Question

In the figure(not drawn to scale), ABCD is a rhombus, FHI is an equilateral triangle and CEFG is a trapezium with EC parallel to FG. DCFI, BCG and GFH are straight lines. Find x and y respectively.

• A. ${100}^o,{46}^o$
• B. ${48}^o,{60}^o$
• C. ${72}^o,{120}^o$
• D. ${66}^o,{58}^o$

Answer 1

Answer: (B)

${48}^o,{60}^o$

given that :

ABCD is a rhombus

$\Rightarrow \angle BAD=\angle DCB={72}^{\circ }$ opposite angles in a rhombus are equal

$\angle GCF=\angle DCB={72}^{\circ }$ vertically opposite angles

$\Delta FIH$ is a equilateral triangle

$\Rightarrow \angle HFI={60}^{\circ }$

$\Rightarrow \angle GFC=\angle HFI={60}^{\circ }$ vertically opposite angles

given that EC parallel to FG and FC is transversal

$\Rightarrow \angle GFC=\angle y={60}^{\circ }$ ailternate interior angles

In $\Delta GFC$

sum of all angles of a triangle $={180}^{\circ }$

$\Rightarrow \angle x+\angle GFC+\angle GCF={180}^{\circ }$

$\Rightarrow \angle x+{72}^{\circ }+{60}^{\circ }={180}^{\circ }$

$\Rightarrow \angle x={48}^{\circ }$