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Solve for ∠PTS: Isosceles Triangles 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? Options: A 72°, B 80°, C 90°, D 100°, E 108°
Correct Answer
C 90°
Detailed Solution Steps
1
Step 1: Analyze isosceles triangle △PTQ. Since PT = QT and ∠PTQ = 36°, we know the base angles ∠TPQ and ∠TQP are equal. Using the triangle angle sum theorem (sum of angles in a triangle is 180°), calculate ∠TQP = (180° - 36°) ÷ 2 = 72°.
2
Step 2: Identify ∠TQR as the supplementary angle of ∠TQP. ∠TQR = 180° - 72° = 108°. Now analyze isosceles triangle △TQR where QT = QR, so base angles ∠QTR and ∠QRT are equal. Calculate ∠QRT = (180° - 108°) ÷ 2 = 36°.
3
Step 3: Identify ∠TRS as the supplementary angle of ∠QRT. ∠TRS = 180° - 36° = 144°. Analyze isosceles triangle △TRS where RT = RS, so base angles ∠RTS and ∠RST are equal. Calculate ∠RTS = (180° - 144°) ÷ 2 = 18°.
4
Step 4: Calculate ∠PTS by adding ∠PTQ, ∠QTR and ∠RTS. ∠PTS = 36° + 36° + 18° = 90°.
Knowledge Points Involved
1
Isosceles Triangle Properties
An isosceles triangle has two equal sides, and the angles opposite those equal sides (base angles) are congruent. This property is used to find unknown angles when two sides are known to be equal.
2
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is always 180°. This theorem is the core formula for calculating unknown angles in a triangle when one or two angles are known.
3
Supplementary Angles
Two angles are supplementary if their sum is 180°. When two angles form a straight line, they are supplementary, which is used to find exterior angles adjacent to triangle interior angles.
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