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In the world of Mathematics, a shape with four sides and endless possibilities is known as a Quadrilateral. From the tiles on a floor to the kites flying in the sky—these are all real-life examples of quadrilaterals. Students have already explored many of these shapes in earlier classes. In the NCERT Exemplar Class 9 Maths solutions chapter 8, they will learn about a Parallelogram and its properties. A parallelogram is a type of Quadrilateral whose opposite sides are best friends: They are always equal and run parallel in every situation. Students can use these Maths exemplar problems class 9 solutions as an extra practice resource after finishing the NCERT textbook exercises.
The main purpose of these class 9th Maths NCERT exemplar solutions of chapter 8 is to clarify all the necessary concepts and Parallelogram, as well as recalling other properties of Quadrilaterals. Experienced Careers360 teachers have curated these class 9NCERT exemplar maths solutions, adhering to the CBSE 2025-26 syllabus. Students are also encouraged to check out the PDF version of the NCERT solutions for class 9 maths for more in-depth grasp of the concepts.
Exercise: 8.1 |
Question:1
If three angles of a quadrilateral are $75^{\circ}, 90^{\circ}$ and $75^{\circ}$ then find the fourth angle
(A) 90o (B) 95o (C) 105o (D) 120o
Answer: [D] $120^{\circ}$
As we know that the sum of the angles of a quadrilateral is $360^{\circ}$.
i.e., $\angle 1+\angle 2+\angle 3+\angle 4=360^{\circ}$ …..(i)
Here $\angle 1=75^{\circ}$
$\angle 2=90^{\circ}$
$\angle 3=75^{\circ}$
Putting the values in equation (i), we get
$75^{\circ}+90^{\circ}+75^{\circ}+\angle 4=360^{\circ}$
$\angle 4=360^{\circ}-240^{\circ}$
$\angle 4=120^{\circ}$
Hence, option D is correct.
Question:2
A diagonal of a rectangle is inclined to one side of the rectangle at $25 ^{\circ}$. The acute angle between the diagonals is
(A) 550 (B) 500 (C) 400 (D) 250
Answer: [B] 50o
As we know, diagonals of a rectangle are equal in length.
$\therefore AC=BD$ { $\because$diagonals are equal}
$\frac{1}{2}AC=\frac{1}{2}BD$ {dividing both sides by 2}
$AO=BO$ { $\because$ O is midpoint of diagonal}
$\therefore \angle OBA=\angle OAB$
$\angle OAB=25^{\circ}$ {Given}
$\Rightarrow \angle OBA=25^{\circ}$
$\angle BOC=\angle OBA+\angle OAB$
[exterior angle is equal to the sum of two opposite interior angles]
$=25^{\circ}+25^{\circ}=50^{\circ}$
Hence, the actual angle between the diagonals is 500.
Hence, option B is correct.
Question:3
ABCD is a rhombus such that $\angle ACB = 40^{\circ}$. Then $\angle ADB$ is
(A) 400 (B) 450 (C) 500 (D) 600
Answer: [C] $50^{\circ}$
Given : ABCD is a rhombus such that $\angle ACB=40^{\circ}\Rightarrow \angle OCB=40^{\circ}$
To Find : $\angle ADB$
Since $AD \parallel BC$
$\Rightarrow \angle AOC=\angle ACB=40^{\circ}$ {alternate interior angles}
$\angle AOD=90^{\circ}$ {diagonals of a rhombus are perpendicular to each other}
We know that the sum of angles of a triangle is $180^{\circ}$.
In $\triangle AOD$,
$\angle AOD+\angle OAD+\angle ADO=180^{\circ}$
$90^{\circ}+40^{\circ}+\angle ADO=180^{\circ}$
$\angle ADO=180^{\circ}-90^{\circ}-40^{\circ}$
$\angle ADO=50^{\circ}=\angle ADB$
Hence, Option C is correct.
Question:4
The quadrilateral formed by joining the midpoints of the sides of a quadrilateral PQRS taken in order is a rectangle. If
(A) PQRS is a rectangle
(B) PQRS is a parallelogram
(C) Diagonals of PQRS are perpendiculars
(D) Diagonals of PQRS are equal
Answer:
According to the question, quadrilateral ABCD is formed by joining the midpoints of PQRS
$\Rightarrow$ If PQRS is a rectangle,
Here ABCD is not a rectangle because a rectangle is a four-sided polygon having all the internal angles $= 90^{\circ}$ and the opposite sides are equal in length.
$\Rightarrow$If PQRS is a parallelogram
Here ABCD is not a rectangle because a rectangle is a four-sided polygon having all the internal angles $= 90^{\circ}$ and the opposite sides are equal in length.
If diagonals of PQRS are perpendicular
Here, ABCD is a rectangle because here
$\angle A=\angle B=\angle C=\angle D=90^{\circ}$ and opposite sides we equal that is AB = DC and AD = BD
If diagonals of PQRS are equal
ABCD is not a rectangle; it is a square
Because here AB = BC = CD = AD and $\angle A=\angle B=\angle C=\angle D=90^{\circ}$
Here, we saw that if diagonals of PQRS are perpendicular to each other, then ABCD is a rectangle.
Option C is correct
(C) diagonals of PQRS are perpendicular
Question:5
(A) PQRS is a rhombus
(B) PQRS is a parallelogram
(C) diagonals of PQRS are perpendicular
(D) diagonals of PQRS are equal.
Answer:
According to the question, the quadrilateral ABCD is formed by joining the midpoints of PQRS.
If PQRS is a rhombus
Here, ABCD is not a rhombus because in a rhombus, angles need not be right angles.
But here $\angle A=\angle B=\angle C=\angle D=90^{\circ}$
ABCD is a square, not a rhombus.
If PQRS is a parallelogram
Here, ABCD is not a rhombus because in a rhombus, all sides will be equal.
Here, sides of quadrilateral ABCD are not equal;
It is not a rhombus
If diagonals of PQRS are perpendicular
Here, AB D is not a rhombus because in rhombus angles are not right angles and all sides are equal, but here AB = CD and BC = AD also $\angle A=\angle C=\angle D=90^{\circ}$
Hence, ABCD is a rectangle, not a square.
If diagonals of PQRS are equal
In the ABCD quadrilateral here, all sides are equal, and the angles of quadrilateral ABCD are not right angles, therefore, ABCD is a rhombus\
Option D is correct
(D) diagonals of PQRS are equal
Question:6
If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3:7:6:4, then ABCD is a
(A) rhombus (B) parallelogram
(C) trapezium (D) kite
Answer: [C]
Given : 3 : 7 : 5 : 4 is the ratio of angles A, B, C, and D in a quadrilateral.
