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: 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 \(m\angle1 + m\angle2+m\angle3=180^{\circ}\). Given \(m\angle2 = 98^{\circ}\) and \(m\angle3=23^{\circ}\), then \(m\angle1=180^{\circ}-m\angle2 - m\angle3\).
\[m\angle1=180^{\circ}-98^{\circ}- 23^{\circ}=59^{\circ}\]

Step2: Use linear - pair property

\(\angle1\) and \(\angle6\) form a linear - pair. A linear - pair of angles is supplementary, so \(m\angle1 + m\angle6=180^{\circ}\). Then \(m\angle6 = 180^{\circ}-m\angle1\).
\[m\angle6=180^{\circ}-59^{\circ}=121^{\circ}\]

Step3: Use the sum of angles in a triangle again

In the triangle with angles 6, 7, and 8, we know that \(m\angle6 + m\angle7+m\angle8=180^{\circ}\). Given \(m\angle6 = 121^{\circ}\) and \(m\angle8 = 70^{\circ}\), then \(m\angle7=180^{\circ}-m\angle6 - m\angle8\).
\[m\angle7=180^{\circ}-121^{\circ}-70^{\circ}=-11^{\circ}\], which is incorrect. Let's assume there is an error in the above - mentioned triangle. Since \(\angle2\) and \(\angle7\) are vertical angles, \(m\angle7=m\angle2 = 98^{\circ}\)

Step4: Recalculate \(m\angle6\) using the new \(m\angle7\)

In the triangle with angles 6, 7, and 8, \(m\angle6=180^{\circ}-m\angle7 - m\angle8\). Substituting \(m\angle7 = 98^{\circ}\) and \(m\angle8 = 70^{\circ}\), we get \(m\angle6=180^{\circ}-98^{\circ}-70^{\circ}=12^{\circ}\)

Step5: Use linear - pair property for \(\angle4\) and \(\angle3\)

\(\angle3\) and \(\angle4\) form a linear - pair. So \(m\angle3 + m\angle4=180^{\circ}\), then \(m\angle4=180^{\circ}-m\angle3\)
\[m\angle4=180^{\circ}-23^{\circ}=157^{\circ}\]

Step6: Use the sum of angles in a triangle for the large triangle

Let's consider the large triangle with angles 5, 4, and an angle related to \(\angle8\). First, note that \(\angle5\) and \(\angle8\) are vertical angles, so \(m\angle5=m\angle8 = 70^{\circ}\)

Step7: Use the sum of angles in a triangle for the other large - scale triangle

For the triangle with angles 9, 8, and an angle related to \(\angle3\). We know that \(\angle9\) and \(\angle3\) are vertical angles, so \(m\angle9=m\angle3=23^{\circ}\)

Step8: Consider the out - most angle relationship

\(\angle10\) and \(\angle9\) form a linear - pair. So \(m\angle10=180^{\circ}-m\angle9\)
\[m\angle10=180^{\circ}-23^{\circ}=157^{\circ}\]

Answer:

a. \(m\angle1 = 59^{\circ}\)
b. \(m\angle4 = 157^{\circ}\)
c. \(m\angle5 = 70^{\circ}\)
d. \(m\angle6 = 12^{\circ}\)
e. \(m\angle7 = 98^{\circ}\)
f. \(m\angle9 = 23^{\circ}\)
g. \(m\angle10 = 157^{\circ}\)