Exercise 8 Page 565

We are on top of Mount Rainier about 2.73 miles above sea level at point B. We want to find the measure of CD. To do so we will begin by showing congruent triangles. By the External Tangent Congruence Theorem, BC is congruent with BD. By the Reflexive Property of Congruence, BA is congruent with BA. Consequently, by the Hypotenuse Leg Theorem, △ ACB is congruent with △ ADB. △ ACB ≅ △ ADB Therefore, because the corresponding parts of congruent triangles are congruent, we can conclude that ∠ CBA ≅ ∠ DBA. Now we will find the measure of ∠ CBD so we can use the Circumscribed Angle Theorem. We just need to find m∠ ABD because m∠ ABD is one-half of m∠ CBD. Since we have the lengths of BA and AD for right triangle △ ADB, to find the angle we will use the sine ratio of m∠ ABD. sin(m∠ ABD)=AD/AB ⇓ sin(m∠ ABD)=4000/4002.73 From here, to find m∠ ABD we will use the inverse sine and calculate the value of it with the help of a calculator. m∠ ABD &=sin^(- 1)(4000/4002.73) &≈ 88 With this information we will find m∠ CBD. 2m∠ ABD=m∠ CBD 2(88)=176 Next, we will find mCD by the Circumscribed Angle Theorem. Notice that since the measure of minor arc is the measure of its central angle, the measure of m∠ CAD is equal to the measure of mCD. Therefore, the measure of CD is 4^(∘).