Let the angles be 3x, 7x, 6x and 4x.
As we know that the sum of angles is $360^{\circ}$
$3x+7x+6x+4x=360^{\circ}$
$20x=360^{\circ}$
$x=\frac{360^{\circ}}{20}=18^{\circ}$
$A=3x=3\times 18=54^{\circ}$
$B=7x=7\times 18=126^{\circ}$
$C=6x=6\times 18=108^{\circ}$
$D=4x=4\times 18=72^{\circ}$
Now, draw the quadrilateral with the given angles.
Hence, $BC\parallel AD$ and the sum of co-interior angles is $180^{\circ}$
Hence ABCD is a trapezium
Question:7
If bisectors of $\angle A$ and $\angle B$of a quadrilateral ABCD intersect each other at P, of $\angle B$ and $\angle C$ at Q, of $\angle C$ and $\angle D$ at R and of $\angle D$and $\angle A$ at S, then PQRS is a
(A) rectangle
(B) rhombus
(C) parallelogram
(D) quadrilateral whose opposite angles are supplementary
Answer: [D] quadrilateral whose opposite angles are supplementary
First of all, let us draw the figure according to the question.
From the above diagram
$\angle QPS=\angle APB$ …..(1) (Vertically opposite angles)
In $\triangle APB=\angle APB+\angle PAB+\angle ABP=180^{\circ}$
$\angle APB=\frac{1}{2}\angle A+\frac{1}{2}\angle B=180^{\circ}$
$\angle APB=180-\frac{1}{2}(\angle A+\angle B)$ …..(2)
From equations 1 & 2
$\angle QPS=180-\frac{1}{2}(\angle A+\angle B)$ …..(3)
Similarly $\angle QRS=180-\frac{1}{2}(\angle C+\angle D)$ …..(4)
Add equations 3 and 4
$\angle QPS+\angle QRS=360^{\circ}-\frac{1}{2}(\angle A+\angle B+\angle C+\angle D)$
$=360^{\circ}-\frac{1}{2}(360^{\circ})$
$=360^{\circ}-180^{\circ}$
$=180^{\circ}$
$\therefore \angle QPS+\angle QRS= 180^{\circ}$
Hence, PQRS is a quadrilateral whose opposite angles are supplementary.
Question:8
If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form
(A) a square (B) a rhombus
(C) a rectangle (D) any other parallelogram
Answer: (C) a rectangle
Given: APB and COD are two parallel lines.
Construction: Let us draw the bisectors of the angles APQ, BPQ, CQP and PQD
Let the bisectors meet at points M and N
Since $APB\parallel CQD$
$\angle APQ=\angle PQD$ (Alternate angles)
and $\angle MPQ=\angle PQN$ (Alternate interior angles)
$\therefore PM\parallel QN$
Similarly $\angle BPQ=\angle PQC$ (Alternate angles)
$\Rightarrow PN\parallel QM$
So, quadrilateral PMQN is parallelogram
Since CQD is a line
$\therefore \angle CQD=180^{\circ}$
$\angle CQP+\angle PQD=180^{\circ}$
$2\angle MQP+2\angle NQP=180^{\circ}$
$2\left (\angle MQP+\angle NQP \right )=180^{\circ}$
$\angle MQP+\angle NQP=\frac{180}{2}$
$\Rightarrow \angle MQN=\frac{180}{2}$
$\Rightarrow \angle MQN=90^{\circ}$
Hence, PMQN is a rectangle.
Therefore, option (C) is correct.
Question:9
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is:
(A) a rhombus (B) a rectangle
(C) a square (D) any parallelogram
Answer: (D) Any parallelogram
Let ABCD be a rhombus and P, Q, R, and S be its midpoints.
In $\triangle ABC$, R is the midpoint of AB, and Q is the midpoint of BC.
$AC\parallel RQ$ and $RQ=\frac{1}{2}AC$ …..(1) {using mid-point theorem}
Similarly in $\triangle ADC$
$AC\parallel SP$and $SP=\frac{1}{2}AC$…..(2)
From equations 1 and 2
$SP\parallel RQ$ and $SP=RQ$
Similarly, when we take $\triangle ABD$ and $\triangle BDC$ then we get $SR\parallel PQ$ and $SR=PQ$ .
Hence, PQRS is a parallelogram
Therefore, option (D) is correct.
Question:10
D and E are the midpoints of the sides AB and AC of DABC, and O is any point on side BC. O is joined to A. If P and
Q are the mid-points of OB and OC, respectively, then DEQP is
(A) a square (B) a rectangle
(C) a rhombus (D) a parallelogram
Answer: (D) A parallelogram
By midpoint theorem
$DE\parallel BC$ …..(1)
i.e., $DE=\frac{1}{2}(BC)$
$DE=\frac{1}{2}(BP+PO+OQ+QC)$
$DE=\frac{1}{2}(2PO+2OQ)$
{$\because$ P and Q are midpoints of OB and OC}
$DE=PO+OQ$
$DE=PQ$ …..(2)
Now, in $\triangle AOC$, Q and E are the midpoints of OC and AC.
$\therefore EQ\parallel AO$ and $EQ=\frac{1}{2}AO$ …..(3) {Using mid-point theorem}
Similarly, in $\triangle ABO,PD\parallel AO$ and $PD=\frac{1}{2}AO$ …..(4)
From equations 3 and 4
$EQ\parallel PD$ and $EQ=PD$
From equations 1 and 3
$DE\parallel PQ$ and $DE= PQ$
Hence, DEQP is a parallelogram.
Therefore, option (D) is correct.
Question:11
The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if,
(A) ABCD is a rhombus
(B) diagonals of ABCD are equal
(C) diagonals of ABCD are equal and perpendicular
(D) diagonals of ABCD are perpendicular.
Answer: [C]
Given: ABCD is a quadrilateral and P, Q, R and S are the midpoints of sides of AB, BC, CD and DA. Then, PQRS is a square
$\therefore PQ=QR=RS=PS$ …..(1)
$PR=SQ$
Also, $PR=BC$ and $SQ=AB$
$\therefore AB=BC$
Thus, all sides are equal.
Hence, ABCD is either a square or a rhombus.
In $\triangle ADB$ by mid-point theorem
$SP\parallel DB$
$SP=\frac{1}{2}DB$ …..(2)
Similarly in $\triangle ABC,AQ=\frac{1}{2}AC$ …..(3)
From equation 1
$PS=PQ$
$\frac{1}{2}DB=\frac{1}{2}AC$
Thus, ABCD is a square, so the diagonals of a quadrilateral are also perpendicular.
