Chapter 9.6, Problem 24WE

To determine

To calculate: Measure of the four non-overlapping arcs.

Answer to Problem 24WE

The four arcs are 90, 86, 84 and 100.

Explanation of Solution

Given information: 

Quadrilateral angle${80}^{\circ },{90}^{\circ },{94}^{\circ }and{96}^{\circ }$

Formula used: 

The measure of an angle formed by two tangent is equal to 1/2 the difference of the two arc

Calculation:

Let the angles be${80}^{\circ },{90}^{\circ },{94}^{\circ }and{96}^{\circ }$, and let the points of tangency be x (between A and B), y, z, w in order.

The measure of an angle formed by two tangents is equal to 1/2 the difference of the two arcs.

  $\begin{array}{l}A=\frac{1}{2}(arcXYZ-arcXW)\\ B=\frac{1}{2}(arcYZX-arcYX)\\ C=\frac{1}{2}(arcZWY-arcZY)\\ D=\frac{1}{2}(arcWXZ-arcWZ)\end{array}$

Let a=arc(XY), b=arc(YZ), c=arc(ZW) and d=arc(WX)

So,

  $\begin{array}{l}80=\frac{1}{2}(a+b+c-d)\\ 90=\frac{1}{2}(a+b+c-a)\\ 94=\frac{1}{2}(a+b+c-b)\\ 96=\frac{1}{2}(a+b+c-c)\end{array}$

As the a =90

So,

  $\begin{array}{l}70=b+c-d\mathrm{....}(1)\\ 270=b+c+d\mathrm{...}(2)\\ 98=-b+c+d\mathrm{...}(3)\\ 102=b-c+d\mathrm{...}(4)\\ Adding(1)and(2)\\ 340=2b+2c\mathrm{...}(5)\\ Adding(3)and(4)\\ 200=2d\\ d=100\\ Adding(2)and(3)\\ 368=2c+2d\\ 2c=168\\ c=84\\ \text{substituting in (5)}\\ \text{b}=\text{86}\end{array}$

Therefore, the Quadrilateral angle${80}^{\circ },{90}^{\circ },{94}^{\circ }and{96}^{\circ }$.