$ \mathrm{ABC} $ is an isosceles triangle in which altitudes $ \mathrm{BE} $ and $ \mathrm{CF} $ are drawn to equal sides $ \mathrm{AC} $ and $ \mathrm{AB} $ respectively (see Fig. 7.31). Show that these altitudes are equal. "
Given:
$ABC$ is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively.
To do:
We have to show that these altitudes are equal.
Solution:
Let us consider $\triangle AEB$ and $\triangle AFC$
We know that,
From Angle-Angle-Side:
if two angles of a triangle with a non-included side are equal to the corresponding angles and non-included side of the other triangle, they are considered to be congruent.
$AB=AC$ (given)
We have,
$\angle A$ as the common angle of $AEB$ and $AFC$
$\angle AEB=\angle AFC$
Therefore,
$\triangle AEB \cong \triangle AFC$
We also know
From corresponding parts of congruent triangles: If two triangles are congruent, all of their corresponding sides must be equal.
Therefore,
$BE=CF$.
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