Angles of a Parallelogram

Q.

A parallelogram which can be inscribed in a circle must necessarily be a :

1. rectangle

2. Rhombus

3. Trapezoid

4. Square

Q.

Adjacent angles of a parallelogram are __________.

Q. Question 161
Find the values of x and y in the following parallelogram.

Q. Question 133
Two adjacent angles of a parallelogram are in the ratio 1 : 3. Find its angles.

Q. In a parallelogram, the opposite angles are ___________ .

1. complementary
2. supplementary
3. equal
4. add up to ${360}^{\circ }$

Q. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

1. False
2. True

Q. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

1. False
2. True

Q. Ratio of two adjacent sides of a parallelogram is 3 : 4, and its perimeter is 112 cm. Find the length of its each side.

1. None of the above
2. 9 cm , 12 cm​
3. 15 cm , 20 cm​
4. 24 cm , 32 cm​

Q. ABCD is a parallelogram and $\angle A={75}^{\circ }.$ Find $\angle B$.

1. ${45}^{\circ }$
2. ${105}^{\circ }$
3. ${60}^{\circ }$
4. ${75}^{\circ }$

Q. $ABCD$ is trapezium in which $AB||CD$ and $AD||BC$, then BD=AC

1. True
2. False

Q. Suppose $E$, $F$ are the midpoints respectively of the oblique sides $PS$, $QR$ of the trapezium $PQRS$. Prove that $EF$ is parallel to $SR$ and $EF=\frac{PQ+RS}{2}$.

Q. Two adjacent angles of a parallelogram are ${\left(3x-4\right)}^{\circ }$ and ${\left(3x+16\right)}^{\circ }$. Find the measure of each of its angles.

Q. $ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$. Then which of the following alternatives is correct?

1. $CQ=DA$
2. $AP=CQ$
3. $DQ=PB$
4. $AP=BC$

Q.

In figure the sides $AB$ and $AC$ of $\Delta ABC$ are produced to point $E$ and $D$ repectively. If bisector $BO$ and $CO$ of $\angle CBE$ and $\angle BCD$ respectively meet a point $O$, then $\angle BOC={90}^{\circ }-\frac{1}{2}\angle BAC$

Q. If one angle of a parallelogram is a right angle, then prove that it is a rectangle.

Q. If PQRS is a parallelogram with PR $=$ QS, then PQRS is a

1. rectangle
2. parallelogram
3. rhombus
4. trapezium

Q. In a parallelogram the angle bisectors of two adjacent angles intersect at angles.

1. acute
2. right
3. obtuse

Q. $P$ and $Q$ are the points of trisection of the diagonal $BD$ of a parallelogram $ABCD.$ Prove that $CQ$ is parallel to $AP$. Prove also that $AC$ bisects $PQ$.

Q.

If ABCD is a parallelogram, then the angle bisectors of $\angle A$ and $\angle D$ meet at ‘O’. The measure of $\angle AOD$ is ${45}^{\circ }$​

1. 
2. 

Q. $P$ and $Q$ are point of trisection of the diagnoal $BD$ of a parallelogram $ABCD$. Prove that $CQ||AP$.

Q. When proving that a quadrilateral is a parallelogram by using slopes you must find

1. the slopes of all four sides
2. the lengths of all four sides
3. the slopes of two opposite sides
4. none of these
5. both the lengths and slopes of all four sides

Q. Show that the bisectors of angles of a parallelogram form a rectangle

Q. In the below given figure of parallelogram, AF and BE are angle bisectors of $\angle A$ and $\angle B$ respectively. Find the measure of $x$.

$\left[4Marks\right]$

Q. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

1. False
2. True

Q. Assertion :If $\hat{u}$ and $\hat{v}$ are unit vectors inclined at an angle $\alpha$ and $\hat{a}$ is a unit vector bisecting the angle between them, then $\hat{a}=\frac{\hat{u}+\hat{v}}{2\cos \left(\alpha /2\right)}$ Reason: If $ABC$ is an isosceles triangle with $AB=AC=1$, then the vector representing bisector of angle $A$ is $\frac{\overrightarrow{AB}+\overrightarrow{AC}}{2}$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Adjacent angles of a parallelogram are ___.

1. supplementary
2. complementary
3. at right angle
4. not related

Q.

In figure, if line PQ and RS intersect at point T, such that ∠ PRT = 40 ° , ∠ RPT = 95 ° and ∠ TSQ = 75 °, find ∠ SQT.

Q. In a parallelogram ABCD, P is a point in interior of parallelogram ABCD. If $ar\left(|{|}^{gm}ABCD\right)=18c{m}^2$, then $\left[ar\right(△APD)+ar(△CPB\left)\right]$ is:

1. $9c{m}^2$
2. $12c{m}^2$
3. $18c{m}^2$
4. $15c{m}^2$

Q. In the given figure, ABCD is a parallelogram. And X and Y are points on the diagonal BD such that $DX=BY$. Prove that:
(i) AXCY is a parallelogram
(ii) AX=CY, AY=CX
(iii) $△AYB\cong △CXD$
(iv) $△AXD\cong △CYB$