Question 312·Easy·Math

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In the figure, $\triangle ABC$ is shown with a ray from $C$ passing through point $D$ . The measures of several angles are labeled.

Which choice is the value of $x$ ?

A triangle with one side extended points to the exterior-angle theorem: the outside angle equals the two inside angles at the other corners added together. Read the angle labels rather than judging their sizes by eye, then write that relationship and use a Desmos regression to find the unknown. The trap is treating the outside angle as the triangle’s inside angle at that corner.

Hints

Hint 1
An exterior angle is made outside a triangle when one side extends past a corner. At $C$ , which labeled angle is outside the triangle, and which two labeled angles are at the other corners?
Hint 2
The exterior angle equals the sum of the two interior angles away from it. Write that relationship using the labels at $A$ and $B$ . What value of $x$ makes the two sides equal?

Step-by-step

Use the exterior angle
Step 1
Connect the outside angle to the two far angles
The ray through $D$ extends side $AC$ , so the $120^\circ$ angle at $C$ is an exterior angle: it lies outside the triangle. An exterior angle equals the two interior angles away from it added together. Those angles are $45^\circ$ at $A$ and $x^\circ$ at $B$ , so $120=45+x$ . The $120^\circ$ angle is not the angle inside the triangle at $C$ .
Step 2
Find the angle at B
Type $45+x_1\sim120$ in Desmos. The $\sim$ tells Desmos to find the unknown $x_1$ that makes both sides equal. Under PARAMETERS, it shows $x_1=75$ . So the angle at $B$ measures $75^\circ$ , and $x=75$ . Choice B.

Question 312·Easy·Math

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