Understanding Quadrilaterals - 8th math

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Angle Sum Property


Angle Sum Property of a Polygon

Angle Sum Property of a Triangle

The sum of all the three angles of a triangle is equal to 1800.

Example (i):

understanding quadrilaterals triangle

Here, angle A + angle B + angle C = 1800

If angle A = 600 and angle B = 450, then angle C = ?

We know that, sum of all the three angles of a triangle = 1800

Therefore, angle A + angle B + angle C = 1800

⇒ 600 + 450 + angle C = 1800

⇒ 1050 + angle C = 1800

⇒ angle C = 1800 – 1050

⇒ angle C = 750 Answer

Example (ii) : In the given triangle if two angles are 450 and 750, then find other angles.

understanding quadrilaterals angle sum property of a triangle

Here, angle c = 450 and angle y = 750

And, since angle c and angle n are vertically opposite angles hence they are equal.

Thus, angle n = angle c = 450

Now, we know that, angle c and angle m are supplementary

Thus, angle c + angle m = 1800

⇒ 450 + angle m = 1800

⇒ angle m = 1800 – 450

⇒ angle m = 1350

And, since angle m and angle g are vertically opposite angles hence they are equal.

Thus, angle m = angle g = 1350

And, since angle c and angle n are vertically opposite angles hence they are equal.

Thus, angle n = angle c = 450

And, since angle y and angle b are vertically opposite angles hence they are equal.

Thus, angle b = angle y = 750

Now, angle y and angle b are supplementary as they together form a straight line.

Thus, angle y + angle a = 1800

⇒ 750 + angle a = 1800

⇒ angle a = 1800 – 750

⇒ angle a = 1050

And, since angle a and angle d are vertically opposite angles hence they are equal.

Thus, angle a = angle d = 1050

Now, in the given triangle,

Angle a = 1050, angle c = 450

And we know that, sum of all the three angles of a triangle = 1800

Therefore, angle a + angle c + angle h = 1800

⇒ 1050 + 450 + angle h = 1800

⇒ 1500 + angle h = 1800

⇒ angle h = 1800 – 1500

⇒ angle h = 300

Now, since angle h and angle q are vertically opposite angle and hence are equal

Thus, angle q = angle h = 300

Now, angle h and angle r together form a straight line and hence are supplementary

Thus, angle h + angle r = 1800

⇒ 300 + angle r = 1800

⇒ angle r = 1800 – 300

⇒ angle r = 1500

Now, angle r and angle p are vertically opposite angles, thus they are equal

Thus, angle p = angle r = 1500

Angle sum property of a quadrilateral

Sum of all the four angles of a quadrilateral is equal to 3600

understanding quadrilaterals angle sum property of a quadrilateral

This means, in a quadrilateral

Angle a + angle b + angle a + angle b = 3600

Example (i): If three angles of a quadrilateral are 650, 950 and 800 respectively find the fourth angle of the given quadrilateral.

understanding quadrilaterals angle sum property of a quadrilateral

Solution

We know that, sum of all the four angles of a quadrilateral = 3600

Therefore, 650 + 950 + 800 + angle D = 3600

⇒ 2400 + angle D = 3600

⇒ angle D = 3600 – 2400

⇒ angle D = 1200 Answer

Example (ii): If four angles of a pentagon are 650, 750, 950 and 700 respectively, then find the fifth unknown angle.

understanding quadrilaterals angle sum property of a pentagon

Solution

We know that, Sum of all the angles of a polygon = 3600

Therefore, in the given figure of pentagon

650 + 750 + 950 + 700 + a = 3600

⇒ 3050 + a = 3600

⇒ a = 3600 – 3050

⇒ a = 550 Answer

Exterior Angle Sum Property of a Polygon

The sum of all the exterior angles of a polygon is equal to 3600.

understanding quadrilaterals angle sum property of a polygon

In the given figure of quadrilateral,

`/_a+/_b+/_c+/_d =360^o`

Example (i) If the four exterior angles of a pentagon are 500, 650, 600 and 750 respectively, then find the fifth exteriror angle of that pentagon.

