Exercise 23 Page 567

We are flying in a hot air balloon about 1.2 miles above the ground. We want to find the measure of XZ. To find it we will use the Circumscribed Angle Theorem. To do so we need to find m∠ ZWX. Thus, we will begin by showing congruent triangles. Note that WX and WZ are tangent, so by the Tangent to Circle Theorem WX⊥YZ and WZ⊥XY. Moreover, by the External Tangent Congruence Theorem WZ is congruent with WX. Lastly, by the Reflexive Property of Congruence WY ≅ YW. Consequently, by the Hypotenuse Leg Theorem △ WZY is congruent with △ WXY. △ WZY ≅ △ WXY Therefore, because the corresponding parts of congruent triangles are congruent, we can conclude that ∠ ZWY ≅ ∠ XWY. Now we will find the measure of ∠ ZWX to use the Circumscribed Angle Theorem. We just need to find m∠ YWX, because m∠ YWX is one-half of m∠ ZWX. To find the angle, since we have the lengths of WY and XY for right triangle △ WXY, we will use the sine ratio of m∠ YWX. sin(m∠ YWX)=XY/WY ⇓ sin(m∠ YWX)=4000/4001.2 From here, to find m∠ ZWX we will use the inverse sine and calculate the value of inverse sine with the help of a calculator. m∠ YWX &=sin^(- 1)(4000/4001.2) &≈ 89 With this information, we will find m∠ ZWX. 2m∠ YWX=m∠ ZWX 2(89)=178 Next, we will find mZX by the Circumscribed Angle Theorem. Notice that since the measure of minor arc is the measure of its central angle, the measure of m∠ ZYX is equal to the measure of mZX. Therefore, the measure of ZX is 2^(∘).