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Circles — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsCircles3 Marks
Question
If two non - parallel sides of a trapezium are equal, prove that it is cyclic.
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Answer

Given: Non parallel sides AD and BC of a trapeziumare equal.
To prove : ABCD is a cyclic trapezium.
Construction: Draw DK $\perp$ AB and CP$\perp$ AB 
Proof:  In it $\Delta ADK$ and $\Delta BCP$
AD=BC
DK=CP [Distance between || sides]
DKA=CBP [ each 90${}^\circ$ ]
By RHS criteria of congruency,
$\triangle$DKA $\cong$$\triangle$CPB
$\angle A=\angle B $
$\angle ADK=\angle BCP\\ \angle ADK+90{}^\circ =\angle BCP+90{}^\circ $
$\angle ADC=\angle BCD$
$\angle D=\angle C$ ................(ii)
$\angle A+\angle B+\angle C+\angle D=360{}^\circ [\angle A= \angle B , \angle C= \angle D]$
$\angle B+\angle B+\angle D+\angle D=360{}^\circ \\ \angle B+\angle D=180{}^\circ $
Hence, ABCD is a cyclic trapezium.

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