Two horizontal parallel lines, L_1 and L_2, are drawn with L_1 above L_2. A point P is loc… — Armed Services Vocational Aptitude Battery (ASVAB)

Two horizontal parallel lines, $L_1$ and $L_2$ , are drawn with $L_1$ above $L_2$ . A point $P$ is located between the two lines. Point $A$ is on $L_1$ and point $B$ is on $L_2$ , such that $P$ is positioned to the right of both $A$ and $B$ . Segment $AP$ and segment $BP$ form a zigzag pattern. The interior angle at $A$ , below $L_1$ and opening towards $P$ , measures $(5x - 4)^\circ$ . The interior angle at $B$ , above $L_2$ and opening towards $P$ , measures $(3x + 16)^\circ$ . The angle $\angle APB$ , opening towards the left, measures $(11x - 42)^\circ$ . What is the measure of the complement of the smaller of the two angles at $A$ and $B$ ?

A
$44^\circ$
B
$52^\circ$
$20^\circ$ Cevap
D
$110^\circ$

Cevap

The correct answer is

20^\circ

Based on the properties of parallel lines intersected by transversals, an auxiliary parallel line drawn through the vertex of the zigzag shows that the middle angle equals the sum of the two outer angles. Setting up the equation

(5x - 4) + (3x + 16) = 11x - 42

yields

x = 18

. Substituting this value back in gives angles of

86^\circ

and

70^\circ

. The smaller angle is

70^\circ

, and its complement is found by subtracting it from

90^\circ

, which gives

20^\circ

Adım Adım Çözüm

Apply geometric properties to set up an equation.

(5x - 4) + (3x + 16) = 11x - 42

By drawing an auxiliary line through point P parallel to L1 and L2, the angle at P is split into two alternate interior angles. Therefore, the angle opening left equals the sum of the two angles opening right.

Combine like terms on the left side of the equation.

8x + 12 = 11x - 42

Combining the

x

terms (

5x + 3x = 8x

) and the constants (

-4 + 16 = 12

) simplifies the equation.

Solve for x.

x = 18

Subtract

8x

from both sides to get

12 = 3x - 42

. Add

42

to both sides to get

54 = 3x

. Divide by

3

to find

x = 18

Substitute x back into the expressions to find the measures of the angles at A and B.

Angle at A =

86^\circ

, Angle at B =

70^\circ

Angle at A:

5(18) - 4 = 90 - 4 = 86^\circ

. Angle at B:

3(18) + 16 = 54 + 16 = 70^\circ

Identify the smaller angle and calculate its complement.

20^\circ

The smaller angle is

70^\circ

. The complement of an angle is

90^\circ

minus the angle, so

90^\circ - 70^\circ = 20^\circ

Anahtar Kavram

Parallel Lines and Transversals (Zigzag Theorem)

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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