Question

Difficulty: MediumLines and Angles

In the geometric plane, line kk is parallel to line mm. A transversal line tt intersects line kk at point PP and line mm at point QQ. Ray PRPR extends along line kk to the right of PP, and ray QSQS extends along line mm to the right of QQ. The measure of interior angle RPQ\angle RPQ is represented by (3x+20)(3x + 20)^\circ and the measure of interior angle PQS\angle PQS is represented by (2x+10)(2x + 10)^\circ. Which of the following statements must be true? Select all such statements.

  1. The value of xx is 3030.Answer
  2. The measure of angle RPQ\angle RPQ is 110110^\circ.Answer
  3. The acute angle formed by line tt and line kk at point PP measures 7070^\circ.Answer
  4. D
    The value of xx is 1212.
  5. E
    Angles RPQ\angle RPQ and PQS\angle PQS are complementary angles.

Answer

The true statements are that x=30x = 30, the measure of angle RPQ\angle RPQ is 110110^\circ, and the acute angle formed by line tt and line kk at point PP measures 7070^\circ.
Lines kk and mm are parallel, so interior angles on the same side of transversal tt (angles RPQ\angle RPQ and PQS\angle PQS) are supplementary. Solving (3x+20)+(2x+10)=180(3x + 20) + (2x + 10) = 180 gives x=30x = 30. Substituting x=30x = 30 yields RPQ=110\angle RPQ = 110^\circ and PQS=70\angle PQS = 70^\circ. The adjacent angle to RPQ\angle RPQ at point PP is 180110=70180^\circ - 110^\circ = 70^\circ, which is acute.

Step-by-Step Solution

1
Set up the geometric equation using the parallel line angle relationship.
(3x+20)+(2x+10)=180(3x + 20) + (2x + 10) = 180
When two parallel lines are cut by a transversal, consecutive interior angles on the same side of the transversal are supplementary (their sum is 180180^\circ).
2
Solve the linear equation for xx.
5x+30=180    5x=150    x=305x + 30 = 180 \implies 5x = 150 \implies x = 30
Combine like terms and isolate xx using basic algebra.
3
Calculate the specific angle measures.
m RPQ=3(30)+20=110\angle RPQ = 3(30) + 20 = 110^\circ and m PQS=2(30)+10=70\angle PQS = 2(30) + 10 = 70^\circ
Substitute x=30x = 30 into the given algebraic angle expressions.
4
Determine the supplementary acute angle at point PP.
180110=70180^\circ - 110^\circ = 70^\circ
Angles along a straight line form a linear pair and sum to 180180^\circ.

Key Concept

Parallel Lines and Consecutive Interior Angles
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