Triangle Angles and Exterior Angles

13.2: Triangle Angles and Exterior Angles. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 63 parts Practice Flashcards

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Triangle angle sum

Objective: Recover a missing interior angle.

Prerequisite skills: straight angles and triangle angle sums. The three interior angles of a Euclidean triangle sum to 180°. An equilateral triangle has three 60° angles; an isosceles triangle has equal angles opposite its equal sides. Common error: subtracting from 360° for a triangle. Restrictions and interpretation: The exterior angle uses the two remote interior angles, not the adjacent interior angle.

Formulas

$A+B+C=180°$

Worked examples

Try itA triangle has interior angles 56° and 55°. What is the third angle in degrees?Show answer

Subtract both known angles from 180°: C=180−56−55=69°. The result is positive. Therefore the requested answer is 69.

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Triangle angle sum

Objective: Recover a missing interior angle.

Formulas

$A+B+C=180°$

Worked examples

Try itA triangle has interior angles 56° and 55°. What is the third angle in degrees?Show answer

Subtract both known angles from 180°: C=180−56−55=69°. The result is positive. Therefore the requested answer is 69.

Exterior-angle theorem

Objective: Find an exterior angle from two remote interior angles.

Extending one triangle side forms an exterior angle supplementary to the adjacent interior angle. Combining that fact with the triangle sum shows that the exterior equals the sum of the two remote interior angles. Common error: adding the adjacent interior angle instead of a remote one. Restrictions and interpretation: The exterior angle uses the two remote interior angles, not the adjacent interior angle.

Formulas

exterior angle=remote angle₁+remote angle₂

Worked examples

Try itA triangle has remote interior angles 50° and 47° relative to a specified exterior angle. What is that exterior angle?Show answer

The exterior-angle theorem gives E=50+47=97°. The adjacent interior angle is 83°, so their sum checks as 180°. Therefore the requested answer is 97.

Isosceles angle constraints

Objective: Use equality of base angles in an algebraic model.

Equal sides in a triangle face equal angles. If an isosceles triangle has vertex angle V, its two base angles each equal (180−V)/2. Identify the vertex between the equal sides rather than guessing from orientation. Common error: using 180−V for each base angle without dividing by two. Restrictions and interpretation: The exterior angle uses the two remote interior angles, not the adjacent interior angle. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

base angle=(180°−vertex angle)/2

Worked examples

Try itAn isosceles triangle has equal sides AB and AC and vertex angle A = 76°. What is angle B in degrees?Show answer

Since AB=AC, angles B and C are equal. Their sum is 180−76=104°, so each is 52°. Therefore the requested answer is 52.

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