Two parallel lines, L_1 and L_2, are intersected by three distinct transversals. The first… — Armed Services Vocational Aptitude Battery (ASVAB)

Two parallel lines, $L_1$ and $L_2$ , are intersected by three distinct transversals. The first transversal creates a pair of alternate interior angles that measure $(4x - 3y)^\circ$ and $(2x + y)^\circ$ . The second transversal creates a pair of consecutive (same-side) interior angles that measure $(3x + y)^\circ$ and $(5y + 36)^\circ$ . The third transversal creates a pair of corresponding angles. One of these corresponding angles measures $(x + y + z)^\circ$ , and the angle forming a linear pair with it measures $108^\circ$ . What is the value of $z$ ?

Cevap: 36

Cevap

By applying geometric theorems for transversals (alternate interior angles are equal, same-side interior angles sum to 180), a system of equations reveals x = 24 and y = 12. Using the linear pair property, the final angle expression (x + y + z) equals 72, which isolates z to exactly 36.

Adım Adım Çözüm

Set up an equation using the alternate interior angles.

4x - 3y = 2x + y \Rightarrow 2x = 4y \Rightarrow x = 2y

Alternate interior angles formed by a transversal intersecting parallel lines are equal.

Set up an equation using the consecutive (same-side) interior angles.

(3x + y) + (5y + 36) = 180 \Rightarrow 3x + 6y + 36 = 180 \Rightarrow 3x + 6y = 144

Consecutive interior angles formed by a transversal intersecting parallel lines are supplementary (sum to 180 degrees).

Solve the system of equations by substituting

x = 2y

into the second equation.

3(2y) + 6y = 144 \Rightarrow 6y + 6y = 144 \Rightarrow 12y = 144 \Rightarrow y = 12

. Since

x = 2y

x = 24

Substitution efficiently solves a two-variable linear system to find the individual angle components.

Use the linear pair to determine the measure of the corresponding angle, then solve for

z

(x + y + z) = 180 - 108 = 72

. Substituting known values:

24 + 12 + z = 72 \Rightarrow 36 + z = 72 \Rightarrow z = 36

Angles forming a linear pair are supplementary. Finding the target angle allows substitution of

x

and

y

to isolate

z

Anahtar Kavram

Complex systems of transversal angle relationships (alternate interior, consecutive interior, linear pairs)

Tahmini Süre:3m 0s

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