Question 50·Medium·Lines, Angles, and Triangles
In the figure shown, lines and are parallel. Two transversals intersect at point .
Which choice is the value of ?
When you see parallel lines cut by transversals, first transfer angle measures using corresponding or alternate interior angles (whichever matches the marked positions). Then, if a triangle is formed, use the triangle angle sum of . Keeping these two ideas separate—“move angles across parallel lines” and then “sum triangle angles”—helps avoid mixing up equal vs. supplementary angles.
Hints
Connect the angles on to angles on
Because and are parallel, use angle relationships created by a transversal (especially alternate interior angles) to transfer a labeled angle on to an angle on .
Think triangle
The segments from to and form triangle . Identify its three interior angles in terms of .
Use the triangle sum
Once you have the three interior angles of triangle , set their sum equal to and solve.
Desmos Guide
Enter the triangle-sum equation
In Desmos, enter
(2x+12)+(x+18)+(3x+6)=180
Find the solution
Desmos will show the solution for on the graph or in the expression line. Use that value of as the answer choice.
Step-by-step Explanation
Use parallel lines to identify the triangle’s base angles
Since , alternate interior angles formed by each transversal with and are equal.
Therefore, the base angle at in triangle is , and the base angle at in triangle is .
Write the triangle angle-sum equation
The interior angles of triangle sum to :
Solve for
Combine like terms:
Therefore, the value of is .