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Difficulty: HardLines and Angles

In the geometric plane, line l1l_1 is parallel to line l2l_2. Point AA lies on line l1l_1 and point BB lies on line l2l_2. Point CC is located between lines l1l_1 and l2l_2 such that CC lies to the right of both AA and BB.

The acute angle between segment ACAC and the ray extending to the right from AA along line l1l_1 measures x∘x^\circ.

The acute angle between segment BCBC and the ray extending to the right from BB along line l2l_2 measures y∘y^\circ.

The interior angle ∠ACB=118∘\angle ACB = 118^\circ, and y=2x−14y = 2x - 14.

Line kk passes through point CC and is perpendicular to segment ACAC. Line kk intersects line l2l_2 at point EE, where point EE lies to the right of point BB.

What is the measure, in degrees, of the acute angle ∠CEB\angle CEB formed by line kk and line l2l_2?

  1. A
    16∘16^\circ
  2. B
    44∘44^\circ
  3. 46∘46^\circAnswer
  4. D
    62∘62^\circ
  5. E
    74∘74^\circ

Answer

The measure of the acute angle ∠CEB\angle CEB is 46∘46^\circ.
By the parallel lines angle property, drawing a parallel line through vertex CC shows that ∠ACB=x+y=118∘\angle ACB = x + y = 118^\circ. Substituting y=2x−14y = 2x - 14 yields 3x−14=1183x - 14 = 118, giving x=44∘x = 44^\circ. Since line l1l_1 is parallel to line l2l_2, segment ACAC intersects line l2l_2 at an acute angle of 44∘44^\circ. Line kk is constructed perpendicular to segment ACAC, so the acute angle formed by line kk and line l2l_2 is complementary to 44∘44^\circ, which gives 90∘−44∘=46∘90^\circ - 44^\circ = 46^\circ.

Step-by-Step Solution

1
Set up the parallel lines zig-zag relationship to find xx and yy.
∠ACB=x+y=118∘\angle ACB = x + y = 118^\circ
By drawing an auxiliary line through CC parallel to l1l_1 and l2l_2, the interior angle ∠ACB\angle ACB facing left equals the sum of the alternate interior angles xx and yy.
2
Substitute the given algebraic relation y=2x−14y = 2x - 14 into the sum equation.
x+(2x−14)=118  ⟹  3x−14=118  ⟹  3x=132  ⟹  x=44∘x + (2x - 14) = 118 \implies 3x - 14 = 118 \implies 3x = 132 \implies x = 44^\circ
Solving the linear system gives the exact value of angle xx.
3
Determine the angle that line ACAC makes with line l2l_2.
Line ACAC intersects line l2l_2 at an acute angle of 44∘44^\circ.
Since l1∥l2l_1 \parallel l_2, alternate interior angles formed by transversal line ACAC are equal (x=44∘x = 44^\circ).
4
Calculate the acute angle ∠CEB\angle CEB between line kk and line l2l_2.
∠CEB=90∘−44∘=46∘\angle CEB = 90^\circ - 44^\circ = 46^\circ
Line kk is perpendicular to segment ACAC, so the angle it forms with line l2l_2 is the complementary angle to the angle line ACAC forms with line l2l_2.

Key Concept

Parallel Line Angle Relationships and Perpendicular Line Complements
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Topics

Lines and Angles

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