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Isosceles Triangles — MATHEMATICS STD 9 — Question

ICSE BoardEnglish MediumSTD 9MATHEMATICSIsosceles Triangles3 Marks
Question
In $\triangle ABC , \text{AB} = \text{AC} ; \text{BE} \perp \text{AC}$ and $\text{CF} \perp \text{AB}$.


Prove that:
$(i) \text{BE} =\text{ CF}$
$(ii) \text{AF} = \text{AE}$
✓

Answer

$(i)$
In $\triangle AEB$ and $\triangle AFC,$
$\angle A = \angle A .......[$Common$]$
$\angle AEB = \angle AFC = 90^\circ ......[$Given: $\text{BE} \perp \text{AC}]......[$Given: $\text{CF} \perp \text{AB}]$
$\text{AB} =\text{ AC} .......[$Given$]$
$\Rightarrow \triangle AEB ≅ AFC .......[\text{AAS}]$
$\therefore \text{BE}= \text{CF} .......[$C.p.c.t$]$
$(ii)$ Since $\triangle AEB ≅ AFC$
$\angle ABE = \angle AFC$
$\therefore \text{AF} = \text{AE} ........[$congruent angles of congruent triangles$]$

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