Question 9 of 10 Multiple Choice: Please select the best answer and click “submit.” Which angle in △DEF has the largest measure? A. ∠D B. ∠E C. ∠F D. Cannot be determined

Answer: ∠F

Explanation: Subject: Math — Geometry (triangle side–angle relationships). Key fact: in any triangle, the largest angle lies opposite the longest side. The side lengths shown are DF = 6, FE = 7, and DE = 11, so DE is the longest side; therefore the angle opposite DE, which is ∠F, is the largest. (You can confirm with the Law of Cosines: ∠F ≈ 115.4°, ∠D ≈ 35.0°, ∠E ≈ 29.6°.)

Steps:

1. Read the side labels from the diagram: DF = 6, FE = 7, DE = 11.
2. Identify the longest side: DE = 11.
3. Use the triangle fact: the largest angle is opposite the longest side.
4. The angle opposite DE is ∠F, so ∠F is the largest.
5. Optional check (Law of Cosines):

• cos(∠F) = (DF^2 + FE^2 − DE^2) / (2·DF·FE) = (36 + 49 − 121) / 84 = −36/84 = −3/7 → ∠F ≈ arccos(−3/7) ≈ 115.4°
• cos(∠D) = (DF^2 + DE^2 − FE^2) / (2·DF·DE) = (36 + 121 − 49) / 132 = 108/132 ≈ 0.81818 → ∠D ≈ 35.0°
• ∠E = 180° − 115.4° − 35.0° ≈ 29.6°

These confirm ∠F is largest.