ABCD is a quadrilateral in which measures of angle D and angle C are 60 degree, and 100 degree respectively. If the internal bisectors of angle A and angle B meet at P, then measure of angle APB isa)800b)900c)1000d)1100Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Answer

Suppose the point p lies on the DC. now A+B+C+D=360 we know the Angles C and D so A+B=200 ....In triangle APB angle PAB = A/2 and PBC = B/2 therefore angle PAB + PBA + P =180 this imply
A/2 + B/2 + P = 180
(A + B)/2 + P = 180
P = 180 -100
P=80

Answer

Solution:To find the measure of angle APB, we need to first find the measures of angle A and angle B.Given:- Measure of angle D = 60 degrees- Measure of angle C = 100 degreesWe know that the sum of the angles in a quadrilateral is 360 degrees. So, we can find the measure of angle A and angle B using the following equation:Measure of angle A + Measure of angle B + Measure of angle C + Measure of angle D = 360 degreesSubstituting the given values, we have:Measure of angle A + Measure of angle B + 100 degrees + 60 degrees = 360 degreesSimplifying the equation, we get:Measure of angle A + Measure of angle B = 200 degreesNow, let's consider the internal bisectors of angle A and angle B, which meet at point P. The internal bisector of an angle divides the angle into two equal parts.Since the internal bisector of angle A meets at point P, it divides angle A into two equal parts, each measuring 30 degrees.Similarly, the internal bisector of angle B meets at point P, dividing angle B into two equal parts, each measuring 50 degrees.Now, let's find the measure of angle APB:Measure of angle APB = Measure of angle A + Measure of angle B = 30 degrees + 50 degrees = 80 degreesTherefore, the measure of angle APB is 80 degrees.Hence, the correct answer is A: 80 degrees.

Answer

Given information:
- ABCD is a quadrilateral
- The measures of angle D and angle C are 60 degrees and 100 degrees respectively

To find:
- The measure of angle APB

Approach:
1. Find the measures of angles A and B.
2. Use the angle bisector theorem to find the ratio of the lengths of the sides of triangle APB.
3. Use the Law of Sines to find the measure of angle APB.

Solution:
1. Finding the measures of angles A and B:
- In quadrilateral ABCD, the sum of the measures of the interior angles is 360 degrees.
- Given that angle D = 60 degrees and angle C = 100 degrees, we can find the measures of angles A and B.
- Let angle A = x and angle B = y.
- x + 60 + y + 100 = 360 (sum of interior angles of quadrilateral ABCD)
- x + y = 200 (equation 1)

2. Using the angle bisector theorem:
- The angle bisector theorem states that in a triangle, the ratio of the lengths of the sides opposite two congruent angles is equal.
- Let the length of side AD be a and the length of side CD be b.
- By the angle bisector theorem, we have:
a/b = AP/PB (equation 2)

3. Using the Law of Sines:
- In triangle APB, we can use the Law of Sines to find the measure of angle APB.
- The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
- Let the length of side AB be c.
- By the Law of Sines, we have:
c/sin(APB) = a/sin(B) = b/sin(A) (equation 3)

4. Solving the equations:
- From equation 1, we have: x + y = 200
- From equation 2, we have: a/b = AP/PB
- From equation 3, we have: c/sin(APB) = a/sin(B) = b/sin(A)
- We can rewrite equation 2 as: a = (AP/PB) * b
- Substituting this value of a in equation 3, we get: c/sin(APB) = ((AP/PB) * b)/sin(B) = b/sin(A)
- Simplifying, we get: sin(APB) = (c * sin(B))/(AP/PB)
- Using the fact that sin(x) = sin(180 - x), we can write: sin(APB) = (c * sin(180 - APB))/(AP/PB)
- Cross-multiplying, we get: (AP/PB) * sin(APB) = c * sin(180 - APB)

5. Simplifying and solving the equation:
- We know that sin(180 - APB) = sin(APB)
- Therefore, we have: (AP/PB) * sin(APB) = c * sin(APB)
- Canceling the common factor of sin(APB), we get: AP/PB = c

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