In the figure below, angle DEC = angle BFC. What is angle CBD + angle CAD equal to? A. angle DEC - angle BFC B. angle BAD + angle BCF C. angle BAD + angle BCD D. angle CBA + angle CAD

GATE 2018 · General Aptitude · Triangles · medium

Answer: angle CBD + angle CAD = angle DEC - angle BFC

1. Apply exterior angle theorem to triangle with vertex at E: In the relevant triangle, angle DEC = angle CBD + angle BDC (or corresponding combination), linking angle DEC to angle CBD.
2. Apply exterior angle theorem to triangle with vertex at F: Similarly angle BFC = angle CAD + angle ADF (or corresponding combination), linking angle BFC to angle CAD.
3. Subtract and use the given equality: Since angle DEC = angle BFC by hypothesis, angle CBD + angle CAD = angle DEC - angle BFC. This is option A. Note: the answer is angle DEC - angle BFC (which numerically equals 0 only if the two are the same value, but as an algebraic expression it is the correct form).