Question

1. if ( mangle 2 = 98^{circ}, mangle 3 = 23^{circ} ) and ( mangle 8 = 70^{circ} ), find each measure.

a. ( mangle 1= )
b. ( mangle 4= )
c. ( mangle 5= )
d. ( mangle 6= )
e. ( mangle 7= )
f. ( mangle 9= )
g. ( mangle 10= )

Explanation:

Step1: Find \(m\angle1\)

Vertical angles are equal. So \(m\angle1 = 180^{\circ}-m\angle2 - m\angle3\). Substitute \(m\angle2 = 98^{\circ}\) and \(m\angle3=23^{\circ}\).
\(m\angle1=180^{\circ}-98^{\circ}-23^{\circ}=59^{\circ}\)

Step2: Find \(m\angle4\)

Since the two lines are parallel (assumed from the figure's symmetry), alternate - interior angles. Let's use triangle angle - sum or parallel - line properties. Another approach: if we consider the larger triangle or parallel - line relations. But if we assume the two segments are parallel (from the figure's structure), and using the fact that \(m\angle4 = m\angle1\) (alternate - interior angles if lines are parallel). So \(m\angle4 = 59^{\circ}\)

Step3: Find \(m\angle5\)

\(m\angle5\) and \(m\angle4\) are supplementary (linear pair). So \(m\angle5=180^{\circ}-m\angle4\). Substitute \(m\angle4 = 59^{\circ}\), then \(m\angle5 = 121^{\circ}\)

Step4: Find \(m\angle6\)

\(m\angle6=m\angle8 = 70^{\circ}\) (alternate - interior angles if lines are parallel)

Step5: Find \(m\angle7\)

In a triangle, sum of angles is \(180^{\circ}\). Consider the triangle with \(\angle2,\angle3,\angle7\) and \(\angle8\) relation. Another way: using the fact that \(m\angle7=180^{\circ}-m\angle2-(m\angle6)\). Substitute \(m\angle2 = 98^{\circ}\) and \(m\angle6 = 70^{\circ}\), \(m\angle7=180^{\circ}-98^{\circ}-70^{\circ}=12^{\circ}\)

Step6: Find \(m\angle9\)

\(m\angle9=m\angle3 = 23^{\circ}\) (alternate - interior angles if lines are parallel)

Step7: Find \(m\angle10\)

\(m\angle10=m\angle5 + m\angle6\) (exterior - angle property of a triangle). Substitute \(m\angle5 = 121^{\circ}\) and \(m\angle6 = 70^{\circ}\), \(m\angle10=191^{\circ}\) (wait, no. Correct approach: \(m\angle10\) is an exterior angle. If we consider the non - parallel line case. Wait, another approach: \(m\angle10=m\angle4 + m\angle6\) (exterior - angle property). Since \(m\angle4 = 59^{\circ}\) and \(m\angle6 = 70^{\circ}\), \(m\angle10=129^{\circ}\)

Answer:

a. \(m\angle1 = 59^{\circ}\)
b. \(m\angle4 = 59^{\circ}\)
c. \(m\angle5 = 121^{\circ}\)
d. \(m\angle6 = 70^{\circ}\)
e. \(m\angle7 = 12^{\circ}\)
f. \(m\angle9 = 23^{\circ}\)
g. \(m\angle10 = 129^{\circ}\)