Question

1. if m∠2 = 98°, m∠3 = 23° and m∠8 = 70°, find the measure of each missing angle.

a. m∠1 =
b. m∠4 =
c. m∠5 =
d. m∠6 =
e. m∠7 =
f. m∠9 =
g. m∠10 =

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°.

Step2: Find \(m\angle1\)

In the left - hand triangle with angles \(\angle1\), \(\angle2 = 98^{\circ}\), and \(\angle3=23^{\circ}\), we use the angle - sum property. So \(m\angle1=180-(98 + 23)=180 - 121 = 59^{\circ}\).

Step3: Find \(m\angle4\)

\(\angle1\) and \(\angle4\) are vertical angles. Vertical angles are equal. So \(m\angle4=m\angle1 = 59^{\circ}\).

Step4: Find \(m\angle5\)

In the triangle with \(\angle4\), \(\angle5\), and \(\angle8 = 70^{\circ}\), using the angle - sum property, \(m\angle5=180-(59 + 70)=180 - 129 = 51^{\circ}\).

Step5: Find \(m\angle6\)

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

Step6: Find \(m\angle7\)

In the triangle with \(\angle3\), \(\angle6\), and \(\angle7\), using the angle - sum property, \(m\angle7=180-(23 + 110)=180 - 133 = 47^{\circ}\).

Step7: Find \(m\angle9\)

\(\angle7\) and \(\angle9\) are vertical angles. So \(m\angle9=m\angle7 = 47^{\circ}\).

Step8: Find \(m\angle10\)

\(\angle2\) and \(\angle10\) are supplementary (linear pair). So \(m\angle10=180 - 98 = 82^{\circ}\).

Answer:

a. \(m\angle1 = 59^{\circ}\)
b. \(m\angle4 = 59^{\circ}\)
c. \(m\angle5 = 51^{\circ}\)
d. \(m\angle6 = 110^{\circ}\)
e. \(m\angle7 = 47^{\circ}\)
f. \(m\angle9 = 47^{\circ}\)
g. \(m\angle10 = 82^{\circ}\)