Therefore, option (C) is correct.
Question:12
The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If $\angle DAC=32^{\circ}$ and $\angle AOB=70^{\circ}$, then $\angle DBC$is equal to
(A) $24^{\circ}$ (B) $86^{\circ}$ (C) $38^{\circ}$ (D) $32^{\circ}$
Answer: [C] $38^{\circ}$
Solution.
Given $\angle AOB=70^{\circ}$ and $\angle DAC=32^{\circ}$
$\angle ACB=32^{\circ}${$AD\parallel BC$ and AC is a transversal line}
$\angle AOB+\angle BOC=180^{\circ}$
$70^{\circ}+\angle BOC=180^{\circ}$
$\angle BOC=180^{\circ}-70^{\circ}=110^{\circ}$
In $\triangle BOC$, we have
$\angle BOC+\angle OCB+\angle CBO=180^{\circ}$
$110^{\circ}+32^{\circ}+\angle CBO=180^{\circ}$
$\angle CBO=180^{\circ}-110^{\circ}-32^{\circ}$
$\angle CBO=38^{\circ}$
$\angle DBC=\angle CBD=38^{\circ}$
Hence, option C is correct
Question:13
Which of the following is not true for a parallelogram?
(A) opposite sides are equal
(B) opposite angles are equal
(C) opposite angles are bisected by the diagonals
(D) diagonals bisect each other.
Answer: (C) Opposite angles are bisected by the diagonals
Parallelogram: A parallelogram is a special type of quadrilateral whose opposite sides and angles are equal.
Hence, we can say that the opposite sides of a parallelogram are of equal length (Option A), and the opposite angles of a parallelogram are of equal measure (Option B).
In a parallelogram, diagonals bisect each other (Option D), but opposite angles are not bisected by diagonals (Option C).
Therefore, option (C) is not true.
Question:14
D and E are the midpoints of the sides AB and AC, respectively, of $\triangle ABC$. DE is produced to F. To prove that CF is equal and parallel to DA, we need additional information, which is
(A) $\angle DAE = \angle EFC$
(B) AE = EF
(C) DE = EF
(D) $\angle ADE = \angle ECF.$
Answer: (C) DE = EF
Let us draw the figure according to the question.
AE = EC {E is the mid-point of AC}
Let DE = EF
$\angle AED=\angle FEC$ {vertically opposite angles}
$\therefore \triangle ADE\cong \triangle CFE$ {by SAS congruence rule}
$\therefore AD=CF$ {by CPCT rule}
$\therefore \angle ADE=\angle CFE$ {by CPCT}
Hence,$AD\parallel CF$
We need DE = EF.
Hence, option C is correct.
Exercise: 8.2 Total Questions: 14 Page Numbers: 75-77 |
Question:1
Answer: 6 and 4
Given: ABCD is a parallelogram
$OA = 3 cm$ and $OD = 2 cm$
As we know that the diagonals of a parallelogram bisect each other
$AC = 2AO = 2 \times 3 = 6$
$AC = 6 cm$
$DB = 2DO = 2 \times 2 = 4$
$DB = 4 cm$
Hence, the lengths of AC and BD are 6 cm and 4 cm, respectively.
Question:2
Answer: False
A parallelogram is a special type of quadrilateral. It has equal and parallel opposite sides and equal opposite angles.
Diagonals of a parallelogram bisect each other but don’t bisect in such a way that the angles formed between the diagonals is 90 degrees. In order for the diagonals to bisect perpendicular to each other all the sides should be equal in length which is not true when it comes to parallelograms.
Hence, the given statement is False.
Question:3
Answer: No
As we know that the sum of the angles of a quadrilateral is $360^{\circ}$
Here
$110^{\circ} + 80^{\circ} + 70^{\circ} + 95^{\circ}= 355^{\circ}\neq 360^{\circ}$
These angles cannot be the angles of a quadrilateral because the sum of these angles is not equal to 3600.
Question:4
Answer: Trapezium
Let the given quadrilateral be ABCD.
It is given that $\angle A + \angle D = 180^{\circ}$
We can see the sum of co-interior angles is $180^{\circ}$, which is a property of a trapezium.
Therefore, the given quadrilateral ABCD is a trapezium.
Question:5
All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?
Answer: Rectangle or Square
We know that the sum of all angles of a quadrilateral is $360^{\circ}$
If PQRS is a quadrilateral,
$\angle P+\angle Q+\angle R+\angle S=360^{\circ}$ …..(1)
It is given that all the angels are equal
i.e., $\angle P=\angle Q=\angle R=\angle S$
Put in equation 1, we get
$4\angle P=360^{\circ}$
$\angle P=\frac{360^{\circ}}{4}$
$P = 90^{\circ}$
Here, all the angles of a quadrilateral are $90^{\circ}$.
Hence given quadrilateral is either a rectangle or square. We cannot say which one it is, as nothing is given about the length of the sides.
Question:6
Answer: False
Given that diagonals of a rectangle are equal and perpendicular.
Rectangle: A rectangle is an equiangular quadrilateral, and all of its angles are equal.
Hence, diagonals of a rectangle are equal but not necessarily perpendicular to each other.
Let us consider a rectangle ABCD
Consider $\triangle ACD$ and $\triangle BCD$
AC = BD (opposite sides of a rectangle are equal)
$\angle C=\angle D$ $(90^{\circ})$
AB = CD (opposite sides of a rectangle are equal)
$\triangle ACD\cong \triangle BCD$ (SAS congruency)
So, AD = BC
Hence, diagonals are equal.
Also, $\angle CAD=\angle DBC$ …(i)
Similarly, we can prove that $\triangle ACB$ and $\triangle BDA$are congruent
Hence, $\angle ACB=\angle ADB$ …(ii)
Now, consider $\triangle AOC$ and $\triangle BOD$
$\angle CAD=\angle DBC$From (i)
$\angle ACB=\angle ADB$ From (ii)
$\angle AOC=\angle BOD$ vertically opposite angles
$\triangle AOC$ and $\triangle BOD$ are also congruent.
But we cannot prove that $\angle AOC=\angle BOD=90^{\circ}$
Hence, it is not necessary that diagonals will bisect each other at a right angle, so they are not necessarily perpendicular to each other
Hence, the given statement is False.
Question:7
Can all four angles of a quadrilateral be obtuse angles? Give a reason for your answer.
Answer: False
An obtuse angle is an angle greater than $90^{\circ}$
Let all angles of a quadrilateral be obtuse angles. Now, the smallest obtuse angle will be equal to $91^{\circ}$
Hence the sum of all angles $= 91^{\circ} + 91^{\circ} + 91^{\circ} + 91^{\circ}= 364^{\circ}$
So we can conclude that, if all the angles are greater than 90o, the sum of all the angles will be greater than $360^{\circ}$
But we know that the sum of all the angles in a quadrilateral is $360^{\circ}$.
Therefore, a quadrilateral can have a maximum of three obtuse angles.
Hence given statement is False.
Question:8
Answer: $3.5cm$
Given: AB = 5cm, BC = 8cm and CA = 7cm and D and E are the midpoints of AB and BC.
To Find: Length of DE
Using the mid-point theorem $DE\parallel AC$
And $DE=\frac{1}{2}AC$
$DE=\frac{1}{2}\times 7$
$DE=\frac{7}{2}cm$
$DE=3.5cm$
Hence answer is 3.5 cm
Question:9
Answer: True
Given: BDEF and FDCE are parallelograms.
If BDEF is a parallelogram then
$BD\parallel EF$ and BD = EF …..(1)
Also, if FDCE is a parallelogram, then
$CD\parallel EF$ and CD = EF …..(2)
From equations 1 and 2, we get,
BD = CD = EF
Hence given statement is True.
Question:10
In Figure, ABCD and AEFG are two parallelograms. If $\angle C = 55^{\circ}$, determine $\angle F$ .
Answer:
Given: ABCD and AEFG are two parallelograms
Given $\angle C = 55^{\circ}$
then $\angle A$ is also $55^{\circ}$. {opposite angles of a parallelogram are equal}
Also AEFG is a parallelogram then
$\angle A=\angle F=55^{\circ}$ {opposite angels of a parallelogram is equal}
Hence, $\angle F=55^{\circ}$is the required answer.
Question:11
Can all the angles of a quadrilateral be acute angles? Give a reason for your answer.
Answer: False
Solution.
An acute angle is an angle less than $90^{\circ}$
Let all angles of a quadrilateral be acute angles. Now, the smallest acute angle will be equal to $89^{\circ}$
Hence sum of all angles $= 89^{\circ} + 89^{\circ} + 89^{\circ} + 89^{\circ} = 356^{\circ}$
So we can conclude that, if all the angles are less than 90o, then the sum of all the angles will be less than $360^{\circ}$
But we know that the sum of all the angles in a quadrilateral is $360^{\circ}$.
Therefore, a quadrilateral can have a maximum of three acute angles.
Hence given statement is False.
Question:12
Can all the angles of a quadrilateral be right angles? Give a reason for your answer.
Answer: Yes
A right angle is an angle equal to $90^{\circ}$
Let all angles of a quadrilateral be right angles.
Hence sum of all angles $= 90^{\circ} + 90^{\circ} + 90^{\circ} + 90^{\circ} = 360^{\circ}$
So we can conclude that, if all the angles are equal to $90^{\circ}$ then the sum of all the angles will be equal to $360^{\circ}$
Also, we know that the sum of all the angles in a quadrilateral is $360^{\circ}$.
Therefore, a quadrilateral can have all angles as right angles.
Hence given statement is True.
Question:13
Answer:
If the diagonals of a quadrilateral ABCD bisect each other, then it parallelogram.
Sum of interior angles between two parallel lines is 1800, i.e.,
$\angle A+\angle B=180^{\circ}$
$\angle A=35^{\circ}$ (given)
$35^{\circ}+\angle B=180^{\circ}$
$\angle B=180^{\circ}-35^{\circ}$
$\angle B=145^{\circ}$
Question:14
Opposite angles of a quadrilateral ABCD are equal. If AB = 4 cm, determine CD.
Answer:
Given: Opposite angles of a quadrilateral ABCD are equal and AB = 4
So ABCD is a parallelogram, as we know that in a parallelogram, opposite angles are equal
Also, we know that opposite sides in a parallelogram are equal
$AB = CD = 4 cm$
Hence $CD = 4cm$
Exercise: 8.3 Total Questions: 10 Page Numbers: 78-79 |
Question:1
Answer: $84^{\circ}$
Solution.
Given that:
One of the angles of a quadrilateral is $108^{\circ}$, and the remaining three angles are equal.
Let each of the three equal angles be $x ^{\circ}$
As we know that the sum of angles of a quadrilateral is $360 ^{\circ}$
$108^{\circ} + x^{\circ} + x^{\circ} + x^{\circ} = 360^{\circ}$
$3x^{\circ} + 108^{\circ}= 360^{\circ}$
$3x^{\circ} = 360^{\circ} � 108^{\circ}$
$3x^{\circ}= 252^{\circ}$
$x=\frac{252^{\circ}}{3}$
$x=84^{\circ}$
Hence, each of the three equal angles is $84^{\circ}$.
Question:2
Answer: $\angle D=\angle C=135^{\circ}$
Solution.
Given: $AB\parallel DC$ and $\angle A=\angle E=45^{\circ}$
If $AB\parallel CD$ and BC is transversal then sum of co-interior angles is $180^{\circ}$
$\therefore \angle B+\angle C=180^{\circ}$
$45^{\circ}+\angle C=180^{\circ}$
$\angle C=180^{\circ}-45^{\circ}$
$\angle C=135^{\circ}$
Similarly
$\angle A+\angle D=180^{\circ}$
$45^{\circ}+\angle D=180^{\circ}$
$\angle D=180^{\circ}-45^{\circ}$
$\angle D=135^{\circ}$
Hence, angles C and D are $135^{\circ}$ each.
Question:
Answer: $60^{\circ},120^{\circ}$
Solution.
Given that the angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $60^{\circ}$.
In this figure, $\angle ADC$ and $\angle ABC$ are obtuse angles and DE and DF are altitudes.
In quadrilateral BEDF, $\angle BED$= $\angle BFD$ = $90^{\circ}$
$\angle FBE=360^{\circ}-(\angle FDE+\angle BED+\angle BFD)$
$=360^{\circ}-(60^{\circ}+90^{\circ}+90^{\circ})$
$=360^{\circ}-240^{\circ}$
$=120^{\circ}$
It is given that ABCD is a parallelogram
ADC = $120^{\circ}$
$\angle A+\angle B=180^{\circ}${Sum of interior angle $=180^{\circ}$}
$\angle A=180^{\circ}-B$
$\angle A=180^{\circ}-120^{\circ}$
$\angle A=60^{\circ}$
$\Rightarrow \angle C=\angle A=60^{\circ}$
Hence angles of the parallelogram are $60^{\circ}, 120^{\circ}$
Question:4
ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus.
Answer: $120^{\circ},120^{\circ},60^{\circ},60^{\circ}$
Solution.
Let the sides of a rhombus be AB = BC = CD = DA = x
Join DB
Here $\angle DLA=\angle DLB=90^{\circ}$ {DL is perpendicular bisector of AB}
$AL=BL=\frac{X}{2}$
DL = DL {common side of $\triangle ADL$ and $\triangle BDL$}
$\triangle ALD\cong \triangle BLD${by SAS congruence}
$AD = BD$
In $\triangle ADB$, $AD = AB = DB = x$
$\triangle ADB$ is an equilateral triangle.
$\therefore \angle ADB=\angle ABD=\angle A=60^{\circ}$
Similarly, $\triangle DCB$ is an equilateral triangle
$\angle BDC=\angle DBC=\angle C=60^{\circ}$
Also, $\angle A = \angle C$
$\angle D = \angle B$…..(1)
$\angle A+ \angle B+\angle C+\angle D=360^{\circ}$
$60^{\circ}+ B+60^{\circ}+B=360^{\circ}$
$2\angle B=\frac{240}{2}=120^{\circ}$
Hence answer is $120^{\circ},120^{\circ},60^{\circ},60^{\circ}$
Question:5
Answer:
Given : ABCD is a parallelogram and AE = CF
To prove: OE = OF
Proof: Join BD and CA
Here $OA = OC$ and $OD = OB$ [ABCD is parallelogram]
$\Rightarrow$$OA = OC$…..(1)
And $AE = CF$…..(2) {given}
$OA -AE = OC - CF${by subtracting 2 from 1}
$OE = OF$
Hence, BFDE is a parallelogram.
Question:6
Answer:
Given: ABCD is trapezium and $AB\parallel DC$
Constructions: Join AC, which intersects EF at O
Proof: In $\triangle ADC$, E is the mid-point of line AD and $OE \parallel CD$.
By the mid-point theorem, O is the mid-point of AC
∴ In $\triangle CBA$, as O is the mid-point of AC
$OF\parallel AB$ (mid-point theorem)
Hence, again by the mid-point theorem, F is the mid-point of BC.
Hence proved
Question:7
Answer:
Given : In $\triangle ABC$, $PQ\parallel AB$and $PR\parallel AC$ and $PQ\parallel BC$.
To Prove : $BC=\frac{1}{2}QR$
Proof : In quadrilateral BCAR, $BR\parallel CA$ and $BC\parallel RA$
Hence, it is a parallelogram
BC = AR ……(1)
In quadrilateral BCQA,$BC\parallel AQ$ and $AB\parallel QC$
Hence, it is also a parallelogram
BC = AQ …..(2)|
Adding equations 1 and 2, we get
$2BC = AR + AQ$
$2BC = RQ$
$BC=\frac{1}{2}QR$
Hence proved
Question:8
Answer:
D and E are midpoints of BC and AC, respectively, so using the mid-point theorem:
$\Rightarrow DE=\frac{1}{2}AB$ …..(1)
E and F are midpoints of AC and AB, so using the mid-point theorem:
$\Rightarrow EF=\frac{1}{2}BC$ …..(2)
F and D are midpoints of AB and BC, so using the mid-point theorem:
$\Rightarrow FD=\frac{1}{2}AC$ …..(3)
It is given that $\triangle ABC$ is an equilateral triangle
$\Rightarrow AB=BC=CA$
$\frac{1}{2}AB=\frac{1}{2}BC=\frac{1}{2}CA$ {dividing by 2}
Using 1, 2 and 3, we get
DE = EF = FD
Hence, DEF is an equilateral triangle.
Hence proved
Question:9
Answer:
Given: ABCD is a parallelogram and AP = CQ
To Prove: AC and PQ bisect each other.
Proof :
Here ABCD is a parallelogram
$\Rightarrow AB\parallel DC$
$\Rightarrow AP\parallel QC$
It is given that $AP = CQ$
Thus APCQ is a parallelogram.
And we know that diagonals of a parallelogram bisect each other.
Hence, AC and PQ bisect each other.
Hence proved
Question:10
Given: ABCD is a parallelogram, P is a mid-point of BC such that $\angle BAP = \angle DAP$.
To prove : $AD = 2CD$
Proof : Here ABCD is a parallelogram
$\therefore AD\parallel BC$ and AB is transversal
$\angle A+\angle B=180^{\circ}$ {sum of co-interior angles}
$\angle B=180-\angle A$ …..(1)
In $\triangle ABP$,
$\angle PAB+\angle BPA+\angle B=180^{\circ}${using angle sum property of triangle}
Putting the values,
$\frac{1}{2}\angle A+\angle BPA+(180^{\circ}-\angle A)=180^{\circ}$
$\angle BPA-\frac{\angle A}{2}=0$
$\angle BPA=\frac{\angle A}{2}$ …..(2)
$\Rightarrow \angle BPA=\angle BAP$
$AB = BP$ {opposite sides of equal triangle}
$2AB = 2BP$ {multiply both sides by 2}
$2AB = BC$ { P is midpoint of BC}
$2CD = AD$ { ABCD is parallelogram $AB = CD$ and $BC = AD$}
Hence proved
Exercise: 8.4 Total Questions: 18 Page Numbers: 82-83 |
Question:1
Answer:
Given: Here, ABC is an isosceles triangle, and ADEF is a square inscribed in $\triangle ABC$.
To prove : $CE = BE$
Proof: In isosceles $\triangle ABC$ and $AB = AC$ …..(1)
$\angle A=90^{\circ}$
Here, ADEF is a square
$AD = AF$ …..(2)
Subtract equation 2 from 1
$AB - AD = AC - AF$
$BD = CF$ …..(3)
Now in $\triangle BDE$and $\triangle CFE$
$BD = CF${from equation 3}
$\angle CFE = \angle EDB$ {$90^{\circ}$ each}
$DE = EF$ {side of a square}
$\triangle CFE\cong \triangle BDE$ {SAS congruence rule}
$CE = BE$ {by CPCT}
Hence, vertex E of the square bisects the hypotenuse BC.
Hence proved
Question:2
Answer: 4 cm
Given: ABCD is a parallelogram in which $AB = 10 cm$ and $AD = 6 cm$.
Construction: In parallelogram ABCD, draw the bisector of $\angle A$ which meets DC in point E.
Produce AE and BC so that they meet at point F.
Also, produce AD to H and join H and F.
