Q: Diagonals of a parallelogram are perpendicular to each other. Is this statement true?

There are six general and important properties of parallelograms to know:There are six important properties of parallelograms to know:Opposite sides are congruent (AB = DC).Opposite angels are congruent (D = B).Consecutive angles are supplementary (A + D = 180°).If one angle is right, then all angles are right.The diagonals of a parallelogram bisect each other. Each diagonal of a parallelogram separates it into two congruent triangles.These properties does not say anything about angle between diagonals . So diagonals \$may\$ \$be\$ perpendicular \$may\$ \$be\$ \$not\$.Special type of parallelogram with perpendicular diagonals are square and rhombus.So correct answer is option is C

Q: The following figure is a parallelogram.Find \$x + y\$. (Lengths are in cm)

\$ GUNS\$ is a parallelogram.The opposite sides of the parallelogram are equalSo \$3x=18\Longrightarrow x=6\$ and \$3y-1=26\Longrightarrow y=9\$ So \$x=6\$ and \$y=9\$ \$\Rightarrow x+y=15\$

Q: State true or false:Is it possible to have a regular polygon whose each exterior angle is \$\displaystyle 20^{\circ}\$

\$\text{Each exterior angle of regular polygon of n sides is given by}\dfrac{360}{n} degree\$\$\text{According to question}\$\$\dfrac{360}{n} degree=20^0\$\$\Rightarrow n=\dfrac{360}{20}\$\$\Rightarrow n=18\$\$\text{Clearly there is a polygon of sides 18 whose each exterior angle is 20}^0\$

Q: Find the unknown angle \$x\$ in the above figure.

The given figure is a regular pentagon since it has \$5\$ sides and all are equal We know that the sum of the internal angles \${ S }_{ n }\$ of an \$n\$ sided convex polygon is \$\left( n-2 \right) \times { 180 }^{ o }\$Here \$n=5,\$ i.e \${ S }_{ n }=\left( 5-2 \right) \times { 180 }^{ o }={ 540 }^{ o }.......(i)\$In a regular polygon, all the sides are equal and all the angles are equal to each other\$\therefore 5x=540^{o}\$\$\Rightarrow x=108^{o}\$

Q: Given a parallelogram ABCD. Prove that \$OC = OA\$

Given:- \$ ABCD\$ is a parallelogram with \$AC \& BD\$ as diagonals.\$ ABCD\$ is a parallelogram. \$ \therefore AD\parallel BC\Longrightarrow \angle BDA=\angle CBD\$ and \$ \angle DAC=\angle ACB\$ ...... (alternate angles) \$ or \angle ODA=\angle CBO\$ and \$\angle DAO=\angle OCB........(i)\$Now between \$\Delta ADO\$ and \$\Delta COB\$ we have \$ AD=BC\$ ...... (opposite side of a parallelogram) \$ \angle ODA=\angle CBO\$ and \$\angle DAO=\angle OCB\$ ..... by \$(i)\$\$ \therefore\$ By ASA test \$\Delta AOD\$ and \$\Delta COB\$ are congruent.\$ \therefore OC=OA\$

Question 1

Diagonals of a parallelogram are perpendicular to each other. Is this statement true?

Accepted Answer

There are six general and important properties of&#10;parallelograms to know:There are six important properties of&#10;parallelograms to know:Opposite sides are congruent (AB = DC).Opposite angels are congruent (D = B).Consecutive angles are supplementary (A + D =&#10;180&#176;).If one angle is right, then all angles are right.The diagonals of a parallelogram bisect each other.&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;Each diagonal of a parallelogram separates it into&#10;two congruent triangles.These properties does not say anything about angle between diagonals . So diagonals $may$ $be$ perpendicular $may$ $be$ $not$.Special type of parallelogram with perpendicular diagonals are square and rhombus.So correct answer is option is C

Question 2

The following figure is a parallelogram.Find $x + y$. (Lengths are in cm)

Accepted Answer

$ GUNS$ is a parallelogram.The opposite sides of the parallelogram are equalSo $3x=18\Longrightarrow x=6$ and $3y-1=26\Longrightarrow y=9$ So $x=6$ and $y=9$ $\Rightarrow x+y=15$

Question 3

State true or false:A square has its opposite sides parallel, so, it is a parallelogram.

Accepted Answer

A parallelogram is a closed figure having 4 sides and 4 angles. It's opposite sides are equal and parallel. A square is a closed figure which has 4 sides & 4 angles.  It's all sides are equal. It's opposite sides are parallelSo, we can say that it's opposite sides are equal and parallel. So, a square is a parallelogram.

Question 4

State true or false:Is it possible to have a regular polygon whose each exterior angle is $\displaystyle 20^{\circ}$

Accepted Answer

$\text{Each exterior angle of regular polygon of n sides is given by}\dfrac{360}{n} degree$$\text{According to question}$$\dfrac{360}{n} degree=20^0$$\Rightarrow n=\dfrac{360}{20}$$\Rightarrow n=18$$\text{Clearly there is a polygon of sides 18 whose each exterior angle is 20}^0$

Question 5

Identify the number of convex polygons from the given figures.

Accepted Answer

In a convex polygon, none of the internal angles is a reflex angle. Instead, all the internal angles are less than $180^{\circ}$. Hence, Figure 2 is the only convex polygon.

Question 6

Find the unknown angle $x$ in the above figure.

