Question 360·Easy·Math

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Two parallel hiking trails are shown on a map. A straight footpath crosses both trails, forming angles.

At the intersection with the first trail, the angle located between the two trails and to the right of the footpath measures $65^\circ$ . At the intersection with the second trail, the angle located between the two trails and to the right of the footpath is labeled $x^\circ$ .

What is the value of $x$ ?

A footpath crossing two parallel trails is a transversal, a line that crosses both. Check where the angles sit before deciding whether they match or add. Angles between the trails and on the same side are called same-side interior angles. They add to $180^\circ$ rather than match. Write their sum, then subtract the known angle.

Hints

Hint 1
The word interior means between the parallel trails. Check each labeled angle at its own intersection: are they on the same side of the footpath or on opposite sides?
Hint 2
Same-side interior angles make a $180^\circ$ total. Write a sum using both angle measures. Setting them equal would use the wrong parallel-line relationship.

Step-by-step

Use same-side interior angles
Step 1
Identify the angle relationship
No Desmos needed. This angle rule leaves only one subtraction. The $65^\circ$ angle is below Trail 1, and $x^\circ$ is above Trail 2, so both lie between the parallel trails. Both are right of the footpath. They're same-side interior angles, which add to $180^\circ$ .
Step 2
Write the angle sum
The label $x^\circ$ means $x$ degrees. Put the two angles into their $180^\circ$ sum:
$65+x=180.$
Step 3
Find the angle at Trail 2
Subtract $65$ from both sides:
$\begin{aligned}x&=180-65\x&=115.\end{aligned}$
So the angle at Trail 2 measures $115^\circ$ . Choice C.

Lessons that teach this

SAT Geometry and TrigonometryBeginnerCoreUse line and angle relationships

Question 360·Easy·Math

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