CAT 2021 Slot 3 QA Q5 Geometry

Question 5 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

In a triangle ABC , ∠ BCA =50°. D and E are points on AB and AC, respectively, such that AD = DE. If F is a point on BC such that BD = DF, then ∠ FDE, in degrees, is equal to

A
100
B
80
C
96
D
72

Answer: (B) 80 Geometry MCQ

Solution

Let \(\angle A = \alpha, \angle B = \beta\), so \(\alpha + \beta = 180^\circ - 50^\circ = 130^\circ\).
\(AD = DE\) makes \(\angle DEA = \alpha\), so \(\angle ADE = 180^\circ - 2\alpha\). \(BD = DF\) makes \(\angle DFB = \beta\), so \(\angle BDF = 180^\circ - 2\beta\).
Angles on line AB at D: \(\angle FDE = 180^\circ - (180^\circ - 2\alpha) - (180^\circ - 2\beta) = 2(\alpha + \beta) - 180^\circ = 80^\circ\).

Shortcut. Take \(\angle FDE = 2(180^\circ - C) - 180^\circ = 180^\circ - 2C\).