) and \(\angle8\): let's see, the exterior angles: above the upper line and below the lower line, on opposite sides of transversal. Wait, maybe I made a mistake earlier. Let's use linear pair and vertical angles. \(\angle1\) and \(\angle2\) are supplementary: \( m\angle2 = 180 - 99 = 81^\circ \). \(\angle2\) and \(\angle6\) are corresponding angles (since two parallel lines, transversal), so \( m\angle6 = m\angle2 = 81^\circ \). \(\angle6\) and \(\angle8\) are vertical angles, so \( m\angle8 = m\angle6 = 81^\circ \). Or \(\angle1\) and \(\angle3\) are vertical (\( m\angle3 = 99^\circ \)), \(\angle3\) and \(\angle7\) are corresponding (\( m\angle7 = 99^\circ \)), \(\angle7\) and \(\angle8\) are supplementary (\( m\angle8 = 180 - 99 = 81^\circ \)). So both ways, \( m\angle8 = 81^\circ \).

Answer:

\( m\angle6 = \boldsymbol{81^\circ} \), \( m\angle8 = \boldsymbol{81^\circ} \)