Question

In the given figure, angleBPC = 19degree. arc AB = arc BC = arc CD
Then, the measure of angleAPD is ?​

Answer 1

Answer:

arc (AB) = arc (CD)

angleAPB=angleAPB +angle BPC+angleCPD

Answer 2

The measure of angle APD is approximately 176.83 degrees.

Arc AB = arc BC = arc CD implies that angles ABD and BCD are congruent, and each is equal to one-third of the central angle subtended by arc ABCD.

Angle BPC = 19 degrees is an inscribed angle that subtends arc BC.

Using these observations, we can construct the following diagram:

       C

      / \

     /   \

    /     \

   /       \

  /         \

 /           \

A-----B-----D

Let x be the measure of angle BCD (and ABD). Then, the central angle subtended by arc ABCD is 3x, and the measure of arc BC is also 3x (since arc AB = arc BC = arc CD). Therefore, the measure of angle BPC (which subtends arc BC) is also 3x.

Since angle BPC = 19 degrees, we have:

3x = 19

x = 6.33 degrees (rounded to two decimal places)

Now, consider the triangle APD:

Angle PAD is an inscribed angle that subtends arc AD.

Arc AD is equal to arcs AB, BC, and CD combined, which is 3 times arc BC (i.e., 3x).

Angle APD is an exterior angle of triangle PAD.

Using these facts, we can apply the exterior angle theorem to triangle PAD:

angle APD = angle PAD + angle PDA

Since angle PAD subtends arc AD, its measure is twice the measure of arc BC, or:

angle PAD = 2x = 12.67 degrees (rounded to two decimal places)

To find angle PDA, we can use the fact that the sum of angles in a triangle is 180 degrees:

angle PDA = 180 - angle PAD - angle A

Angle A is half of angle BCD (since arc AB = arc BC), so:

angle A = x/2 = 3.17 degrees (rounded to two decimal places)

Therefore:

angle PDA = 180 - 12.67 - 3.17 = 164.16 degrees (rounded to two decimal places)

Finally, we can calculate angle APD:

angle APD = angle PAD + angle PDA = 12.67 + 164.16 = 176.83 degrees (rounded to two decimal places)

Therefore, the measure of angle APD is approximately 176.83 degrees.

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