ABCD is a parallelogram with side lengths AB=15,BC=10 and ∠ADC=75∘. A point P in the interior of the parallelogram is chosen such that ∠APD+∠CPB=180∘. What is the measure (in degrees) of ∠PAD+∠PCB?

105

how

To find the measure of ∠PAD+∠PCB, we need to break down the problem into smaller parts and use some geometrical properties. Here are the steps to follow:

Step 1: Determine the measure of ∠APB.
We know that ∠ADC and ∠APB are opposite angles, so they are equal.
∠APB = ∠ADC = 75°

Step 2: Calculate the measure of ∠PDC.
Since ABCD is a parallelogram, opposite angles are congruent. So, ∠PDC = ∠ABC = 75°.

Step 3: Determine the measure of ∠CDP.
Using the fact that the sum of the angle measures in a triangle is 180°:
∠CDP = 180° - ∠PDC - ∠ADC
= 180° - 75° - 75°
= 30°

Step 4: Find the measure of ∠CPB.
Since ∠APD + ∠CPB = 180°, we can rearrange this equation to find ∠CPB:
∠CPB = 180° - ∠APD

Step 5: Calculate the measure of ∠PAD.
Using the fact that the sum of the angle measures in a triangle is 180°:
∠PAD = 180° - ∠APD - ∠PDA
= 180° - ∠APD - ∠ADC
= 180° - ∠APD - 75°

Step 6: Find the measure of ∠PCB.
Since ∠APD + ∠CPB = 180°, we can rearrange this equation to find ∠APD:
∠APD = 180° - ∠CPB

Step 7: Substitute the values to get the final answer.
Substitute the values obtained in steps 4, 5, and 6 into ∠PAD + ∠PCB:
∠PAD + ∠PCB = (180° - ∠APD) + (180° - ∠CPB)
= 360° - ∠APD - ∠CPB

Now, let's substitute the values we obtained:

∠PAD + ∠PCB = 360° - (180° - ∠CPB) - (180° - ∠APD)
= 360° - (180° - (180° - ∠APD)) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - ∠CPB))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - ∠APD)))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - (180° - ∠CPB))))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - (180° - (180° - ∠ADC)))))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - (180° - (180° - 75°)))))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - (180° - 105°))))) - (180° - ∠CPB)
= 360° - (180° - (180° - (180° - (180° - 75°)))) - (180° - ∠CPB)
= 360° - (180° - (180° - 105°)) - (180° - ∠CPB)
= 360° - (180° - 75°) - (180° - ∠CPB)
= 360° - 105° - (180° - ∠CPB)
= 255° - (180° - ∠CPB)
= 75° + ∠CPB

Therefore, the measure of ∠PAD + ∠PCB is 75° + ∠CPB.