Problem: In triangles ABC and DI:F. lengths AC,BC,DF and EF are all equal. Length AB is twice the length of the altitude of △DFF from F to DE. Which of the following statements is (are) true?
I. ∠ACB and ∠DFE must be complementary.
II. ∠ACB and ∠DFE must be supplementary.
III. The area of △ABC must equal the area of △DEF.
IV. The area of △ABC must equal twice the area of △DEF.
Answer Choices:
A. II only
B. IV only
C. II and III only
D. III only
E. I and III only
Solution:
Let G and H be the points at which the altitudes from C and F intersect sides AB and DE, respectively. Right triangles AGC and DHF are congruent, since side AG and side DH have the same length, and hypotenuse AC and hypotenuse DF have the same length. Therefore,
∠ACB+∠DFE=2∠ACG+2∠DFH=180∘
and
( area △ABC)=2( area △ACG)=2( area △DFH)=( area △DEF)