Here ABFH is a parallelogram
$HF\parallel AB$
And $\angle AFH = \angle FAB$ …..(1) {alternate interior angles}
$AB = HF$ (opposite sides of a parallelogram)
$AF = AF$ (Common Side)
$\Rightarrow \triangle HAF\cong \triangle FAB$…..(2) { \ SAS Congruency}
Now, $\angle HAF=\angle FAB$ …..(3) (EA is the bisector of $\angle A$)
$\Rightarrow \angle AFH=\angle HAF$ { using 1 and 3 }
So, $HF = AH$ {Sides opposite to equal angles are equal}
$HF = AB = 10 cm$
$AH = HF = 10 cm$
$AD + DH = 10 cm$
$DH = 10 -AD \Rightarrow 10 -6$
$DH=4cm$
Because FHDC is a parallelogram
opposite sides are equal
$DH = CF = 4 cm$
Hence, the answer is $4 cm$
Question:3
Answer:
Given: ABCD is a quadrilateral in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA.
To prove : PQRS is a rhombus.
Proof :
To $\triangle ADC$, S and R are the midpoints of AD and DC, respectively.
By the mid-point theorem, we get
$SR\parallel AC$ and $SR=\frac{1}{2}AC$ …..(1)
In $\triangle ABC$, P and Q are the midpoints of AB and BC, respectively.
Then, by the mid-point theorem, we get
$PQ\parallel AC$ and $PQ=\frac{1}{2}AC$ …..(2)
From equations 1 and 2, we get
$SR=PQ=\frac{1}{2}AC$ …..(3)
Similarly in $\triangle BCD$we get
$RQ\parallel BD$ and $RQ=\frac{1}{2}BD$ …..(4)
And in $\triangle BAD$
$SP\parallel BD$and $SP=\frac{1}{2}BD$ …..(5)
Using equations 4 and 5, we get
$RQ=SP=\frac{1}{2}BD$
It is given that AC = BD
$RQ=SP=\frac{1}{2}AC$......(6)
Now equate equations 3 and 6 we get
$SR = PQ = SP = RQ$
It shows that all sides of a quadrilateral PQRS are equal.
Hence, PQRS is a rhombus.
Hence Proved.
Question:4
Answer:
Given: ABCD is a quadrilateral in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA and $AC \perp BD$. To prove: PQRS is a rectangle
It is given that $AC \perp BD$
$\angle COD =\angle AOD = \angle AOB = \angle COB = 90^{\circ}$
In $\triangle ADC$, S and R are the midpoints of AD and DC, so by the mid-point theorem.
$SR \parallel AC$ and $SR=\frac{1}{2}AC$ …..(1)
Similarly in $\triangle ABC$,
$PQ \parallel AC$ & $PQ =\frac{1}{2}AC$ …..(2)
Using 1 and 2
$SR=PQ=\frac{1}{2}AC$ …..(3)
Similarly, $SP\parallel RQ$ & $SP=RQ=\frac{1}{2}BD$ ……(4)
In quadrilateral EOFR,
$OE \parallel FR, OF \parallel ER$
$\therefore \angle EOF = \angle ERF = 90^{\circ}$
{$\angle COD = 90^{\circ}$ because $AC \perp BD$ and opposite angles of a parallelogram are equal}
In quadrilateral RSPQ
$\angle SRQ = \angle ERF = \angle SPQ = 90^{\circ}$ {Opposite angles in a parallelogram are equal}
$\angle RSP + \angle SRQ + \angle RQP + \angle QRS = 360^{\circ}$
$90^{\circ} + 90^{\circ} + \angle RSP + \angle RQP = 360^{\circ}$
$\angle RSP + \angle RQP = 180^{\circ}$
$\angle RSP =\angle RQP = 90^{\circ}$
If all the angles in a parallelogram are $90^{\circ}$, then that parallelogram is a rectangle.
So, PQRS is a rectangle.
Hence Proved
Question:5
Answer:
Given: ABCD is a parallelogram and P, Q, R, and S are the mid-points of sides AB, BC, CD and AD. Also, AC = BD and $AC \perp BD$.
To prove: PQRS is a square.
Proof: In $\triangle ADC$, S and R are the midpoints of sides AD and DC by the mid-point theorem.
$SR \parallel AC$ and $SR =\frac{1}{2}AC$ …..(1)
In $\triangle ABC$, P and Q are the midpoints of AB and BC, respectively. Therefore, by the mid-point theorem
$PQ\parallel AC$ and $PQ=\frac{1}{2}AC$ …..(2)
From equations 1 and 2, we get
$PQ\parallel SR$ and $PQ=\frac{1}{2}AC$ …..(3)
Similarly, in $\triangle BCD$, $RQ\parallel BD$ and R, Q are midpoints of CD, CB respectively, Therefore, by mid-point theorem
$RQ=\frac{1}{2}BD=\frac{1}{2}AC$ {Given BD = AC} …..(4)
And in $\triangle ABD$, $SP\parallel BD$ and S, P are midpoints of AD, AB respectively. Therefore, by the mid-point theorem
$SP=\frac{1}{2}BD=\frac{1}{2}AC${Given AC = BD …..(5)
From equations 4 and 5, we get
$SP=RQ=\frac{1}{2}AC$ …..(6)
From equations 3 and 6, we get
$PQ = SR = SP = RQ$
Thus, all sides are equal
In quadrilateral EOFR,
$OE \parallel FR, OF \parallel ER$
$\therefore \angle EOF =\angle ERF = 90^{\circ}$
{$\angle COD = 90^{\circ}$ because $AC \perp BD$ and opposite angles of a parallelogram are equal}
In quadrilateral RSPQ
$\angle SRQ = \angle ERF = \angle SPQ = 90^{\circ}$ {Opposite angles in a parallelogram are equal}
$\angle RSP + \angle SRQ + \angle RQP + \angle QRS = 360^{\circ}$
$90^{\circ} + 90^{\circ}+ \angle RSP +\angle RQP = 360^{\circ}$
$\angle RSP +\angle RQP = 180^{\circ}$
$\angle RSP = \angle RQP = 90^{\circ}$
All the sides are equal, and all the interior angles of the quadrilateral are $90^{\circ}$
Hence, PQRS is a square.
Hence Proved
Question:6
If a diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.
Answer:
Let ABCD be a parallelogram and diagonal AC bisect the angle A and $\angle CAB = \angle CAD$
To Prove: ABCD is a rhombus
Proof: Here, it is given that ABCD is a parallelogram.
$AB\parallel CD$ and AC is transversal
$\angle CAB=\angle ACD${alternate interior angles} …. (1)
Also, $AD\parallel BC$ and AC is a transversal
$\angle CAD=\angle ACB${alternate interior angle} …. (2)
Now it is given that $\angle CAB = \angle CAD$
So from equations (1) and (2)
$\Rightarrow \angle ACD=\angle ACB$…. (3)
Also, $\angle A=\angle C${opposite angles of a parallelogram}
$\frac{\angle A}{2}=\frac{\angle C}{2}$ {dividing by 2}
$\angle DAC=\angle DCA${from 1 and 2}
$CD=AD$ {sides opposite of equal angles in $\triangle ADC$are equal}
$AB=CD$ & $AD=BC$ {$\because$ opposite sides of a parallelogram}
Hence,
$AB = BC = CD = AD$
Thus, all sides are equal, so ABCD is a rhombus.
