Relationships between Triangles

In figure shown below, PSDA is a $\parallel \mathrm{gm}$. Points Q and R are taken on PS such that $PQ=QR=RS$ and $PA\parallel QB\parallel RC$. Prove that area $\left(PQE\right)=$area $\left(CFD\right)$.

In a trapezium ABCD, $AB\parallel DC$ and L is the mid point of BC (Figure shown below). Through L, a line $MN\parallel AD$ is drawn to meet AB in M and DC produced in N. Prove that area $\left(ABCD\right)=$area $\left(AMND\right)$.

In Fig., $PQRS$ is a square and $LM$ is parallel to $PR$. $N$ is the mid point of $LM$. Prove that $QN$ on producingwill pass through $S$.

In Fig., $PQRS$ is a square and $LM$ is parallel to $PR$. $N$ is the mid point of $LM$. Prove that $QN$ bisects $\angle PQR$

In Fig., $PQRS$ is a square and $LM$ is parallel to $PR$. $N$ is the mid point of $LM$. Prove that $PL = RM$

In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. 

Triangle $ABC$ is similar to triangle $PQR$. If bisector of $\angle \mathrm{A}$ meets $BC$ at point $D$ and bisector of $\angle \mathrm{P}$ meets $QR$ at a point $M$ prove that $\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AD}}{\mathrm{PM}}$.

If $P$is any point on the bisector of angle $BAC$, prove that the perpendicular drawn from $P$to $AB$ and $AC$are equal. See Fig. 

In Fig., $ABCD$ is a ${\parallel }^{gm}$ and $X$ is the mid point of $BC$. The line $AX$ produced meets $DC$ produced at $Q$.The parallelogram $ABPQ$ is completed. Prove that $DC=CQ=QP$

In Fig.,$ABCD$  is a ${\parallel }^{gm}$and $X$ is the mid point of $BC$. The line $AX$ produced meets $DC$ produced at $Q$.The parallelogram $ABPQ$  is completed. Prove that $∆ABX$ $\cong$$△QCX$

In a parallelogram $ABCD$, $M$ is the mid point of $AC$. Fig. , $X$ and $Y$ are the points on $AB$ and $DC$ respectively such that $AX=CY$. Prove that $XMY$ is a straight line. 

In given figure $AB, EF$ and $DC$ are perpendiculars to the straight line $BFC$. If $AB=x, CD=y$ and $EF=z$, then prove that $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ 

The sides AB and AC of $△ABC$ are equal and D and E are points on AC and AB respectively such that $\angle ADB and \angle AEC$ are right angles.

Prove that the triangles AEC and ADB are congruent.

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 Fig. $10.32$, it is given that $AB=BC=25m. If AE=7 m and CD=24 m,$ find the length of DE and also show that the triangle ABE and $△BDC$ are congruent.

From any point on the bisector of an angle of a triangle, perpendiculars are drawn to the arms of the angle, prove that the perpendiculars are equal. 

In Fig.,$PA=PC$  and $AB=BC$. Prove that $∆APD\cong ∆CPD$

In Fig., $AM$ is a median of $∆ABC$. $BL$ and $CN$ are perpendiculars drawn from $B$ and $C$ respectively of $AM$ and $AM$ produced. Prove that $BL=CN$.

In Fig, in parallelogram $ABCD$ the angles $A$and $C$ are obtuse. Points $M$ and $N$ are taken on the diagonal $BD$ such that the angles $NAD$ and $MCB$ are right angles. Prove that $CM=AN$.