Tenth Grade · Math · week 13

Triangle Sum, Exterior Angle Theorem, and classifying triangles

Classify triangles by sides and angles, and apply the Triangle Sum Theorem and Exterior Angle Theorem.

Triangles are classified two ways: by sides (scalene = no two sides congruent; isosceles = at least two sides congruent; equilateral = all three sides congruent) and by angles (acute = all three angles less than 90 degrees; right = exactly one 90-degree angle; obtuse = exactly one angle greater than 90 degrees; equiangular = all three angles congruent). Every triangle earns one label from each list, like "isosceles right triangle."

The Triangle Sum Theorem states that the interior angles of any triangle always add to 180 degrees -- knowing any two angles instantly gives the third. Extend one side of a triangle past a vertex and you create an exterior angle, adjacent to (and supplementary with) the interior angle at that vertex. The Exterior Angle Theorem says that exterior angle equals the SUM of the two remote interior angles (the two angles not next to it) -- a shortcut that skips a subtraction step and reappears constantly in proofs.

📋 For the teacher
Teach it (10–15 min):
  • Tear the three corners off a paper triangle and line them up along a straight edge -- physically seeing them form a straight line is far more convincing than being told the theorem.
  • Build a quick reference chart with all seven classification names (3 by side, 4 by angle) and classify several triangles by BOTH criteria at once, like "scalene right."
  • Show the Exterior Angle Theorem as a shortcut, not a brand-new fact: derive it once from the Triangle Sum Theorem and a linear pair, so it feels earned rather than memorized.
  • Have them practice spotting which two angles are "remote" to a given exterior angle -- always the two vertices NOT touching that exterior angle.
Talk about it:
  • Why must an equiangular triangle always have three 60-degree angles?
  • How could you find an exterior angle two different ways, and why do they agree?
  • Can a triangle have two right angles? Use the Triangle Sum Theorem to explain.

Mastery looks like: They apply the Triangle Sum Theorem and Exterior Angle Theorem to find unknown angle measures, and classify triangles correctly by both sides and angles.

Common stumbles: Forgetting a triangle can be classified by side AND by angle at the same time; picking the wrong two angles as "remote" for the Exterior Angle Theorem.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

AdditionSee groups combine — with counters, blocks, and the number line.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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