AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Solve for ∠PTS with Isosceles Triangles and Angle Sum Theorem | 9th Grade Math
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°, the base angles ∠TPQ and ∠TQP are equal. Calculate them using the triangle angle sum theorem: (180° - 36°) ÷ 2 = 72°. So ∠TPQ = ∠TQP = 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 the base angles ∠QTR and ∠QRT are equal. Calculate them: (180° - 108°) ÷ 2 = 36°. So ∠QTR = ∠QRT = 36°.
3
Step 3: Calculate ∠PTR by adding ∠PTQ and ∠QTR: 36° + 36° = 72°. Then find ∠RTS using the supplementary angle of ∠QRT: ∠TRS = 180° - 36° = 144°.
4
Step 4: Analyze isosceles triangle △RTS where RT = RS. Calculate the base angles ∠RTS and ∠RST: (180° - 144°) ÷ 2 = 18°.
5
Step 5: Calculate ∠PTS by adding ∠PTR and ∠RTS: 72° + 18° = 90°.
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. This property is used to find unknown angles when two sides are known to be equal, or to find unknown sides when two angles are congruent.
2
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is always 180°. This theorem is the foundational rule for calculating unknown interior angles in a triangle when one or two angles are known.
3
Supplementary Angles
Two angles are supplementary if the sum of their measures is 180°. When two angles form a linear pair (share a common side and form a straight line), they are supplementary. This is used to calculate angles adjacent to a straight line.
Loading solution...
