Exercise 6.3

State whether the following pair of triangles is similar or not.

State whether the pair triangles are similar or not. Write the similarity criterion used by you for answering the question and also write the pair of similar triangles in the symbolic form.

In the figure, altitudes $\mathrm{}\mathrm{A}\mathrm{D}$ and $\mathrm{}\mathrm{C}\mathrm{E}\mathrm{}$ of $\mathrm{}∆\mathrm{A}\mathrm{B}\mathrm{C}$ intersect each other at the point $\mathrm{}\mathrm{P}$. Show that $∆\mathrm{A}\mathrm{B}\mathrm{D}~∆\mathrm{C}\mathrm{B}\mathrm{E}$.

In Fig. altitudes $\mathrm{}\mathrm{A}\mathrm{D}$ and $\mathrm{}\mathrm{C}\mathrm{E}\mathrm{}$ of $\mathrm{}∆\mathrm{A}\mathrm{B}\mathrm{C}$ intersect each other at the point $\mathrm{}\mathrm{P}$. Show that $∆\mathrm{A}\mathrm{E}\mathrm{P}~∆\mathrm{A}\mathrm{D}\mathrm{B}$.

In the figure, altitudes $\mathrm{}\mathrm{A}\mathrm{D}$ and $\mathrm{}\mathrm{C}\mathrm{E}\mathrm{}$ of $\mathrm{}∆\mathrm{A}\mathrm{B}\mathrm{C}$ intersect each other at the point $\mathrm{}\mathrm{P}$. Show that $∆\mathrm{PDC}~∆\mathrm{BEC}$.

In the figure, $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{}\mathrm{A}\mathrm{M}\mathrm{P}$ are two right triangles, right-angled at $\mathrm{}\mathrm{B}$ and $\mathrm{}\mathrm{M}\mathrm{}$ respectively. Prove that:
$\frac{\mathrm{C}\mathrm{A}}{\mathrm{P}\mathrm{A}}=\frac{\mathrm{B}\mathrm{C}}{\mathrm{M}\mathrm{P}}$

$\mathrm{C}\mathrm{D}$ and $\mathrm{}\mathrm{G}\mathrm{H}$ are respectively the bisectors of $\angle \mathrm{A}\mathrm{C}\mathrm{B}$ and $\angle \mathrm{E}\mathrm{G}\mathrm{F}$ such that $\mathrm{}\mathrm{D}$ and $\mathrm{}\mathrm{H}$ lie on sides $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{}\mathrm{F}\mathrm{E}$ of $△\mathrm{A}\mathrm{B}\mathrm{C}$ and $△\mathrm{E}\mathrm{F}\mathrm{G}$ respectively. If $△\mathrm{A}\mathrm{B}\mathrm{C}~△\mathrm{F}\mathrm{E}\mathrm{G}$, show that $△\mathrm{D}\mathrm{C}\mathrm{B}~△\mathrm{H}\mathrm{G}\mathrm{E}$.

$\mathrm{C}\mathrm{D}$ and $\mathrm{}\mathrm{G}\mathrm{H}$ are respectively the bisectors of $\angle \mathrm{A}\mathrm{C}\mathrm{B}$ and $\mathrm{}\mathrm{\angle }\mathrm{E}\mathrm{G}\mathrm{F}$ such that $\mathrm{}\mathrm{D}$ and $\mathrm{}\mathrm{H}$ lie on sides $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{}\mathrm{F}\mathrm{E}$ of $△\mathrm{A}\mathrm{B}\mathrm{C}$ and $△\mathrm{E}\mathrm{F}\mathrm{G}$ respectively. If $△\mathrm{A}\mathrm{B}\mathrm{C}~△\mathrm{F}\mathrm{E}\mathrm{G}$, show that $△\mathrm{D}\mathrm{C}\mathrm{A}~△\mathrm{H}\mathrm{G}\mathrm{F}$