Question

1. if ( mangle2 = 98^{circ}), ( mangle3 = 23^{circ}) and ( mangle8 = 70^{circ}), find the measure of each missing angle.

a. ( mangle1=)
b. ( mangle4=)
c. ( mangle5=)
d. ( mangle6=)
e. ( mangle7=)
f. ( mangle9=)
g. ( mangle10=)

Explanation:

Step1: Find \(m\angle1\)

\(\angle1\) and \(\angle2\) are supplementary (linear - pair). So \(m\angle1=180^{\circ}-m\angle2\).
Since \(m\angle2 = 98^{\circ}\), then \(m\angle1=180^{\circ}-98^{\circ}=82^{\circ}\).

Step2: Find \(m\angle4\)

In a triangle, the sum of interior angles is \(180^{\circ}\). Consider the triangle with angles \(\angle2\), \(\angle3\), and \(\angle4\).
So \(m\angle4=180^{\circ}-m\angle2 - m\angle3\).
Substitute \(m\angle2 = 98^{\circ}\) and \(m\angle3 = 23^{\circ}\), then \(m\angle4=180^{\circ}-98^{\circ}-23^{\circ}=59^{\circ}\).

Step3: Find \(m\angle5\)

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

Step4: Find \(m\angle6\)

\(\angle6\) and \(\angle8\) are supplementary (linear - pair). So \(m\angle6=180^{\circ}-m\angle8\).
Since \(m\angle8 = 70^{\circ}\), then \(m\angle6=180^{\circ}-70^{\circ}=110^{\circ}\).

Step5: Find \(m\angle7\)

In a triangle, the sum of interior angles is \(180^{\circ}\). Consider the triangle with angles \(\angle6\), \(\angle7\), and \(\angle8\).
Let's re - check using another approach. We know that for the triangle with angles related to \(\angle2\), \(\angle3\), \(\angle4\) and the other triangle. But using the exterior - angle property or the sum of angles in a more complex figure (assuming the figure is composed of two triangles and some linear - pairs).
Another way: Consider the fact that \(\angle7\) can be found from the angle - sum of a triangle. Let's assume a triangle where we know two angles. But more simply, if we consider the relationship between the angles.
We know that \(\angle7\) and the angle adjacent to it (from the first part) and using the sum of angles around a point. But a better way:
We know that for the triangle with \(\angle6\), \(\angle7\) and the angle formed by the intersection. Using the angle - sum formula \(m\angle7=180^{\circ}-m\angle6 - m\angle3\) (assuming a non - standard triangle, but if we consider the overall figure as two triangles sharing a common side).
\(m\angle7=180^{\circ}-110^{\circ}-23^{\circ}=47^{\circ}\).

Step6: Find \(m\angle9\)

\(\angle9\) and \(\angle4\) are alternate interior angles (assuming parallel lines, if the figure is composed of parallel - line - cut - by - transversals). Or using the angle - sum of a larger figure.
Another approach: If we consider the two - triangle figure. The sum of angles around the intersection point.
\(m\angle9=m\angle7 = 47^{\circ}\) (alternate interior angles, if the lines are parallel).

Step7: Find \(m\angle10\)

\(\angle9\) and \(\angle10\) are supplementary (linear - pair). So \(m\angle10=180^{\circ}-m\angle9\).
Since \(m\angle9 = 47^{\circ}\), then \(m\angle10=180^{\circ}-47^{\circ}=133^{\circ}\).

Answer:

a. \(m\angle1 = 82^{\circ}\)
b. \(m\angle4 = 59^{\circ}\)
c. \(m\angle5 = 121^{\circ}\)
d. \(m\angle6 = 110^{\circ}\)
e. \(m\angle7 = 47^{\circ}\)
f. \(m\angle9 = 47^{\circ}\)
g. \(m\angle10 = 133^{\circ}\)