Question

In circle A, the measure of ∠BAD is 148°. Circle A with angle BAD measuring 148 degrees, points B, C, and D lie on circle A. If ∠BC is 105°, what is ∠CD?

Answer 1

To find ∠CD in circle A, we use ∠BAD's measure being half the sum of the arcs it intercepts. Solving, we find that ∠CD measures 95.5°.

Step-by-step explanation:

To find the measure of ∠CD in circle A, given that ∠BAD is 148° and ∠BC is 105°, we first need to understand the relationships between the angles and arcs in a circle. Since ∠BAD is an angle formed by two chords BA and BD, its measure is half the sum of the measures of the arcs it intercepts, which are arc BC and arc CD.

Therefore, we can write the equation as:

∠BAD = ½(arc BC + arc CD)

Substituting the given values yields:

148° = ½(105° + arc CD)

After multiplying both sides by 2 to get rid of the fraction, we have:

296° = 105° + arc CD

And by subtracting 105° from both sides, we obtain:

arc CD = 296° - 105°

arc CD = 191°

Finally, the measure of ∠CD is half of the measure of its intercepted arc, thus:

∠CD = ½(arc CD)

∠CD = ½(191°)

∠CD = 95.5°

So, the measure of ∠CD is 95.5°