Question 54·Medium·Geometry and Trigonometry

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In the figure shown, lines $r$ and $s$ are parallel.

What is the value of $y$ ?

When parallel lines are cut by a transversal, first look for equal-angle relationships (corresponding or alternate interior) to solve for any variables in the angle expressions. After you know the actual angle measures, interpret the angle at the intersection of two transversals by relating each line’s tilt to the same parallel reference line; the acute angle between the transversals is often the sum of those two acute tilts.

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Start with transversal $m$

Use the fact that $r$ and $s$ are parallel to identify a pair of equal angles formed by line $m$ .

Solve for $x$

Set the two expressions on line $m$ equal to each other and solve the resulting linear equation.

Break $y$ into two parts

At point $P$ , think of $y$ as the sum of the acute angles that lines $m$ and $n$ make with a horizontal line parallel to $r$ and $s$ .

Desmos Guide

Solve for $x$ by graphing two lines

In Desmos, enter y = x + 10 and y = 3x - 20. Click the intersection point and note the $x$ -coordinate (call it $t$ ).

Substitute the intersection value

Create a value for that coordinate, for example t = 15 (use the intersection’s $x$ -value). Then enter a = t + 10 and b = 2t + 5 to compute the two angle measures.

Add the angles to get $y$

Enter a + b. This sum is the measure of $y$ .

Step-by-step Explanation

Use parallel lines to solve for $x$

Along transversal $m$ , the angle labeled $(x+10)^\circ$ at line $r$ and the angle labeled $(3x-20)^\circ$ at line $s$ are alternate interior angles, so they are equal:

\begin{aligned} x+10 &= 3x-20 \\ 30 &= 2x \\ x &= 15 \end{aligned}

Find the two angle measures at the top line

Substitute $x=15$ :

$(x+10)^\circ = (15+10)^\circ = 25^\circ$
$(2x+5)^\circ = (2\cdot 15+5)^\circ = 35^\circ$

Relate $y$ to those angles

At point $P$ , angle $y$ is formed by ray $PB$ (on line $n$ ) and ray $PC$ (on line $m$ ) on the right side of the intersection.

Ray $PB$ makes a $35^\circ$ angle with the horizontal parallel line, and ray $PC$ makes a $25^\circ$ angle with the horizontal parallel line, on opposite sides of that horizontal. Therefore,

y = 35^\circ + 25^\circ = 60^\circ

So, the value of $y$ is $60^\circ$ .

Question 54·Medium·Geometry and Trigonometry

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation