Criteria for Similarity of Triangles

If in two triangles, two sides of one triangle are proportional to the two sides of the other triangle, and an angle between the two proportional sides of first triangle is equal to an angle between the two proportional sides of second triangle then the triangles are similar.

In triangle $\mathrm{ABC}, \mathrm{AD}$ is perpendicular to side $\mathrm{BC}$ and ${\mathrm{AD}}^2=\mathrm{BD}\times \mathrm{DC}$, then the $\angle \mathrm{BAC}$ is

In the figure, $\mathrm{PQRS}$ is a parallelogram with $\mathrm{PQ}=16\mathrm{cm}$ and $\mathrm{QR}=10\mathrm{cm}.$ $\mathrm{L}$ is a point on $\mathrm{PR}$ such that $\mathrm{RL}:\mathrm{LP}=2:3.$ $\mathrm{QL}$ produced meets $\mathrm{RS}$ at $\mathrm{M}$ and $\mathrm{PS}$ produced at $\mathrm{N}.$ The length of $\mathrm{PN}$ is

In the given figure, $\mathrm{AB}\mathit{\parallel }\mathrm{DC}, \mathrm{BO}=6\mathrm{cm}$ and $\mathrm{DQ}=8\mathrm{cm}$; find: $\mathrm{BP}\times \mathrm{DO}.$

In the given figure, find $\mathrm{PT}$ given that $l_1\parallel l_2$

In $△MNP, \angle MNP=90°, \text{seg} NQ\perp \text{seg} MP$. If $MQ=9, QP=4$, then find $NQ$.

In a right-angled $△PQR$ angled at Q, prove that  $P{Q}^2+Q{R}^2=P{R}^2$

In Fig. $\angle ACB=90° \mathrm{and} CD\perp AB.$

Prove that $\frac{B{C}^2}{A{C}^2}=\frac{{\displaystyle BD}}{AD}$

In the figure, $△PQR$ and $△QRS$ are on the same side of the common base $QR$. Prove that line $SP$ is the perpendicular bisector of $QR$

In the figure, find the value of $x$ in terms of $a, b, c$.

If $AB\perp BC$, $DC\perp BC$ and $DE\perp AC$, then prove that $△CED~△ABC$.

Prove that four triangles obtained by joining midpoints of three sides of a triangle are similar to the original triangle.

The length of the shadow of a vertical pole $12 \mathrm{m}$ high from the ground is $8 \mathrm{m}$. At the same time the shadow of the vertical minaret is $56 \mathrm{m}$ long. If the height of the minaret is $k \mathrm{m}$, find the value of $k$.

A girl of height $90 \mathrm{cm}$ is walking away from the base of a long pole at a speed of $1.2 \mathrm{m}/\mathrm{s}$. If the bulb is put at the height of $3.6 \mathrm{m}$ above the ground, if the length of her shadow after $4$ seconds is $k \mathrm{m}$, find the value of $k$.

In the given figure $PQ\parallel RS$. Then prove that $△POQ~△SOR$.

You have to choose two similar triangles out of the following triangles which are similar to each other if $\angle CBE=\angle CAD$.

In the adjoining figure, $△PRQ~△TRS$, state which angles should be equal to each other in the given pair of similar triangles?

What do you understand by equiangular triangles? What relation may be among them?

If two sides of a triangle are proportional to two sides of another triangles and one angle of first triangle is equal to one angle of other triangle, then both triangles are similar. Is this statement true? Give reasons to justify your answer.

In $∆ABC$ and $∆DEF, \angle A=\angle D, \angle B=\angle F$, state whether $∆ABC~∆DEF$? Give proper reason in support to your answer.