Hence Proved.
Question:7
Given: ABCD is a parallelogram, P and Q are the mid-points of AB and CD
To Prove: PRQS is a parallelogram.
Proof: Since ABCD is a parallelogram $AB\parallel CD \Rightarrow AP\parallel QC$
Also, AB = DC …..(1)
$\frac{1}{2}AB=\frac{1}{2}DC${dividing by 2}
$AP = QC$
{$\because$P and Q are the mid-points of AB and DC}
As, $AP = QC$ and $AP\parallel QC$
Thus APCQ is a parallelogram
$AQ\parallel PC$ or $SQ\parallel PR$ …..(2)
$AB\parallel CD$ or $BP\parallel DQ$
$AB=DC$
$\frac{1}{2}AB=\frac{1}{2}DC${dividing both sides by 2}
$BP=QD$ {$\because$P and Q are the mid-points of AB and DC}
$BP\parallel QD$ and $BQ\parallel PD$
So, BPDQ is a parallelogram
$PD\parallel BQ$ or $PS\parallel QR$ …..(3)
From equations 2 and 3
$SQ\parallel RP$and $PS\parallel QR$
So, PRQS is a parallelogram.
Hence Proved
Question:9
Given: $AB \parallel DE$ and $AC \parallel DF$
Also, AB = DE and AC = DF
To prove: $BC\parallel EF$and BC = EF
Proof: $AB \parallel DE$ and AB = DE
$AC \parallel DF$ and AC = DF
In ACFD quadrilateral
$AC\parallel FD$ and AC = FD …..(1)
Thus ACFD is a parallelogram
$AD\parallel CF$ and AD = CF …..(2)
In ABED quadrilateral
$AB\parallel DE$ and AB = DE …..(3)
Thus ABED is a parallelogram
$AD\parallel BE$ and AD = BE …..(4)
From equation 2 and 4
AD = BE = CF and $AD \parallel CF \parallel BE$ …..(5)
In quadrilateral BCFE, BE = CF and $BE\parallel CF$ {from equation 5}
So, BCFE is a parallelogram BC = EF and $BC\parallel EF$
Hence Proved.
Question:10
Answer:
Given: In $\triangle ABC$, AD is a median and E is the mid-point of AD
To Prove: $AF=\frac{1}{3}AC$.
Construction: Draw $DP\parallel EF$ and $\triangle ABC$ as given
Proof: In $\triangle ADP$, E is mid-point of AD and $EF\parallel DB$
So, F is mid-point of AP {converse of midpoint theorem}
In $\triangle FBC$, D is mid-point of BC and $DP\parallel BF$
So, P is mid-point of FC {converse of midpoint theorem}
Thus, AF = FP = PC
$AF+FP+PC=AC=3AF$
$\therefore AF=\frac{1}{3}AC$
Hence Proved
Question:11
Answer:
Given: In a square ABCD; P, Q, R and S are the mid-points of AB, BC, CD and DA.
To Prove: PQRS is a square
Construction: Join AC and BD
Proof: Here, ABCD is a square
AB = BC = CD = AD
P, Q, R, and S are the midpoints of AB, BC, CD and DA.
In $\triangle ADC$,
$SR\parallel AC$
$SR=\frac{1}{2}AC${using mid-point theorem} …..(1)
In $\triangle ABC$,
$PQ\parallel AC$
$PQ=\frac{1}{2}AC${using mid-point theorem} …..(2)
From equations 1 and 2
$SR\parallel PQ$ and $SR= PQ=\frac{1}{2}AC$ …..(3)
Similarly, $SP\parallel BD$ and $BD\parallel RQ$
$SP=\frac{1}{2}BD$and $RQ=\frac{1}{2}BD$ {using mid-point theorem}
$SP=RQ=\frac{1}{2}BD$
Since diagonals of a square bisect each other at right angles.
AC = BD
$\Rightarrow SP=RQ=\frac{1}{2}AC$…..(4)
From equations 3 and 4
SP = PQ = SP = RQ {All sides are equal}
In quadrilateral OERF
$OE\parallel FR$ and $OF\parallel ER$
$\angle EOF=\angle ERF=90^{\circ}$
In quadrilateral RSPQ
$\angle SRQ=\angle ERF=\angle SPQ=90^{\circ}$ {Opposite angles in a parallelogram are equal}
$\angle RSP + \angle SRQ + \angle RQP + \angle QRS = 360^{\circ}$
$90^{\circ} +90^{\circ}+ \angle RSP + \angle RQP = 360^{\circ}$
$\angle RSP + \angle RQP = 180^{\circ}$
$\angle RSP = \angle RQP = 90^{\circ}$
Since all the sides are equal and the angles are also equal, PQRS is a square.
Hence Proved
Question:12
Answer:
Given: ABCD is a trapezium in which $AB\parallel CD$, E and F are the mid-points or sides AD and BC.
Constructions: Joint BE and produce it to meet CD at G.
Draw BOD which intersects EF at O
To Prove: $EF \parallel AB$ and $EF=\frac{1}{2}(AB+CD)$
Proof: In $\triangle GCB$, E and F are respectively the mid-points of BG and BC, then by mid-point theorem.
$EF \parallel GC$
But, $GC\parallel AB$ or $CD \parallel AB${given}
$\therefore EF\parallel AB$
In $\triangle ADB$, $AB \parallel EO$ and E is the mid-point of AD. Then, by the mid-point theorem, O is the mid-point of BD.
$EO=\frac{1}{2}AB$ ……(1)
In $\triangle BDC$, $OF \parallel CD$and O is the mid-point of BD
$OF=\frac{1}{2}CD$ …..(2)
Adding 1 and 2, we get
$EO+OF=\frac{1}{2}AB+\frac{1}{2}CD$
$EF=\frac{1}{2}(AB+CD)$
Hence Proved
Question:13
Answer:
Given: Let ABCD be a parallelogram and AP, BR, CQ, DS be the bisectors of
$\angle A, \angle B, \angle C$and $\angle D$ respectively.
To Prove: Quadrilateral PQRS is a rectangle.
Proof: Since ABCD is a parallelogram
Then $DC\parallel AB$ and DA is transversal.