Solution

Let the figure of pentagon is given below.

understanding quadrilaterals angle sum property of a pentagon-a

And as given in the question, angle a = 500

Angle b = 650

Angle c = 600

And angle d = 750

Then angle e = ?

We know that, sum of all the exterior angles of a pentagon = 3600

Therefore, in the given pentagon,

`50^o\+65^o\+60^o\+75^o\+/_e=360^o`

`:. /_e+250^0=360^o`

`=>/_e=360^0-250^o`

Thus, angle a = 1100 Answer

Example (ii) Find all the unknown angles in the given pentagon.

understanding quadrilaterals angle sum property of a pentagon-example

In ∠DAE and ∠ PAE

∠DAE and ∠ PAE both form a straight line and hence are supplementary angles.

Thus, ∠DAE + 750 = 1800

⇒ ∠DAE = 1800 – 750

⇒ ∠DAE = 1050

Now, since ∠ PAE and ∠ DAO are vertically opposite angles, and hence are equal

Thus, ᩐ DAO = ∠ PAE = 750

And since ∠ PAO and ∠ DAO are vertically opposite angles, and hence are equal

Thus, ∠ PAO = ∠ DAO = 1050

Now, in between angle AEF and angle GEB

∠ AEF and ∠GEB are vertically opposite angles and hence are equal

Thus, ∠ AEF = ∠ GEB = 650

Now, in between ∠GEB and ∠AEB

Both angles GEB and AEB together form a straight line and hence are supplementary.

Thus, ∠ GEB + ∠ AEB = 1800

⇒ 650 + ∠ AEB = 1800

⇒ ∠ AEB = 1800 – 650

⇒ ∠ AEF = 1150

Now, between angles AEB and angle FEG

Both the angles AEB and FEG are vertically opposite angles and hence are equal

Thus, ∠ AEF = ∠ FEG = 1150

Now, between angles EBC and angle HBJ

∠ EBC and ∠ HBJ are vertically opposite angles and hence are equal

Thus, ∠ EBC = ∠ HBJ = 1100

Now, between angles EBH and EBC

Both of the angles EBH and EBC form a straight line and hence are supplementary

Thus, ∠ EBH + ∠ EBC = 1800

⇒ ∠ EBH + 1100 = 1800

⇒ ∠EBH = 1800 – 1100

⇒ ∠ EBH = 700

Now between angles EBH and CBJ

Both the angles EBH and CBJ are vertically opposite angles.

Thus, ∠ EBH = ∠ EBJ = 700

Now, between angles DCB and LCK

Both the angles DCB and LCK are vertically opposite angles

Thus, ∠ECB = ∠ LCK = 1150

Between angles DCB and BCK

Angles DCB and BCK together form a straight line, and hence are supplementary angles.

Thus, ∠ DCB + ∠ BCK = 1800

⇒ 1150 + ∠ BCK = 1800

⇒ ∠ BCK = 1800 – 1150

⇒ ∠ BCK = 650

Between angles BCK and DCL

Angles BCK and DCL are vertically opposite angles, and hence are equal

Thus, ∠ BCK = ∠ DCL = 650

Between angles ADC and NDM

Angles ADC and NDM are vertically opposite angles, and hence are equal

Thus, ∠ ADC = ∠ NDM = 1200

Between angles ADC and CDM

Angles ADC and CDM together form a straight line, and hence are supplementary angles.

Thus, ∠ ADC + ∠ CDM = 1800

⇒ 1200 + ∠ CDM = 1800

⇒ ∠ CDM = 1800 – 1200

⇒ ∠ CDM = 600

Now, between angles CDM and ADN

Since, angles CDM and ADN are vertically opposite angles and hence are equal.

Thus, ∠ CDM = ∠ ADN = 600

Thus,

∠ DAE = 1050

∠ PAO = 1050

∠ OAD = 750

∠ FEA = 650

∠ AEB = 1150

∠ FEG = 1150

∠ EBH = 700

∠ HBJ = 1100

∠ CBJ = 700

∠ BCK = 650

∠ KCL = 1150

∠ DCL = 650

∠ CDM = 600

∠ NDM = 1200

And, ∠ ADN = 600

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