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Difficulty: MediumLines and Angles

Lines L1L_1 and L2L_2 are parallel. Points PP and QQ lie on line L1L_1, and points RR and SS lie on line L2L_2. Segment PSPS and segment QRQR intersect at point TT. If the measure of angle TPQTPQ is 38∘38^\circ and the measure of angle PTQPTQ is 105∘105^\circ, what is the measure, in degrees, of angle TRSTRS?

Answer: 37 degrees

Answer

37
To find the measure of angle TRSTRS, we first find the measure of angle TQPTQP using the property that the sum of interior angles in triangle TPQTPQ is 180∘180^\circ. This gives ∠TQP=180∘−38∘−105∘=37∘\angle TQP = 180^\circ - 38^\circ - 105^\circ = 37^\circ. Since lines L1L_1 and L2L_2 are parallel, the transversal line QRQR creates alternate interior angles that are equal in measure. Therefore, angle TRSTRS is equal to angle TQPTQP, which is 37∘37^\circ.

Step-by-Step Solution

1
Calculate the measure of angle TQPTQP in triangle TPQTPQ.
∠TQP=37∘\angle TQP = 37^\circ
The sum of the angles in a triangle is 180∘180^\circ. Therefore, ∠TQP=180∘−∠TPQ−∠PTQ=180∘−38∘−105∘=37∘\angle TQP = 180^\circ - \angle TPQ - \angle PTQ = 180^\circ - 38^\circ - 105^\circ = 37^\circ.
2
Relate angle TQPTQP to angle TRSTRS using parallel line properties.
∠TRS=37∘\angle TRS = 37^\circ
Since line L1L_1 is parallel to line L2L_2, the transversal line QRQR creates equal alternate interior angles. Thus, ∠TRS=∠TQP=37∘\angle TRS = \angle TQP = 37^\circ.

Key Concept

Alternate interior angles and triangle angle sum theorem
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Topics

Lines and Angles

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