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12After studying this lesson, you can get a good understanding of, drawing and measuring the interior angles and exterior angles of plane calculating the values of exterior angles using the values of interior angles.

You can see above, several types of triangles which you have learned earlier. You have also learnt that the closed plane formed with 3 straight line segments is called a triangle.

Let us learn more about triangles.

12.1 Interior angles of a triangle

A triangle ABC is given in the This has three sides and three angles.The three sides are AB, AC and BC.The three angles are ABC, BCA and CAB.

A

B

C

Equilateral triangle

Isosceles triangle

Right- angled triangle

Acute-angled triangle

Scalene triangle

Obtuse-angled triangle

Triangles

The sum of the magnitudes of the three interior angles of a triangle is1800 '

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Activity 12.1 Draw any triangle you like on a sheet of paper. Cut out the three angles as given in the

Now paste the three angles on a paper as given in the

Examine whether AOB is a straight line by using a ruler.

You have learnt that the sum of the angles on a straight line is 180o

. p + q + r = 1800

Accordingly you can see that, the sum of the three interior angles of a triangle is 1800.Draw any triangle and verify whether this is true.Draw a large triangle in your exercise book and measure the three angles using a protractor. Do you get the sum of the three interior angles as 1800 ?

Example 1In the triangle LMN,

and are equal. Find the value of

R

P

Q

p

r

q

A O Br p q

xo

72o 83o

56o

xo xo xo

xo

xo

(i) (ii) (iii) (iv)

46o

72o

L

N

M

Exercise 12.1 (1) Find the value of x in each of the following triangles.

MLN = 72o

LMN LNM LMN.

MLN + LMN + LNM = 180o

72o + 2LMN = 180o

2LMN = 108o

LMN = 54o

^^ ^

^ ^ ^^

^^

>

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Into how many triangles is the quadrilateral ABCD separated by joining ‘A’ and ‘C’? It is

separated into two triangles.

In ABC"

p + q + r = 180o--------- (1)Similarly in ADC"

t + s + u = 180o--------- (2)

Now by adding (1) and (2)

p + q + r + t + s + u = 180o + 180o

(p + t) + q + (r + s) + u = 360o

Draw any quadrilateral and measure the interior angles using a protractor to examine whether the sum is 360o.

54o

48o yo

134o

yoyo 85o

y46o

30oyo

64o

57o

26o

xo

xo xo

xo

2 xo2 xo

4xo

2xo

3xo

(v) (vi) (vii)

B

A

D

C

p t

q

u

rs

The sum of the interior angles of a quadrilateral is . 360o

12.2 The interior angles of a quadrilateral

^3& Find the value of y in each of the following

^2& What are the values of the angles of a triangle out of the following? (i) 80 , 60 , 40 (ii) 115 , 110 , 15 (iii) 50 , 72 , 48 (iv) 68 , 102 , 10 (v) 110 , 20 , 50 (vi) 45 , 45 , 45

o o o

o o o

o o o

o o o

o o o

o o o

o

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Activity 12.2

Draw any quadrilateral. Cut out the angles separately. Paste the angles about a point so that the

straight edges are together and with no gaps. Do you get the sum of the angles of the

quadrilateral as 360o @

Try this with another quadrilateral.

Example 2

The value of three angles of a quadrilateral are 85o , 52o , and 61o' Find the value of the remaining angle.

The sum of the 4 angles of the quadrilateral = 360o

85o + 52o + 81o + remaining angle = 360o

∴ Value of the remaining angle = 360o − (85o + 52o + 81o) = 360o − 218o

= 142o

Exercise 12.2^1& Using the information given in the

(i) the value of x'(ii) the value of y'(iii) the value of x + y'(iv) the sum of the angles of the quadrilateral ABCD.

A B

C

D

yo xo 85o

75o

46o39o

D

A

B

C

b

a c

d

o

o

o o

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^2& Find the value of x and y in each of the following '

3̂& Select those which are not the four angles of a qudrilateral out of the following.

(i) 90o , 90o , 90o , 90o (ii) 75o , 150o , 75o , 50o

(iii) 65o , 108o , 35o , 132o (iv) 110o , 110o , 70o , 70o

12.3 The exterior angles of a triangleIn the triangle PQR, the side QR is produced to ‘S’.'Angle is an exterior angle of the triangle PQR.

What are the other exterior angles of the triangle PQR?The remaining exterior angles can be drawn by producing the sides RP and PQ. What is shown in the are the exterior angles of the triangle PQR.

Activity 12.3 Draw a triangle and produce the sides as in the above Measure the size of each of the exterior angles using a protractor. Do you get the sum of the three angles as 360o @Examine whether you will get the sum as 360o by cutting out the exterior angles and pasting them as the sum of the interior angles was found'

The sum of the three exterior angles of a triangle is 360o.

P

QR

S

P

Q

R

S

112o

E

A B

CD86o

(i) (ii) (iii) (iv)

69o

31o

65o

xo

yo

126o

84o yo

120o

xo

2xo

yo

C

xoyo

PRS^

o o

yo

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Exercise 12.3^1& Find the value of x in each of the following

^2& Select and write those which can be the values of the exterior angles of a triangle out of the following.

(i) 125o , 98o , 137o (ii) 150o , 130o , 80o (iii) 120o , 15o , 45o (iv) 120o , 120o , 120o

^3& Gayani who drew the exterior angles of a triangle as in the says that "the sum of those exterior angles is not 360o". Find the error in this.

Are angles marked as 1 , 2 , 3 exterior angles?Draw the exterior angles correctly and examine whether you get the sum of the exterior angles as 360o' N

L

M1

23

115o

48o

xo(iii) (iv)

132o

58o

xo

142o

P

Q

R

xo

35o

161o

xo

(i) (ii)100o

o

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dc

ba

c

d

a

b

P S

Q R

Thesumoftheexterioranglesofaquadrilateralis360o.

12.4 The exterior angles of a quadrilateralLookatthequadrilateralPQRS.Examinethewayhowthesidesareproduced.Asinthe triangle, the sides of the quadrilateral also are produced in the same direction to gettheexteriorangles.

Draw a quadrilateral as above and get the exterior angles by producing the sides.Examinewhetherthesumofthoseis360o by measuring each angle using aprotractor.

Cut and separate the exterior angles of a quadrilateralasshowninthefigure.Pastethemsothatthesidesaretogether.

Examine whether the sum of the four exterior anglesis360o asgiveninthefigure.

Activity 12.4

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Exercise 12.4^1& Find the value of a in each of the following

^2& What is the value of an exterior angle of a rectangle @

^3& A quadrilateral is drawn by combining two equilateral triangles. Are all the exterior angles equal@

Find the value of each of the exterior angles.

^4& Find the values of all the interior angles and the exterior angles of the quadrilateral PQRS.

ao

120o

90o

100o

120o

140o

ao

2ao

ao

90o

90o

(i) (ii) (iii)

ao

D

A

B C

Summary

The sum of the three interior angles of a triangle is 180 .o

The sum of the four interior angles of a quadrilateral is 360 .o

The sum of the three exterior angles of a triangle is 360 .o

The sum of the four exterior angles of a quadrilateral is 360 .o

110o

S

R

Q

P120o

100o

o