Questions: Exterior Angle Theorem

Question 1

Triangle ABC has ∠A = 48° and ∠B = 57°. A side is extended past vertex C, forming exterior angle E. What is m∠E?

• A. 75° — the remaining interior angle at C (180° − 48° − 57°)
• B. 105° — the sum of the two remote interior angles, ∠A + ∠B
• C. 132° — the supplement of ∠A
• D. 123° — the supplement of ∠B

Answer: (B)

By the Exterior Angle Theorem, the exterior angle equals the sum of the two remote (nonadjacent) interior angles. The remote angles here are ∠A and ∠B: 48° + 57° = 105°. Option A (75°) is the third interior angle at C — the angle adjacent to E — not E itself. The theorem gives a one-step shortcut: no need to find the third interior angle first and subtract from 180°.

Question 2

A student calculates an exterior angle by first finding the adjacent interior angle (subtracting the two known angles from 180°), then subtracting that from 180°. A second student directly adds the two remote interior angles. Which is true?

• A. Only the first method is valid — the Exterior Angle Theorem applies only to specific triangle configurations
• B. Both methods are valid and always give the same answer; the second method is faster
• C. Only the second method is valid — subtracting twice from 180° introduces errors
• D. The methods may give different answers depending on whether the triangle is acute or obtuse

Answer: (B)

Both methods are correct and always agree — this is precisely what the Exterior Angle Theorem proves. Method 1: E = 180° − C (supplementary). Method 2: E = A + B (the theorem). They give the same result because C = 180° − A − B (angle sum theorem), so 180° − C = 180° − (180° − A − B) = A + B. The second method is simply faster — it skips the intermediate step of finding the third angle.

Question 3

An exterior angle of a triangle is always larger than either of the two remote interior angles individually.

• T. True
• F. False

Answer: True

Since the exterior angle equals the sum of the two remote interior angles (E = A + B) and both A and B are positive (all interior angles of a triangle are between 0° and 180°), E must be greater than each one individually: E = A + B > A (since B > 0) and E = A + B > B (since A > 0). This inequality is actually used to prove the Triangle Inequality theorem.

Question 4

Each vertex of a triangle has exactly one exterior angle.

• T. True
• F. False

Answer: False

Each vertex of a triangle has two exterior angles — one formed by extending each of the two sides that meet at that vertex. However, these two exterior angles are vertical angles (formed by two lines intersecting), so they are always congruent. In practice, 'the' exterior angle at a vertex refers to either one (since they're equal), but geometrically there are two. Each triangle has six exterior angles total, in three congruent pairs.

Question 5

Prove the Exterior Angle Theorem using only the Triangle Angle Sum Theorem and the fact that angles on a straight line sum to 180°.

Model answer: Let the triangle have interior angles A, B, and C, and let E be the exterior angle formed at vertex C by extending one side. (1) Since E and C lie on a straight line, they are supplementary: E + C = 180°. (2) By the Triangle Angle Sum Theorem: A + B + C = 180°. (3) Both expressions equal 180°, so: E + C = A + B + C. (4) Subtracting C from both sides: E = A + B. Therefore the exterior angle equals the sum of the two remote interior angles.

The proof is elegant because it requires only two facts you already know, combined in a single step. The key move is recognizing that both E + C and A + B + C equal 180°, which forces E = A + B. This is also why the theorem is really just a restatement of the angle sum theorem — not an independent fact, but a consequence you can derive in two lines.