Triangles : isoceles and angles inside with other triangleWe 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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Triangles : isoceles and angles inside with other triangleWe 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.

Lilliana Mason · Accepted Answer

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.

mangicele4s · Accepted Answer

Step 1Since 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 2But   ∠  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.

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

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