Accepted Answer

The given figure is a regular pentagon since it has $5$ sides and all are equal We know that the sum of the internal angles ${ S }_{ n }$ of an $n$ sided convex polygon is $\left( n-2 \right) \times { 180 }^{ o }$Here $n=5,$ i.e ${ S }_{ n }=\left( 5-2 \right) \times { 180 }^{ o }={ 540 }^{ o }.......(i)$In a regular polygon, all the sides are equal and all the angles are equal to each other$\therefore 5x=540^{o}$$\Rightarrow x=108^{o}$

Question 7

Given a parallelogram ABCD. Prove that $OC = OA$

Accepted Answer

Given:- $ ABCD$ is a parallelogram with $AC \&  BD$ as diagonals.$ ABCD$ is a parallelogram. $ \therefore  AD\parallel BC\Longrightarrow \angle BDA=\angle CBD$ and $ \angle DAC=\angle ACB$ ...... (alternate angles) $ or  \angle ODA=\angle CBO$ and $\angle DAO=\angle OCB........(i)$Now between $\Delta ADO$ and $\Delta COB$ we have $ AD=BC$  ...... (opposite side of a parallelogram) $ \angle ODA=\angle CBO$ and $\angle DAO=\angle OCB$ ..... by $(i)$$ \therefore$  By ASA test $\Delta AOD$ and $\Delta COB$ are congruent.$ \therefore  OC=OA$

Question 8

Identify the number of concave polygons from the given figures?

Accepted Answer

In a concave polygon, at least one of the internal angles is a reflex angle i.e greater than 180.    So, figure 1 is the only concave polygon.

Question 9

Is it possible to have a regular polygon whose each interior angle is $138^{\circ}$State true or false:

Accepted Answer

Let number of polygon = n (which must be an integer)Sum of all interior angles is $ n \times 138^o $Sum of all interior angles of a polygon is $ 180^o (n-2) $$ => n \times 138^o = 180^o (n-2) $$ => 42^o n = 360^o $ $=> n = \dfrac{60}{7} $ Their is no possibility to have a regular polygon whose each interior angle is $ 138^o  $

Question 10

Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that)What can you say about the angle sum of a convex polygon with the number of sides (n) = 8?

Accepted Answer

\( After\quad analysing\quad the\quad given\quad table\quad it\quad is\quad observed\quad that\quad when\ (i)\quad numbre\quad of\quad sides\quad of\quad the\quad polygon=3\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles={ 180 }^{ o }=\left( 3-2 \right) \times { 180 }^{ o }\ (ii)\quad numbre\quad of\quad sides\quad of\quad the\quad polygon=4\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles={ 360 }^{ o }=\left( 4-2 \right) \times { 180 }^{ o }\ (iii)\quad numbre\quad of\quad sides\quad of\quad the\quad polygon=5\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles={ 540 }^{ o }=\left( 5-2 \right) \times { 180 }^{ o }\ (iv)\quad numbre\quad of\quad sides\quad of\quad the\quad polygon=6\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles={ 720 }^{ o }=\left( 6-2 \right) \times { 180 }^{ o }\ \therefore \quad By\quad induction,\quad it\quad can\quad be\quad concluded\quad that\ When\quad the\quad numbre\quad of\quad sides\quad of\quad the\quad polygon=n\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles=\left( n-2 \right) \times { 180 }^{ o }\ \therefore \quad For\quad a\quad 8\quad sided\quad polygon\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad sum\quad of\quad the\quad internal\quad angles=\left( 8-2 \right) \times { 180 }^{ o }={ 1080 }^{ o }\ Ans-(i)\quad If\quad the\quad numbre\quad of\quad sides\quad of\quad a\quad polygon=n\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \longrightarrow sum\quad of\quad the\quad internal\quad angles=\left( n-2 \right) \times { 180 }^{ o }\ \quad \quad \quad \quad (ii){ 1080 }^{ o }\ \ \ \ \ \  \)

Understanding Quadrilaterals

Maths
11228 Views

Diagonals of a parallelogram are perpendicular to each other. Is this statement true?

The following figure is a parallelogram.
Find $x + y$ . (Lengths are in cm)

State true or false:
A square has its opposite sides parallel, so, it is a parallelogram.

State true or false:
Is it possible to have a regular polygon whose each exterior angle is $2 0^{\circ}$

Identify the number of convex polygons from the given figures.

Find the unknown angle $x$ in the above figure.

Given a parallelogram ABCD. Prove that $O C = O A$

Identify the number of concave polygons from the given figures?

Is it possible to have a regular polygon whose each interior angle is $13 8^{\circ}$
State true or false:

Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that)
What can you say about the angle sum of a convex polygon with the number of sides (n) = 8?

Understanding Quadrilaterals

Maths 11228 Views

Diagonals of a parallelogram are perpendicular to each other. Is this statement true?

The following figure is a parallelogram. Find x+y. (Lengths are in cm)

State true or false:A square has its opposite sides parallel, so, it is a parallelogram.

State true or false:Is it possible to have a regular polygon whose each exterior angle is 20∘

Identify the number of convex polygons from the given figures.

Find the unknown angle x in the above figure.

Given a parallelogram ABCD. Prove that OC=OA

Identify the number of concave polygons from the given figures?

Is it possible to have a regular polygon whose each interior angle is 138∘State true or false:

Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that)What can you say about the angle sum of a convex polygon with the number of sides (n) = 8?

Maths
11228 Views

The following figure is a parallelogram.
Find $x + y$ . (Lengths are in cm)

State true or false:
A square has its opposite sides parallel, so, it is a parallelogram.

State true or false:
Is it possible to have a regular polygon whose each exterior angle is $2 0^{\circ}$

Find the unknown angle $x$ in the above figure.

Given a parallelogram ABCD. Prove that $O C = O A$

Is it possible to have a regular polygon whose each interior angle is $13 8^{\circ}$
State true or false:

Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that)
What can you say about the angle sum of a convex polygon with the number of sides (n) = 8?