In the figure shown, lines and are parallel.
What is the value of ?
Parallel lines crossed by a slanted line create alternate interior angles: angles between the parallels on opposite sides of the slanted line. Set those angles equal, then use a Desmos regression to find the unknown. If the requested angle lies outside a triangle, add the two far interior angles instead of stopping at one marked angle.
Hints
- Hint 1
A transversal is a line that crosses both parallel lines. Along line , the angles at and are between the parallels on opposite sides of . How do their measures compare?
- Hint 2
Once you have the equal-angle equation, change to and to in Desmos. A regression finds the value that makes both sides agree. What appears under PARAMETERS?
- Hint 3
An exterior angle lies outside a triangle where one side continues past a vertex. In triangle , side continues toward . Which two far angles add to ?
Step-by-step
Match angles, then use the exterior angle
Step 1Match the angles on the parallel lines
Because and are parallel, the angles at and lie between the parallels on opposite sides of line . They're alternate interior angles, so they're equal:
- Step 2
Find in Desmos
Type . For this regression, is the number Desmos finds, and asks it to match the two sides. Under PARAMETERS, Desmos shows . That's the value of , not the angle asked for.
- Step 3
Relate to the triangle's angles
At , ray continues ray past , so is an exterior angle of triangle . It equals the two far interior angles added together, at and :
- Step 4
Calculate the requested angle
Keep the regression and type . Desmos uses its found value of and prints . So the angle on the right of measures . Choice C.