$ \mathrm{BE} $ and $ \mathrm{CF} $ are two equal altitudes of a triangle $ \mathrm{ABC} $. Using RHS congruence rule, prove that the triangle $ \mathrm{ABC} $ is isosceles.
Given:
$BE$ and $CF$ are two equal altitudes of a triangle $ABC$.
To do:
By using the RHS congruence rule we have to prove that the triangle $ABC$ is isosceles.
Solution:
Let us consider $\triangle BEC$ and $\triangle CFB$,
We have,
$BE$ and $CF$ are two equal altitudes of a triangle $ABC$.
This implies,
$\angle BEC=\angle CFB=90^o$
and $BE=CF$
Since $BC$ is the common side
we get,
$BC=CB$
We know that According to the RHS rule if the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and one side of another right-angled triangle; then both the right-angled triangle are said to be congruent.
Therefore,
$\triangle BEC \cong \triangle CFB$
We also know that,
The corresponding parts of congruent triangles: If two triangles are congruent, all of their corresponding angles and sides must be equal.
This implies,
$\angle C=\angle B$
Therefore,
As the sides opposite to equal angles is always equal we get,
$AB=AC$.
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