MBApyq

Angle in a triangle with equal segments

CAT 2021 Slot 3 · Quantitative Ability · Hard · Geometry

This is a hard Quantitative Ability question from the CAT 2021 Slot 3 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

All CAT 2021 Slot 3 questionsMore Geometry questionsAll CAT QA

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

  1. A.

    100

  2. B.

    80

  3. C.

    96

  4. D.

    72

Answer

B

Explanation

Let ∠A=α\angle A = \alpha and ∠B=β\angle B = \beta. In △ABC\triangle ABC, α+β+50∘=180∘  ⟹  α+β=130∘\alpha + \beta + 50^\circ = 180^\circ \implies \alpha + \beta = 130^\circ.

In △ADE\triangle ADE, since AD=DEAD = DE, it is an isosceles triangle with ∠AED=∠DAE=α\angle AED = \angle DAE = \alpha. Thus, ∠ADE=180∘−2α\angle ADE = 180^\circ - 2\alpha.

In △BDF\triangle BDF, since BD=DFBD = DF, it is an isosceles triangle with ∠DFB=∠DBF=β\angle DFB = \angle DBF = \beta. Thus, ∠BDF=180∘−2β\angle BDF = 180^\circ - 2\beta.

Since points A, D, B lie on a straight line: ∠BDF+∠FDE+∠ADE=180∘\angle BDF + \angle FDE + \angle ADE = 180^\circ (180∘−2β)+∠FDE+(180∘−2α)=180∘(180^\circ - 2\beta) + \angle FDE + (180^\circ - 2\alpha) = 180^\circ ∠FDE=2(α+β)−180∘=2(130∘)−180∘=260∘−180∘=80∘\angle FDE = 2(\alpha + \beta) - 180^\circ = 2(130^\circ) - 180^\circ = 260^\circ - 180^\circ = 80^\circ.

Related Geometry questions

Practise this under exam conditions

Sign in to solve it with a live timer, the on-screen CAT calculator, and streak and accuracy tracking across every question you attempt.

Solve in the workspace