If the angles of a quadrilateral are in the ratio $1: 2: 3: 4,$ then the measure of the smallest angle is

In the given figure, $ABC$ is an isosceles triangle with $AB = AC$. If $AE = AF$ and $\angle BAE=40°$, then the measure of the angle $FEC$ is

Perimeter of the rectangle in the figure is $36 centimetres$. $AC=\sqrt{164} centimetres.$ What is $AB+BC$? 

In the figure, $\angle B=90°, \angle C=44°$

What is the measure of $\angle A$?

The rod $AC$ of a TV disc antenna is fixed at right angles to the wall $AB$ and a rod $CD$ is supporting the disc as shown in the figure. If $AC=1.5 \mathrm{m}$ long and $CD=3 \mathrm{m}$, find $\sec \theta +\mathrm{cosec}\theta$

Perimeter of the rectangle in the figure is $36 centimetres$. $AC=\sqrt{164} centimetres.$ Find length of $AB$.

In triangle $ABC, \angle A=\angle B=30°, AC=4 centimetres$. What is the length of $BC$?

In the figure, chord $BC$ is extended to $P$. Tangent from $P$ to the circle is $PA$. $AQ$ is the bisector of $\angle BAC$. If $\angle PAC=x$ and $\angle PCA=y$, then prove that $\angle BAC=y–x$.

If $\sin \theta =\frac{12}{13}$, find the value of $\cos \theta \& \tan \theta .$

Let $ABCD$ be a square. An arc of a circle with $A$ as centre and $AB$ as radius is drawn inside the square joining the points $B$ and $D$. Points $P$ on $AB, S$ on $AD, Q$ and $R$ on arc $BD$ are taken such that $PQRS$ is a square. Further suppose that $PQ$ and $RS$ are parallel to $AC$. Then, $\frac{\text{ area }PQRS}{\text{ area }ABCD}$ is