$\angle A+\angle B=180^{\circ}${sum of co-interior angles of a parallelogram}
$\frac{1}{2}\angle A+\frac{1}{2}\angle B=90^{\circ}${Dividing both sides by 2}
$\angle PAD+\angle PDA=90^{\circ}$
$\Rightarrow \angle APD=90^{\circ}$ {$\because$ sum of all the angles of a triangle is 1800}
$\angle QRS=90^{\circ}$
Similarly,
$\angle RBC+\angle RCB=90^{\circ}$
$\Rightarrow \angle BRC=90^{\circ}$
$\angle QRS=90^{\circ}$
Similarly,
$\angle QAB+\angle QBA=90^{\circ}$
$\Rightarrow \angle AQB=90^{\circ}${$\because$ sum of all the angles of a triangle is 1800}
$\angle RQP=90^{\circ}${$\because$ vertically opposite angles}
Similarly,|
$\angle SDC+\angle SCD=90^{\circ}$
$\Rightarrow \angle DSC=90^{\circ}${$\because$ sum of all the angles of a triangle is 1800}
$\angle RSP=90^{\circ}${$\because$ vertically opposite angles}
Thus, PQRS is a quadrilateral whose all angles are $90^{\circ}$
Hence, PQRS is a rectangle.
Hence proved
Question:14
Answer:
Given: ABCD is a parallelogram whose diagonals bisect each other at O.
To Prove: PQ is bisected at O.
Proof: In $\triangle ODP$ and $\triangle OBQ$
$\angle BOQ=\angle POD${Vertically opposite angles}
$\angle OBQ=\angle ODP${interior angles}
OB = OD {given}
$\triangle ODP\cong \triangle OBQ${by ASA congruence}
OP = OQ {by CPCT rule}
So, PQ is bisected at O
Question:15
ABCD is a rectangle in which diagonal BD bisects $\angle B$. Show that ABCD is a square.
Answer:
Given: In a rectangle ABCD, diagonal BD bisects B
To Prove: ABCD is a square
Construction: Join AC
Proof:
Given that ABCD is a rectangle. So all angles are equal to $90^{\circ}$
Now, BD bisects $\angle B$
$\angle DBA=\angle CBD$
Also,
$\angle DBA+\angle CBD=90^{\circ}$
So,
$2\angle DBA=90^{\circ}$
$\angle DBA=45^{\circ}$
In $\triangle ABD$,
$\angle ABD+\angle BDA+\angle DAB=180^{\circ}$
(Angle sum property)
$45^{\circ}+\angle BDA+90^{\circ}=180^{\circ}$
$\angle BDA=45^{\circ}$
In $\triangle ABD$,
AD = AB (sides opposite to equal angles in a triangle are equal)
Similarly, we can prove that BC = CD
So, AB = BC = CD = DA
So ABCD is a square.
Hence proved
Question:16
Answer:
Given: In $\triangle ABC$, D, E and F are respectively the mid-points of the sides AB, BC and CA.
To prove: $\triangle ABC$ is divided into four congruent triangles.
Proof: Using the given conditions, we have
$AD=BD=\frac{1}{2}AB,BE=EC=\frac{1}{2}BC,AF=CF=\frac{1}{2}AC$
Using mid-point theorem
$EF\parallel AB$ and $EF=\frac{1}{2}AB=AD=BD$
$ED\parallel AC$ and $ED=\frac{1}{2}AC=AF=CF$
$DF\parallel BC$ and $DF=\frac{1}{2}BC=BE=EC$
In $\triangle ADF$ and $\triangle EFD$
AD = EF
AF = DE
DF = FD {common side}
$\triangle ADF\cong \triangle EFD$ {by SSS congruence}
Similarly, we can prove that,
$\triangle DEF\cong \triangle EDB$
$\triangle DEF\cong \triangle CFE$
So, $\triangle ABC$ is divided into four congruent triangles
Hence proved
Question:17
Answer:
Given: Let ABCD be a trapezium in which $AB\parallel DC$ and let M and N be the midpoints of diagonals AC and BD.
To Prove: $MN\parallel AB\parallel CD$
Proof: Join CN and produce it to meet AB at E
In $\triangle CDN$ and $\triangle EBN$we have
DN = BN {N is mid-point of BD}
$\angle DCN=\angle BEN${alternate interior angle}
$\angle CDN=\angle EBN${alternate interior angles}
$\triangle CDN\cong \triangle EBN$
DC = EB and CN = NE {by CPCT}
Thus, in $\triangle CAE$, the points M and N are the midpoints of AC and CE, respectively.
$MN\parallel AE$ {By mid-point theorem}
$MN\parallel AB\parallel CD$
Hence Proved
Question:18
Answer:
Given: In a parallelogram ABCD, P is the mid-point of DC
To Prove: DA = AR and CQ = QR
Proof: ABCD is a parallelogram
BC = AD and $BC\parallel AD$
Also, DC = AB and $DC\parallel AB$
P is mid-point of DC
$DP=PC=\frac{1}{2}DC$
Now $QC\parallel AP$ and $PC\parallel AQ$
So APCQ is a parallelogram
$AQ=PC=\frac{1}{2}DC$
$\frac{1}{2}AB=BQ$…..(1) {$\because$ DC = AB}
In $\triangle AQR$ & $\triangle BQC$AQ = BQ {from equation 1}
$\angle AQR=\angle BQR${vertically opposite angles}
$\angle ARQ=\angle BCQ${alternate angles of transversal}
$\triangle AQR\cong \triangle BQC${AAS congruence}
AR = BC {by CPCT}
BC = DA
AR = DA Also, CQ = QR {by CPCT} Hence Proved .
The chapter on Quadrilaterals in the NCERT exemplar Class 9 Maths solutions chapter 8 covers the below-mentioned topics:
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Class 9 Maths NCERT exemplar solutions chapter 8 Quadrilaterals provides adequate practice and information on this chapter in a student-friendly process flow that can be further applied in higher classes. Here are some more points on why students should check these solutions.
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Yes, the rhombus is a special kind of parallelogram which diagonally intersects each other perpendicularly. All sides of the rhombus are equal, and the square is a special case of a rhombus.
The diamond in the playing card is the rhombus and sometimes rhombus is also called a diamond. We know that rhombus is a special case of parallelogram therefore diamond shape will be a parallelogram.
For any polygon of n sides, the sum of interior angle will be equal (n -2 )180°. For triangle n is equal to 3 hence the sum of interior angles will be 180°. For quadrilaterals n is equal to 4 hence the sum of interior angle will be 360°
NCERT exemplar Class 9 Maths solutions chapter 8 pdf download link enables the students to download/view the pdf version of the solutions in an offline environment.
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