Question

$\text{In the given figure, BE and CF are two equal altitudes of}\Delta ABC.\text{Show that (i)}\Delta ABE\cong \Delta ACF,\left(ii\right)AB=AC.$

Answer 1

Congruence of triangles:
Two triangles are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other triangle

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:
There are 4 criteria for congruence of triangles.

Here we use ASA Congruence.

ASA(angle side angle):
Two Triangles are congruent if two angles and the included side of One triangle are equal to two angles & the included side of the other trian gle.

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Given:
$\Delta ABC$in which BE perpendicular to AC & CF perpendicular to AB, such that BE=CF.

To Prove:
i) $\Delta ABE\cong \Delta ACF$
ii) AB=AC

Proof:
(i) In $\Delta ABEand\Delta ACF$,
$\angle A=\angle A$(Common)
$\angle AEB=\angle AFC$ each ${90}^o$
BE = CF (Given)

Therefore, $\Delta ABE\cong \Delta ACF$ (by ASA congruence rule)

(ii) since $\Delta ABE\cong \Delta ACF$
Thus, AB = AC (by CPCT)

Therefore $\Delta ABC$ is an isosceles triangle.