The labels blue means : lengths PQ = QR The label green means: lengths QS = QT. The given angle PQS = 24 (deg)

madeeha1d8 2022-09-25 Answered
Triangles : isoceles and angles inside with other triangle
We have one main isoceles and another one inside of it. I have attached here a diagram, and we wish to find the angle in red:

The labels blue means : lengths P Q = Q R The label green means : lengths Q S = Q T. The given angle P Q S = 24 (deg)
We wish to find angle in red, angle RST.
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Answers (2)

Lilliana Mason
Answered 2022-09-26 Author has 11 answers
Explanation:
Let Q R P ^ = R P Q ^ = x, so R Q P ^ = 180 2 x, so R Q S ^ = 156 2 x, so the sum of the angles Q S T ^ and Q T S ^ is 24 + 2 x, so they are both x + 12, so R T S ^ = 168 x, so R S T ^ = 12.
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mangicele4s
Answered 2022-09-27 Author has 1 answers
Step 1

Since PRQ and STQ are isosceles and share vertex Q, let the circle with center Q and radius QT intersect PQ at U and PR at V, and join TU. Thus, since Q U Q P = Q T Q R then T U R P and arcs   V T = S U so that V S T = S T U.
Step 2
But S Q U = 2 S T U.
Therefore S Q U = 2 V S T that is S Q P = 2 R S T.
Thus under the given conditions, the angle we are looking for is always half the given angle.
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