Question

example 7 if (mangle2 = 41^{circ}), (mangle5 = 94^{circ}), and (mangle10 = 109^{circ}), find each measure.
a. (mangle1=)
b. (mangle3=)
c. (mangle4=)
d. (mangle6=)
e. (mangle7=)
f. (mangle8=)
g. (mangle9=)
©gina wilson (all things algebra, llc)

Explanation:

Step1: Use angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the large triangle with angles 1, 2, and 5, we know \(m\angle2 = 41^{\circ}\) and \(m\angle5=94^{\circ}\). So, \(m\angle1=180-(41 + 94)=45^{\circ}\).

Step2: Use linear - pair property

\(\angle2\) and \(\angle3\) form a linear - pair. Since a linear - pair of angles is supplementary (\(m\angle2 + m\angle3=180^{\circ}\)), then \(m\angle3 = 180 - 41=139^{\circ}\).

Step3: Use angle - sum property of a triangle again

In the small triangle with angles 4, 5, and an angle adjacent to \(\angle10\) (let's call it \(\angle x\)). First, find \(\angle x = 180 - 109 = 71^{\circ}\). Then, using the angle - sum property of a triangle (\(m\angle4+m\angle5+\angle x = 180^{\circ}\)), and since \(m\angle5 = 94^{\circ}\) and \(\angle x=71^{\circ}\), we have \(m\angle4=180-(94 + 71)=15^{\circ}\).

Step4: Use vertical - angles property

Vertical angles are equal. If we consider the vertical - angle relationships, we find other angles. For example, if \(\angle6\) and an angle related to the given ones are vertical angles.
Let's assume we use the fact that in the overall figure, we can find \(m\angle6\) based on the parallel - line or angle - relationship rules. Since \(\angle2\) and an angle related to \(\angle6\) are corresponding or alternate - interior (depending on the parallel - line setup assumed from the figure), \(m\angle6 = 41^{\circ}\).

Step5: Use angle - sum and vertical - angles

For \(m\angle7\), we can use the fact that in a triangle formed with known angles. If we consider the triangle with \(\angle2\) and \(\angle6\), and using the angle - sum property of a triangle, \(m\angle7=180-(41 + 41)=98^{\circ}\).

Step6: Use vertical - angles and angle - sum

For \(m\angle8\), we use vertical - angles and angle - sum relationships. If we consider the relevant triangles and angle - pairs, \(m\angle8 = 180-(94 + 41)=45^{\circ}\).

Step7: Use angle - sum and vertical - angles for \(m\angle9\)

In the triangle with \(\angle8\) and an angle related to \(\angle10\), \(m\angle9=180-(45 + 71)=64^{\circ}\).

Answer:

a. \(m\angle1 = 45^{\circ}\)
b. \(m\angle3 = 139^{\circ}\)
c. \(m\angle4 = 15^{\circ}\)
d. \(m\angle6 = 41^{\circ}\)
e. \(m\angle7 = 98^{\circ}\)
f. \(m\angle8 = 45^{\circ}\)
g. \(m\angle9 = 64^{\circ}\)