Mathematics
In the adjoining figure, ∠ABC = ∠ACB, D and E are points on the sides AC and AB respectively such that BE = CD. Prove that
(i) △EBC ≅ △DCB
(ii) △OEB ≅ △ODC
(iii) OB = OC.
Answer
(i) Given, ∠ABC = ∠ACB.
∴ ∠EBC = ∠DCB.
In △EBC and △DCB,
∠EBC = ∠DCB (Proved)
BE = CD (Given)
BC = BC (Common)
Hence, proved △EBC ≅ △DCB by SAS axiom.
(ii) We know that, △EBC ≅ △DCB.
Subtracting common △OBC from both sides we get,
⇒ △EBC - △OBC ≅ △DCB - △OBC
⇒ △OEB ≅ △ODC
Hence, proved that △OEB ≅ △ODC.
(iii) We know that,
△OEB ≅ △ODC
We know that corresponding angles of congruent triangles are equal.
∴ OB = OC.
Hence, proved that OB = OC.
Related Questions
In the adjoining figure, two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the mid-point of both the line segments AB and CD.
ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C.
In the adjoining figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that △ABC ≅ △DEF.
In the adjoining figure, ∠BCD = ∠ADC and ∠BCA = ∠ADB. Show that
(i) △ACD ≅ △BDC
(ii) BC = AD
(iii) ∠A = ∠B
