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?
To find the measure of ∠PAD+∠PCB, we need to use some properties of parallelograms and angles.
Step 1: Draw the parallelogram ABCD and point P inside the parallelogram.
Step 2: Since ABCD is a parallelogram, opposite angles are congruent. So, we have ∠ADC = ∠BCD.
Step 3: We are given that ∠ADC = 75°. Therefore, ∠BCD = 75° as well.
Step 4: Now, let's use the fact that the sum of angles in a triangle is 180°.
In triangle APD, we have ∠APD + ∠PAD + ∠DAP = 180°. (Equation 1)
Similarly, in triangle CPB, we have ∠CPB + ∠PCB + ∠BCP = 180°. (Equation 2)
Step 5: From the given information, we know that ∠APD + ∠CPB = 180°. We can rewrite this equation as ∠APD = 180° - ∠CPB.
Step 6: Substituting this value into Equation 1, we get (180° - ∠CPB) + ∠PAD + ∠DAP = 180°.
Step 7: Since ∠DAP equals ∠BCD (as they are opposite angles), we can rewrite Equation 6 as (180° - ∠CPB) + ∠PAD + ∠BCD = 180°.
Step 8: Now, we can substitute the given value ∠BCD = 75° into Equation 7 to get (180° - ∠CPB) + ∠PAD + 75° = 180°.
Step 9: Simplifying Equation 8, we have 180° - ∠CPB + ∠PAD + 75° = 180°.
Step 10: Rearranging terms, we get ∠PAD + ∠CPB = 180° - 75° - 75°.
Step 11: Evaluating the right side of Equation 10, we find ∠PAD + ∠CPB = 180° - 150°.
Step 12: Calculating further, we have ∠PAD + ∠CPB = 30°.
Therefore, the measure of ∠PAD + ∠PCB is 30°.