Question

In the circle below, if arc AB = 104 ° and arc CD = 26 °, find the measure of < BPA.

Answer 1

The answer is B 30 degrees

Answer 2

In a circle, if arc AB measures 104° and arc CD measures 26°, the inscribed angle BPA, intercepted by arc AB, is half the arc's measure, resulting in a ∠BPA of 52°. Additionally, exterior ∠BPD is 78°, making the correct answer (a) 78°.

To find the measure of angle BPA in the circle, we can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc.

Let
`\( \angle BPA \)` be the inscribed angle, and let
`\( \text{arc } AB = 104^\circ \).` Since
`\( \angle BPA \)` intercepts arc AB, we have:

`\[ \text{measure of } \angle BPA = \frac{\text{measure of arc } AB}{2} \]`

`\[ \text{measure of } \angle BPA = (104^\circ)/(2) = 52^\circ \]`

So, the measure of
`\( \angle BPA \) is \( 52^\circ \).`

Now, let's look at arc CD. Since C and D are on the other side of the circle,
`\( \angle BPD \)` is an exterior angle to triangle APB. The measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles.

`\[ \text{measure of } \angle BPD = \text{measure of } \angle BPA + \text{measure of arc } CD \]`

`\[ \text{measure of } \angle BPD = 52^\circ + 26^\circ = 78^\circ \]`

So, the measure of
`\( \angle BPD \) is \( 78^\circ \).`

The answer is
`\( \boxed{\text{(a) } 78^\circ} \).`