In △PQR; PQ = PR. A is a point in PQ and B is a point in PR, so that QR=RA=AB=BP.
i) show that angle P:angle R=1:3
ii) find the value of angle Q.

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Solution

The exterior angle of a triangle is the sum of the other two interior angles (because the three angles sum to two right angles, and the interior + exterior angles also sum to two right angles).

Because of the equal sides that are given triangles, PAB, BAR, ARQ are isosceles.

In triangle PAB, angle P = angle PAB,.therefore angle ABR = 2 angle P.

In triangle BAR, angle ARB = angle ABR = 2P, therefore angle BAR = 2 right angles - 4P.

The three angles at point A add up to two right angles, therfore angle QAR = 3P

Triangle QAR is isosceles, therefore angle Q = 3P
Triangle PQR is isosceles, therefore angle R = 3P

2) The angles of PQR add up to two right angles, so P + 3P + 3P = 180
P = 180/7 degrees
Q = 3x180/7 = 540/7 = 77.14 degrees.

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In △PQR; PQ = PR. A is a point in PQ and B is a point in PR, so that QR=RA=AB=BP. i) show that angle P:angle R=1:3 ii) find the value of angle Q. ​

In △PQR; PQ = PR. A is a point in PQ and B is a point in PR, so that QR=RA=AB=BP.
i) show that angle P:angle R=1:3
ii) find the value of angle Q.