Understanding Quadrilaterals - 8th math
NCERT Exercise 3.3-part-2
Understanding Quadrilaterals NCERT Exercise 3.3 Question (6) Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Solution
Given, two adjacent angles of a parallelogram are equal in measure.
Then find the each angles of the parallelogram.
We know that, a square and rectangle has equal measure of adjacent angles.
And, a square and rectangle has angles equal to 900
Thus, each angles of the parallelogram = 900 Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (7) The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Solution
In the given figure of parallelogram HOPE
Given, ∠EHP = 400
And, ∠ O = 700
Then, measure of angles x, y and z = ?
Between angle EAP and angle HPO
Both of the angles EAP and HPO are pair of alternate interior angles, and we know that, pair of alternate interior angles are equal in measure.
Thus, ∠HPO = ∠EHP
⇒ y = 400
Now, between ∠HOP and 700
Both the angles form a linear pair of angle, and hence are supplementary
Thus, ∠HOP + 700 = 1800
⇒ ∠HOP = 1800 – 700
⇒ ∠HOP = 1100
Now, in triangle HOP
We know that, sum of all the three angles of a triangle = 1800
Thus, z + ∠HOP + y = 1800
⇒ z + 1100 + 400 = 1800
⇒ z + 1500 = 1800
⇒ z = 1800 – 1500
⇒ z = 300
Now, between angles PEH and HOP
Both PEH and HOP are the opposite angles of the given parallelogram, and hence are equal
Thus, ∠PEH = ∠HOP
⇒ x = 1100
Thus, x = 1100, y = 400 and z = 300 Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (8) The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
(i)
Solution
In the given parallelogram GUNS
Side GU = 3y – 1
And Side, SN = 26 cm
And, GS = 3x
And, UN = 18 cm
Thus, x and y = ?
We know that, opposite sides of a parallelogram are equal.
Thus, in the given parallelogram,
Side GS = Side UN
⇒ 3x = 18 cm
`=>x=18/3` cm
⇒ x = 6 cm
Now, side GU = side SN
⇒ 3y – 1 = 26 cm
⇒ 3y = 26 + 1
⇒ 3y = 27 cm
`=>y = 27/3` cm
⇒ y = 9 cm
Thus, x = 6 cm and y = 9 cm Answer
(ii)
Solution
In the given parallelogram, let the point where both the diagonals meet is O
Given, RUNS is a parallelogram and RN and SU are diagonals of the parallelogram.
And, OS = 20 cm
OU = y + 7
And, RO = 16 cm
And, ON = x + y
Thus, value of x and y = ?
Now, we know that, diagonals of parallelogram bisect one another.
Thus, in the given parallelogram in the diagonal SU
OU = OS
[∵ O is the middle point of diagonal SU as RN another diagonal bisects it.]
⇒ y + 7 = 20
⇒ y = 20 – 7
⇒ y = 13 cm
Now, in diagonal RN
ON = RO
[∵ O is the middle point of diagonal RN as SU another diagonal bisects it.]
⇒ x + y = 16
After substituting the value of y = 13 cm, we get
x + 13 cm = 16 cm
⇒ x = 16 cm – 13 cm
⇒ x = 3 cm
Thus, x = 3 cm and y = 13 cm Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (9)
In the above figure both RISK and CLUE are parallelograms. Find the value of x.
Solution
In the given figure let meeting point of EC and IS is O.
Now, In parallelogram RISK
Between ∠ RKE and ∠ ISK
Both ∠ SKR and ∠ ISK are adjacent angles of the parallelogram RISK, and hence are supplementary.
Thus, ∠ SKR + ∠ ISK = 1800
⇒ 1200 + ∠ ISK = 1800
⇒ ∠ ISK = 1800 – 1200
⇒ ∠ ISK = 600
Now, in parallelogram CLUE
Between ∠ CLU and ∠ UEC
Both ∠CLU and ∠UEC are opposite angles of parallelogram CLUE and hence are equal.
Thus, ∠ UEC = ∠ CLU
⇒ ∠ UEC = 700
Now, in triangle EOS
We know that, sum of measure of all the three angles of a triangle = 1800
Thus, ∠ SEO + ∠ OSE + x = 1800
⇒ 700 + 600 + x = 1800
⇒ 1300 + x = 1800
⇒ x = 1800 – 1300
⇒ x = 500 Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (10) Explain how this figure is a trapezium. Which of its two sides are parallel?
Solution
In the given quadrilateral,
Given, ∠ L = 800
And, ∠ M = 1000
Thus, find the parallel sides.
Let side NM and KL are parallel
And, ML is the transversal passing through these parallel lines.
Now, Between angle LMN and angle NMS
∠ LMN and ∠ NMS are linear pair, and thus are supplementary
Thus, ∠LMN + ∠ NMS = 1800
⇒ 1000 + ∠ NMS = 1800
⇒ ∠ NMS = 1800 – 1000
⇒ ∠ NMS = 800
Now, In between ∠ KLM and ∠ NMS
Since, ∠ KLM = ∠ NMS = 800
And we know that, pair of corresponding angles is equal, then line are parallel.
Here, ∠ KLM and ∠ NMS are pair of corresponding angles and are equal, thus, Line KL and NM are parallel.
Thus, in the given trapezium side KL and NM are parallel. Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (11) Find m∠C in given figure if `bar(AB)||bar(DC)`.
Solution
In the given quadrilateral ABCD
Given, AB||DC
And, ∠ B = 1200
Thus, ∠C = ?
In the given quadrilateral ∠B and ∠C are adjacent angles.
And, we know that sum of measure of adjacent angles of a quadrilateral = 1800
Thus, ∠ B + ∠ C = 1800
⇒ 1200 + ∠ C = 1800
⇒ ∠ C = 1800 – 1200
⇒ ∠ C = 600
Thus, m∠C = 600 Answer
Understanding Quadrilaterals NCERT Exercise 3.3 Question (12) Find the measure of ∠P and ∠S if `bar(SP)||bar(RQ)` in the given figure. (If you find m∠R, is there more than one method to find m∠P?)
Solution
Given, `bar(SP)||bar(RQ)` in the given quadrilateral PQRS
And, ∠Q = 1300
And, ∠R = 900
Then, ∠ P and ∠ S = ?
We know that, adjacent angles of a quadrilateral are supplementary, that is their sum is equal to 1800
Thus, ∠Q + ∠P = 1800
⇒ 1300 + ∠P = 1800
⇒ ∠P = 1800 – 1300
⇒ ∠P = 500
Similarly, ∠R and ∠S are adjacent angles of the given quadrilateral. And hence they are supplementary, this means measure of their sum = 1800
Thus, ∠R + ∠S = 1800
⇒ 900 + ∠S = 1800
⇒ ∠S = 1800 – 900
⇒ ∠S = 900
More method to find m∠P
In the given quadrilateral,
∠Q = 1300
And, ∠R = 900
Now, since ∠R and ∠S are linear pair, thus measure of their sum = 1800
Hence, ∠S = 900
Now, we know that, sum of measure of all the four angles of a quadrilateral = 3600
Thus, In the given quadrilateral,
∠P + ∠Q + ∠R + ∠S = 3600
⇒ ∠P + 1300 + 900 + 900 = 3600
⇒ ∠P + 3100 = 3600
⇒ ∠P = 3600 – 3100
⇒ ∠P = 500
Thus, m∠P = 500 Answer
Reference: