Question

example 7 if m∠2 = 41°, m∠5 = 94°, and m∠10 = 109°. find each measure. a. m∠1 = d. m∠6 = g. m∠9 = b. m∠3 = e. m∠7 = c. m∠4 = f. m∠8 = ©gina wilson (all things algebra®), llc, 2014

Explanation:

Step1: Use the fact that the sum of angles in a triangle is 180°

In the triangle with angles 1, 2, and 3, we know that \(m\angle1 + m\angle2+m\angle3=180^{\circ}\). Also, we can use linear - pair and angle - sum relationships.

Step2: Find \(m\angle1\)

Since \(\angle1\) and \(\angle5\) are supplementary (a linear pair), \(m\angle1 + m\angle5=180^{\circ}\). Given \(m\angle5 = 94^{\circ}\), then \(m\angle1=180^{\circ}-94^{\circ}=86^{\circ}\).

Step3: Find \(m\angle3\)

In the triangle with \(\angle2\), \(\angle3\) and the third angle, we know that the sum of angles in a triangle is \(180^{\circ}\). First, note that \(\angle1\) and \(\angle5\) are linear - pair. In the inner - triangle with \(\angle2 = 41^{\circ}\), and since \(\angle1 = 86^{\circ}\), then \(m\angle3=180^{\circ}-(m\angle1 + m\angle2)=180^{\circ}-(86^{\circ}+41^{\circ}) = 53^{\circ}\).

Step4: Find \(m\angle4\)

\(\angle4\) and \(\angle10\) are supplementary (a linear pair). So \(m\angle4=180^{\circ}-m\angle10\). Given \(m\angle10 = 109^{\circ}\), then \(m\angle4 = 71^{\circ}\).

Step5: Find \(m\angle6\)

In the triangle with \(\angle4\), \(\angle6\) and another angle, we know that the sum of angles in a triangle is \(180^{\circ}\). Also, using angle - relationships, we find that \(m\angle6=180^{\circ}-(m\angle4 + m\angle5)\). Substituting \(m\angle4 = 71^{\circ}\) and \(m\angle5 = 94^{\circ}\), we get \(m\angle6=180^{\circ}-(71^{\circ}+94^{\circ}) = 15^{\circ}\).

Step6: Find \(m\angle7\)

\(\angle7\) and \(\angle2\) are vertical angles. So \(m\angle7=m\angle2 = 41^{\circ}\).

Step7: Find \(m\angle8\)

In the triangle with \(\angle7\), \(\angle8\) and \(\angle10\), we know that \(m\angle7 + m\angle8+m\angle10=180^{\circ}\). Substituting \(m\angle7 = 41^{\circ}\) and \(m\angle10 = 109^{\circ}\), we get \(m\angle8=180^{\circ}-(41^{\circ}+109^{\circ}) = 30^{\circ}\).

Step8: Find \(m\angle9\)

\(\angle9\) and \(\angle3\) are vertical angles. So \(m\angle9=m\angle3 = 53^{\circ}\).

Answer:

a. \(m\angle1 = 86^{\circ}\)
b. \(m\angle3 = 53^{\circ}\)
c. \(m\angle4 = 71^{\circ}\)
d. \(m\angle6 = 15^{\circ}\)
e. \(m\angle7 = 41^{\circ}\)
f. \(m\angle8 = 30^{\circ}\)
g. \(m\angle9 = 53^{\circ}\)