Chapter 12.3, Problem 7E

a)

To determine

To find: The sum of the measures of the interior angles of the quadrilateral.

Answer to Problem 7E

Sum of the measures of the interior angles of the quadrilateral is ${360}^{\circ }$ .

Explanation of Solution

Given information: Here, the polygon is quadrilateral.

Formula used : The sum of the measure of the all interior angles of a convex n-gon is $\left(n-2\right)\times {180}^{\circ }$

Calculation:

Here, polygon is quadrilateral. i.e. $n=4$

Therefore, the sum of the all interior angle of a quadrilateral is

  $\begin{array}{l}\left(n-2\right)\times {180}^{\circ }\\ \Rightarrow \left(4-2\right)\times {180}^{\circ }\\ \Rightarrow 2\times {180}^{\circ }\\ \Rightarrow {360}^{\circ }\end{array}$

Sum of the interior angles of a quadrilateral is ${360}^{\circ }$ .

b)

To determine

To find: The measurement of $m\angle 1$

Answer to Problem 7E

  $m\angle 1={69}^{\circ }$

Explanation of Solution

Given information:Other three angles of the quadrilaterals are ${107}^{\circ }$ , ${120}^{\circ }$ and ${64}^{\circ }$ .

Concept Used: Sum up the all three angles.

Calculation:

  $\begin{array}{l}{107}^{\circ }+{120}^{\circ }+{64}^{\circ }+m\angle 1={360}^{\circ }\\ \Rightarrow m\angle 1={360}^{\circ }-{291}^{\circ }\\ \Rightarrow m\angle 1={69}^{\circ }\end{array}$

Hence, $m\angle 1={69}^{\circ }$

c)

To determine

To subtract: The value of $m\angle 1$ from ${180}^{\circ }$

Answer to Problem 7E

The value is ${111}^{\circ }$

Explanation of Solution

Given information:The value of $m\angle 1={69}^{\circ }$

Concept Used:we know that the a straight angle is ${180}^{\circ }$

Calculation:

  $\begin{array}{l}m\angle 1+m\angle 2={180}^{\circ }\\ \Rightarrow {69}^{\circ }+m\angle 2={180}^{\circ }\\ \Rightarrow m\angle 2={180}^{\circ }-{69}^{\circ }\\ \Rightarrow m\angle 2={111}^{\circ }\end{array}$

Hence, $m\angle 2={111}^{\circ }$