Question
If two non - parallel sides of a trapezium are equal, prove that it is cyclic.
✓
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$$\triangle$CPB
$\angle A=\angle B $
$\angle ADK=\angle BCP\\ \angle ADK+90{}^\circ =\angle BCP+90{}^\circ $
$\angle ADC=\angle BCD$
$\angle D=\angle C$ ................(ii)
$\angle A+\angle B+\angle C+\angle D=360{}^\circ [\angle A= \angle B , \angle C= \angle D]$
$\angle B+\angle B+\angle D+\angle D=360{}^\circ \\ \angle B+\angle D=180{}^\circ $
Hence, ABCD is a cyclic trapezium.
To prove : ABCD is a cyclic trapezium.
Construction: Draw DK $\perp$ AB and CP$\perp$ AB
Proof: In it $\Delta ADK$ and $\Delta BCP$
AD=BC
DK=CP [Distance between || sides]
DKA=CBP [ each 90${}^\circ$ ]
By RHS criteria of congruency,
$\triangle$DKA $\cong$$\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.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In each of the following determine rational numbers a and b:
$\frac{\sqrt{11}-\sqrt7}{\sqrt11+\sqrt7}=\text{a}-\text{b}\sqrt{77}$
→$\frac{\sqrt{11}-\sqrt7}{\sqrt11+\sqrt7}=\text{a}-\text{b}\sqrt{77}$
A field is in the form of a rectangular length 18m and width 15m. A pit 7.5m long, 6m broad and 0.8m deep, is dug in a corner of the field and the earth taken out is spread over the remaining area of the field. Find out the extent to which the level of the field has been raised.
A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120m and 44m. If one of the sides of the parallelogram measures 66., find its corresponding altitude.
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following case:
f(x) = x2 - 1, x = 1, -1
f(x) = x2 - 1, x = 1, -1
The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank if its length and depth are respectively10m and 2.5m. (Given, 1000 litres = 1m3.)
Factorise:
2(x + y)2 - 9(x + y) - 5
2(x + y)2 - 9(x + y) - 5
Express the following decimals in the form $\frac{\text{p}}{\text{q}},$ where p, q are integers and $\text{q}\neq0.$
$0.\overline{2}$
→$0.\overline{2}$
In each of the following determine rational numbers a and b:
$\frac{5+3\sqrt3}{7+4\sqrt3}=\text{a}+\text{b}\sqrt3$
→$\frac{5+3\sqrt3}{7+4\sqrt3}=\text{a}+\text{b}\sqrt3$
In figure, arms BA and BC of $\angle\text{ABC}$ are respectively parallel to arms ED and EF of $\angle\text{DEF}.$ Prove that $\angle\text{ABC}=\angle\text{DEF}.$
→Locate $\sqrt{17}$ on the number line.
→