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Difficulty: MediumLines and Angles

A transversal line cuts across two parallel lines, forming a pair of same-side interior angles. If the measures of these angles are (7x−5)∘(7x - 5)^\circ and (3x+25)∘(3x + 25)^\circ, what is the measure of the larger angle?

  1. A
    43∘43^\circ
  2. B
    46∘46^\circ
  3. 107∘107^\circAnswer
  4. D
    117∘117^\circ

Answer

The larger angle measures 107∘107^\circ.
Same-side interior angles are supplementary, so setting their sum equal to 180∘180^\circ gives 10x+20=18010x + 20 = 180, solving to x=16x = 16. Substituting this value back into the original expressions yields angle measures of 107∘107^\circ and 73∘73^\circ. The larger of the two is 107∘107^\circ.

Step-by-Step Solution

1
Set up an equation based on the geometric relationship.
(7x−5)+(3x+25)=180(7x - 5) + (3x + 25) = 180
Same-side interior angles formed by parallel lines and a transversal are supplementary, meaning they add up to 180 degrees.
2
Combine like terms to simplify the left side of the equation.
10x+20=18010x + 20 = 180
Combine the variable terms (7x+3x=10x7x + 3x = 10x) and the constant terms (−5+25=20-5 + 25 = 20).
3
Solve the linear equation for xx.
10x=160⇒x=1610x = 160 \Rightarrow x = 16
Subtract 20 from both sides, then divide by 10 to isolate the variable xx.
4
Substitute x=16x = 16 back into the original algebraic expressions to find the measures of both angles.
7(16)−5=112−5=107∘7(16) - 5 = 112 - 5 = 107^\circ and 3(16)+25=48+25=73∘3(16) + 25 = 48 + 25 = 73^\circ
The exact measures must be calculated in order to determine which angle is larger.
5
Compare the two angles and identify the larger one.
107∘107^\circ
Between 107∘107^\circ and 73∘73^\circ, 107∘107^\circ is the larger value.

Key Concept

Same-side (consecutive) interior angles formed by a transversal intersecting parallel lines are supplementary.
Estimated Time:1m 30s
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Topics

Lines and Angles

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