Question

In the following figure, AB = AC and AD is perpendicular to BC. BE bisects angle B and EF is perpendicular to AB.
Prove that:
(i) BD = CD
(ii) ED = EF​

Answer 1

Given:-

• AB = AC
• AD ⊥ BC
• BE bisects angle B
• EF ⊥ AB

To Prove:-

1. BD = CD
2. ED = EF

Solution:-

★ In ∆ ADB and ∆ ADC,

AB = AC [ ∴ Given]

AD = AD [ ∴ Common]

∠DAB = ∠ADC [ ∴ ∠A common]

∆ADB ≅ ∆ADC [ ∴ By SAS congruence criterion]

BD = DC [ ∴ By CPCT]

★ In ∆ EBF and ∆ EBD,

EFB = BDE [ ∴ each 90°]

∠EBF = ∠EBD [ ∴ Given]

EB = BE [ ∴ Common]

∆EBF ≅ ∆EBD [ ∴ By AAS congruence criterion]

EF = ED [ ∴ By CPCT ]

★ Hence Proved!!

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