Question

Prove that a triangle with two congruent altitudes is an isosceles triangle.

Write the complete proof in your paper homework and for online (only) respond to questions or statements (if any) that are parts of your proof or related to it.

Answer 1

Prove that a triangle with two congruent altitudes is an isosceles triangle:

Let $\mathrm{PQR}$ be a triangle with altitudes$\mathrm{PS}$ and $QT$ of equal length(as altitudes are congruent)

Considering the $△\mathrm{PSR}\mathrm{and}△\mathrm{QTR}$ ,

The right- triangles with a common angle i.e,$\angle PRQ$

$$\begin{gathered} \angle \mathrm{RPS}\cong \mathrm{RQT}[\mathrm{complementary}\mathrm{angles}\mathrm{to}\mathrm{the}\mathrm{angle}\mathrm{PRQ}] \\ \mathrm{PS}\cong \mathrm{RT}\left[\mathrm{Edge}\right] \\ \angle \mathrm{PSR}=\angle \mathrm{QTR}={90}^{°}\left[\mathrm{angle}\right] \\ \angle \mathrm{RPS}=\angle \mathrm{RQT}\left[\mathrm{angle}\right] \end{gathered}$$

By $\mathrm{ASA}$ postulate;

$△\mathrm{PSR}\cong △\mathrm{QTR}$

By $\mathrm{CPCT}$[Corresponding Part of Congruent Triangles are equal]

$\mathrm{PR}=\mathrm{QR}$

According to the definition of the triangle:

$\mathrm{\Delta PQR}$ is isosceles triangle

Hence, Proved.