Understanding Quadrilaterals - 8th math

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

cm

⇒ x = 6 cm

Now, side GU = side SN

⇒ 3y – 1 = 26 cm

⇒ 3y = 26 + 1

⇒ 3y = 27 cm

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 .

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 in the given figure. (If you find m∠R, is there more than one method to find m∠P?)

Solution

Given, 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