PRACTICE QUESTIONS COMPETITION WING

In a rhombus $BEST$, if $\angle B=60°$ and $BT=6$ cm, then find the length of the diagonal $TE.$

The following steps are involves in solving the above problem. Arrange them in sequential order.

(A)$\Rightarrow ∆BTE$ is an equilateral triangle.

(B) Join $T$ and $E$.

(C) $\text{In} ∆BET, BT=BE\Rightarrow \angle BTE=\angle BET=\frac{180°-60°}{2}=60° (\therefore \angle B=60°)$

(D) $TE=6$ cm.

The following steps are involved in construction of the incircle for the triangle $XYZ$ in which $\angle Y=90°$, $XZ=6$ cm and $YZ=4$cm.

Arrange them in sequential order from first to the last.

(A) Mark the foot of the perpendicular from $I$ onto $YZ$ as $D.$

(B) Construct the triangle $XYZ$ with $\angle Y=90°, XZ=6 \text{cm and} YZ=4 \text{cm.}$

(C) Draw a circle with $I$ as the centre and $ID$ as radius. This is the required incircle.

(D) Draw the bisectors of $\angle X, \angle Y \text{and }\angle Z$ and mark their point of concurrence as $I.$

In the above figure, $DEF$ is a triangle whose side $DF$ is produced to $G.$ $HF$and $HD$ are the bisectors of $\angle EFG$ and $\angle EDG$, respectively. If $\angle DEF=23\frac{1}{2}°,$ then $\angle DHF$(in degrees)$=$_____.

In the figure below, $\overline{AB}\parallel \overline{FC}, \overline{AE}\parallel \overline{BC} \text{and} \overline{AF}\parallel \overline{BD}.$ If $\angle F=x°, \angle C=y°, \angle EAB=k° \text{and} \angle ABP=p°,$ then which of the following options is correct?

In the above figure, $GH\parallel IJ \text{and} AC\parallel BD, AB\text{ and} CD$ are bisectors of $\angle EAH$ and $\angle FCJ$ respectively. Find $\angle ABD+\angle BDC,$ if $\angle BAC=3\angle BDC.$

In a rhombus $PQRS,$ the diagonals intersect at $O.$ Given that $\angle P=120°$ and $OP=3$ cm, what is the side of the rhombus?

In the figure above (not to scale), $\overline{AD}$ is the angle bisector of $\angle EAF, \angle AFC=110° \text{and} \angle DCF=20°.$ If $\angle DAF=30° \text{and }\angle EBD=10°,$ then $\angle AEB=$_____.

In the figure below, $m\parallel l\parallel n$ and $\overline{PT}\parallel \overline{QR}$. If $\angle TUV=x, \angle QRS=y \text{and} \angle QVW=z$, then which of the following is necessarily true?