Question

1. act/sat in the illustration below, figure abcd represents part of the side support for a shelving unit. in the diagram, $overline{ab} parallel overline{dc}$, $mangle a = 5x$, and $mangle d = 20x + 5$. what is the value of x?

a 7
b 18
c 35
d 90
e 145

Explanation:

Step1: Identify the relationship between angles

Since \( \overline{AB} \parallel \overline{DC} \) and \( \angle B \) is a right angle (\( 90^\circ \)), we know that \( \angle A \) and \( \angle D \) are same - side interior angles. When two parallel lines are cut by a transversal, same - side interior angles are supplementary, meaning their sum is \( 180^\circ \). But we also note that \( \angle B = 90^\circ \) and \( \angle C = 90^\circ \) (since \( ABCD \) has a right angle at \( B \) and \( AB\parallel DC \)). Wait, actually, for the trapezoid - like figure \( ABCD \) with \( AB\parallel DC \) and \( \angle B = 90^\circ \), \( \angle A+\angle D=180^\circ \)? Wait, no. Wait, looking at the diagram, \( AB \) is vertical (since \( \angle B \) is a right angle), \( DC \) is also vertical? Wait, no, the diagram shows \( AB \) and \( DC \) as parallel, \( \angle B = 90^\circ \), so \( AB \) is perpendicular to \( BC \), and \( DC \) is also perpendicular to \( BC \), so \( AB \) and \( DC \) are both perpendicular to \( BC \), so \( AB\parallel DC \). Then, the side \( AD \) is a transversal. So \( \angle A \) and \( \angle D \) are same - side interior angles? Wait, no, actually, \( \angle A \) and \( \angle D \) should be supplementary? Wait, no, let's think again. If \( AB \) and \( DC \) are parallel, and we consider the lines \( AB \) and \( DC \) with transversal \( AD \), then \( \angle A \) and \( \angle D \) are same - side interior angles, so they should be supplementary? Wait, but also, \( \angle B = 90^\circ \) and \( \angle C = 90^\circ \), so the sum of the interior angles of quadrilateral \( ABCD \) is \( 360^\circ \). So \( \angle A+\angle B+\angle C+\angle D = 360^\circ \). Since \( \angle B = 90^\circ \) and \( \angle C = 90^\circ \), then \( \angle A+\angle D=360^\circ-(90^\circ + 90^\circ)=180^\circ \).

Step2: Set up the equation

We know that \( m\angle A = 5x \) and \( m\angle D=20x + 5 \). Since \( \angle A+\angle D = 180^\circ \), we can set up the equation:
\( 5x+(20x + 5)=180 \)

Step3: Solve the equation

First, combine like terms:
\( 5x+20x+5 = 180 \)
\( 25x+5 = 180 \)
Then, subtract 5 from both sides:
\( 25x=180 - 5 \)
\( 25x=175 \)
Then, divide both sides by 25:
\( x=\frac{175}{25}=7 \)

Answer:

A. 7