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Difficulty: EasyLines and Angles

Three straight lines, RR, SS, and TT, all intersect at a single point PP. Line RR is perpendicular to line SS. Line TT intersects line RR such that one of the acute angles formed between line RR and line TT measures 35∘35^\circ. Which of the following degree measures represent angles formed between any two of the intersecting lines at point PP? Select all that apply.

  1. 55∘55^\circAnswer
  2. 90∘90^\circAnswer
  3. 145∘145^\circAnswer
  4. D
    115∘115^\circ
  5. E
    70∘70^\circ

Answer

The degree measures 55∘55^\circ, 90∘90^\circ, and 145∘145^\circ are all valid angle measures formed by the intersecting lines at point PP.
The intersecting lines form several distinct angle measures at point PP. The right angle between lines RR and SS measures 90∘90^\circ. The acute angle between lines TT and SS is complementary to the 35∘35^\circ angle, giving 90∘−35∘=55∘90^\circ - 35^\circ = 55^\circ. The obtuse angle between lines TT and RR along straight line RR is supplementary to 35∘35^\circ, giving 180∘−35∘=145∘180^\circ - 35^\circ = 145^\circ. Therefore, the options stating 55∘55^\circ, 90∘90^\circ, and 145∘145^\circ are all correct.

Step-by-Step Solution

1
Identify the angle measure between perpendicular lines R and S.
Since line R⊥SR \perp S, the angle between them is 90∘90^\circ.
Perpendicular lines intersect at right angles (90∘90^\circ).
2
Calculate the acute angle between line T and line S.
Angle between TT and SS = 90∘−35∘=55∘90^\circ - 35^\circ = 55^\circ.
The given 35∘35^\circ angle between RR and TT and the adjacent angle between TT and SS form the 90∘90^\circ right angle between RR and SS.
3
Determine the supplementary obtuse angles formed by line T with line R and line S.
Supplementary angle to 35∘35^\circ is 180∘−35∘=145∘180^\circ - 35^\circ = 145^\circ. Supplementary angle to 55∘55^\circ is 180∘−55∘=125∘180^\circ - 55^\circ = 125^\circ.
Adjacent angles along a straight line sum to 180∘180^\circ.
4
Compare calculated angle measures with the options.
The valid angle measures are 35∘35^\circ, 55∘55^\circ, 90∘90^\circ, 125∘125^\circ, and 145∘145^\circ. Thus 55∘55^\circ, 90∘90^\circ, and 145∘145^\circ are correct.
Matching calculated angle measures with the choices provided.

Key Concept

Perpendicular line relationships and supplementary angle properties of intersecting lines.
Estimated Time:1m 0s
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Topics

Lines and Angles

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