In a square ABCD, point P is chosen inside ABCD and point Q outside ABCD such that APB and BQC are congruent isosceles triangles with angle APB = angle BQC = 80 degrees. T is a point where BC and PQ meet. Find the size of angle BTQ
since triangles APB and BQC are isosceles with the same vertex angles, sides PB and QB are congruent.
That means triangle BPQ is isosceles, making T the midpoint of PQ.
Angles BTP and BTQ are right angles.
To find the measure of angle BTQ, we can use the fact that the sum of the angles in a triangle is 180 degrees.
Since triangles APB and BQC are congruent isosceles triangles with angle APB = angle BQC = 80 degrees, we know that angle PAB = angle PBA and angle QCB = angle QBC.
Let's call the measure of angle PAB (or PBA) as x. Then, the measure of angle QBC (or QCB) would also be x.
Since angles PAB, PBA, QBC, and QCB are all equal, we can find the measure of angle BAC, which is part of triangle BAC.
The measure of angle BAC can be calculated as follows:
angle BAC = 180 - angle PAB - angle PBA
= 180 - x - x
= 180 - 2x.
Now, let's consider triangle PAQ. The measures of angles BAC, APB, and QBC are all part of this triangle.
Therefore, the measure of angle PAQ can be calculated as follows:
angle PAQ = 180 - angle BAC - angle APB - angle QBC
= 180 - (180 - 2x) - 80 - x
= 2x - x - 80
= x - 80.
Since angle PAQ is a straight line, its measure is 180 degrees. Therefore, we can write the following equation:
x - 80 = 180.
Solving this equation, we find that x = 260.
Now, we can find the measure of angle BTQ in triangle BTQ, by subtracting the measures of angle BQT (which is 80 degrees) and angle BQT from 180 degrees, since the sum of the angles in a triangle is 180 degrees.
angle BTQ = 180 - angle BQT - angle BQT
= 180 - 80 - 80
= 180 - 160
= 20 degrees.
Therefore, the size of angle BTQ is 20 degrees.