Mathematics
On the sides AB and AC of triangle ABC, equilateral triangles ABD and ACE are drawn. Prove that :
(i) ∠CAD = ∠BAE
(ii) CD = BE.
Answer
△ ABC with equilateral triangles ABD and ACE drawn on its sides AB and AC, respectively are is shown below:
(i) Since, ABD and ACE are equilateral triangles.
∴ ∠BAD = ∠CAE (Both equal to 60°)
Adding ∠CAB on both sides we get :
⇒ ∠BAD + ∠CAB = ∠CAE + ∠CAB
⇒ ∠CAD = ∠BAE.
Hence, proved that ∠CAD = ∠BAE.
(ii) In △ CAD and △ BAE,
⇒ AC = AE (△ ACE is equilateral triangle)
⇒ ∠CAD = ∠BAE (Proved above)
⇒ AD = AB (△ ABD is equilateral triangle)
∴ △ CAD ≅ △ BAE (By S.A.S. axiom)
We know that,
Corresponding parts of congruent triangles are equal.
∴ CD = BE.
Hence, proved that CD = BE.
Related Questions
In a triangle ABC, D is mid-point of BC; AD is produced upto E, so that DE = AD. Prove that :
(i) △ ABD and △ ECD are congruent.
(ii) AB = EC
(iii) AB is parallel to EC.
From the given diagram, in which ABCD is a parallelogram, ABL is a line segment and E is mid point of BC.
Prove that :
(i) △ DCE ≅ △ LBE
(ii) AB = BL
(iii) AL = 2DC
A line segment AB is bisected at point P and through point P another line segment PQ, which is perpendicular to AB, is drawn. Show that : QA = QB.
In the following diagrams, ABCD is a square and APB is an equilateral triangle. In each case,
(i) Prove that : △ APD ≅ △ BPC
(ii) Find the angles of △ DPC.
