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Solve for ∠PTS with Isosceles Triangles and Angle Sum Theorem (∠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 using the triangle angle sum theorem (sum of angles in a triangle is 180°). ∠TPQ = ∠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 its base angles are equal: ∠QTR = ∠QRT = (180° - 108°) ÷ 2 = 36°.
3
Step 3: Calculate ∠PRT, which is supplementary to ∠QRT. ∠PRT = 180° - 36° = 144°. Then find ∠TRS, which is vertical to ∠PRT, so ∠TRS = 144°.
4
Step 4: Analyze isosceles triangle △TRS where RT = RS. Calculate its base angles: ∠RTS = ∠RST = (180° - 144°) ÷ 2 = 18°.
5
Step 5: ∠PTS is the same as ∠RTS, so ∠PTS = 18°.
Knowledge Points Involved
1
Isosceles Triangle Properties
In an isosceles triangle, the angles opposite the equal sides are equal. This property is used to calculate unknown base or vertex angles when two sides are known to be congruent.
2
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is always 180°. This theorem is the foundational formula for calculating unknown angles in triangles, used repeatedly in this problem to solve for missing angles.
3
Supplementary Angles
Two angles are supplementary if their sum equals 180°. When two angles form a straight line, they are supplementary, which is used here to find angles outside the interior of a triangle.
4
Vertical Angles
Vertical angles are pairs of opposite angles formed by two intersecting lines, and they are congruent (equal in measure). This is used to relate ∠PRT and ∠TRS in the problem.
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