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Solve for ∠PTS in a composite isosceles triangle diagram with PT=QT=QR, RT=RS and ∠PTQ=36°
Mathematics
Grade 9 (Junior High School)
Question Content
In the diagram shown, PT = QT = QR. Also, RT = RS and ∠PTQ = 36°. What is ∠PTS?
Correct Answer
C 90°
Detailed Solution Steps
1
Step 1: Calculate ∠TPQ and ∠TQP in isosceles △PTQ. Since PT=QT and ∠PTQ=36°, the base angles are equal. ∠TPQ = ∠TQP = (180° - 36°) ÷ 2 = 72°.
2
Step 2: Identify ∠TQR as the exterior angle of △PTQ. ∠TQR = 180° - ∠TQP = 180° - 72° = 108°. Then, in isosceles △TQR where QT=QR, calculate the base angles: ∠QTR = ∠QRT = (180° - 108°) ÷ 2 = 36°.
3
Step 3: Calculate ∠PRT, which is the sum of ∠TPQ and ∠PTQ (exterior angle of △PTQ) or ∠QRT + ∠TQP. ∠PRT = 72° + 36° = 108°. Then find ∠RTS in isosceles △RTS where RT=RS: ∠RTS = ∠RST = (180° - 108°) ÷ 2 = 36°.
4
Step 4: Sum the angles to get ∠PTS: ∠PTS = ∠PTQ + ∠QTR + ∠RTS = 36° + 36° + 18° = 90°.
Knowledge Points Involved
1
Isosceles Triangle Properties
In an isosceles triangle, the two sides of equal length have corresponding equal base angles. The sum of interior angles of a triangle is 180°, so the base angles can be calculated by (180° - vertex angle) ÷ 2. This applies to △PTQ, △TQR, and △RTS in the problem.
2
Exterior Angle Theorem of Triangles
The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. It can also be calculated as 180° minus the adjacent interior angle. This is used to find ∠TQR and ∠PRT efficiently without calculating intermediate angles.
3
Sum of Interior Angles of a Triangle
The total measure of the three interior angles of any triangle is always 180°. This fundamental rule is used to solve for unknown angles in every triangle in the problem.
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