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Solve for ∠PTS with Isosceles Triangle Properties 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
18°
Detailed Solution Steps
1
Step 1: Analyze isosceles triangle △PTQ. Since PT = QT and ∠PTQ = 36°, we calculate the base angles: ∠TPQ = ∠TQP = (180° - 36°) ÷ 2 = 72°.
2
Step 2: Identify ∠TQR as the supplementary angle of ∠TQP. ∠TQR = 180° - 72° = 108°. Now look at isosceles △TQR where QT = QR, so its base angles are ∠QTR = ∠QRT = (180° - 108°) ÷ 2 = 36°.
3
Step 3: Calculate ∠PRT, which is supplementary to ∠QRT. ∠PRT = 180° - 36° = 144°. For isosceles △RTS with RT = RS, its base angles are ∠RTS = ∠RST = (180° - 144°) ÷ 2 = 18°.
4
Step 4: ∠PTS is equal to ∠RTS, so ∠PTS = 18°.
Knowledge Points Involved
1
Isosceles Triangle Properties
In an isosceles triangle, the two sides of equal length are called legs, and the angles opposite these legs (base angles) are congruent. The sum of all interior angles of a triangle is 180°, so we can calculate unknown angles using this property when we know one angle or the equal side relationship.
2
Supplementary Angles
Two angles are supplementary if their sum is 180°. When two angles form a linear pair (share a common side and lie on a straight line), they are supplementary. This is used to calculate angles adjacent to a triangle's exterior angle on a straight line.
3
Triangle Angle Sum Theorem
The total measure of the three interior angles of any triangle is always 180°. This is a fundamental theorem used to derive unknown angles in any triangle, especially combined with properties of special triangles like isosceles triangles.
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