The interior angles of every triangle add to 180°, and that one fact drives everything in this lesson. Extending a side of a triangle creates an exterior angle, and the measure of that exterior angle is equal to the sum of the measures of the two remote interior angles. That result turns many two-step problems into one-step problems. A third result, the Third Angle Theorem, says that if two angles of one triangle are congruent (equal in measure) to two angles of another triangle, then the remaining angles of the two triangles are congruent (equal in measure) as well.
What to do: Drag any of the three orange vertex handles. The vertices snap to grid points, and the move is refused if it would place two vertices at the same point or put all three vertices on one line, because neither shape would be a triangle. Every measure is recomputed from the triangle you have built.
What to notice: the individual angle measures change as you drag, but their sum never changes. The three interior angles of every triangle add to exactly 180°.
What to do: Side BC of triangle ABC is extended past point C to point D, so points B, C, and D all lie on one straight line. Drag the orange handle at A to move vertex A anywhere above the line. Slide the purple handle at C along the line to change the length of side BC; point D always stays on the same line, directly past C. Vertex B stays in place.
What to notice: ∠ACD, the exterior angle, is formed by side AC and the extension CD. Its two remote interior angles are the angles at A and B, the two far corners. Watch the readouts below: the exterior angle ∠ACD is always equal to the sum of its two remote interior angles, and the exterior angle ∠ACD together with the adjacent interior angle ∠ACB is supplementary (their measures add to 180°).
What to do: Each challenge is solved in four steps: (1) choose the theorem that fits, (2) choose the correct equation, (3) solve for the unknown value, and (4) report the measure the question actually asks for. Read the diagram and the question carefully before each step — in some challenges the value of x is not the final answer, because the question asks for a measure. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the challenge then moves on. When all four steps are complete, the full worked solution appears with a substitution check. Press New Challenge for a different challenge.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Measuring Angles in Triangles — Mr. J Geometry. All activities on this page run entirely in your browser.