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Difficulty: MediumProperties of Quadrilaterals

In parallelogram PQRSPQRS, the measures of consecutive angles ∠P\angle P and ∠Q\angle Q are (4x+15)∘(4x + 15)^\circ and (2x+45)∘(2x + 45)^\circ, respectively. What is the measure, in degrees, of ∠R\angle R?

  1. A
    20∘20^\circ
  2. B
    75∘75^\circ
  3. C
    85∘85^\circ
  4. 95∘95^\circAnswer
  5. E
    105∘105^\circ

Answer

95∘95^\circ
In any parallelogram, consecutive interior angles formed by parallel lines and a transversal are supplementary (sum to 180∘180^\circ). Adding the expressions for consecutive angles ∠P\angle P and ∠Q\angle Q gives (4x+15)+(2x+45)=180∘(4x + 15) + (2x + 45) = 180^\circ, which simplifies to 6x+60=180∘6x + 60 = 180^\circ, giving x=20x = 20. Substituting x=20x = 20 into the expression for ∠P\angle P yields 4(20)+15=95∘4(20) + 15 = 95^\circ. Because opposite angles in a parallelogram are congruent, ∠R\angle R has the same measure as ∠P\angle P, which is 95∘95^\circ.

Step-by-Step Solution

1
Set up an equation using the consecutive angle property of parallelograms.
(4x+15)+(2x+45)=180(4x + 15) + (2x + 45) = 180
Consecutive angles in any parallelogram are supplementary (their sum is 180∘180^\circ).
2
Solve the linear equation for xx.
6x+60=180  ⟹  6x=120  ⟹  x=206x + 60 = 180 \implies 6x = 120 \implies x = 20
Combine like terms (4x+2x=6x4x + 2x = 6x and 15+45=6015 + 45 = 60) and isolate xx.
3
Calculate the measure of ∠P\angle P.
m\angle P = 4(20) + 15 = 80 + 15 = 95^\circ$
Substitute x=20x = 20 into the expression given for ∠P\angle P.
4
Determine the measure of ∠R\angle R.
m\angle R = m\angle P = 95^\circ$
Opposite angles in a parallelogram are equal in measure.

Key Concept

Angle Relationships in Parallelograms
Estimated Time:1m 0s
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Topics

Properties of Quadrilaterals

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