RS Aggarwal Solutions: Properties of Triangles | Mathematics (Maths) Class 7 PDF Download

 Page 1


Q u e s t i o n : 1
In a ?ABC, if ?A = 72° and ?B = 63°, find ?C.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
? ?A + ?B + ?C = 180°72° + 63° + ?C = 180° ?C = 45° 
Hence, ?C measures 45°.
Q u e s t i o n : 2
In a ?DEF, if ?E = 105° and ?F = 40°, find ?D.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?DEF:
?D + ?E + ?F = 180° ?D +105°+40° = 180°or ?D = 180°-(105°+40°)or ?D = 35°
Q u e s t i o n : 3
In a ?XYZ, if ?X = 90° and ?Z = 48°, find ?Y.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?XYZ:
?X + ?Y + ?Z = 180°90°+ ?Y +48° = 180° = > ?Y = 180°-138° = 42°
 
Q u e s t i o n : 4
Find the angles of a triangle which are in the ratio 4 : 3 : 2.
S o l u t i o n :
Suppose the angles of the triangle are 4x
o
, 3x
o
 and 2x
o
.
Sum of the angles of any triangle is 180
o
.
? 4x + 3x + 2x = 180
9x = 180
x = 20
Therefore, the angles of the triangle are (4 ×20)°, (3 ×20)° and ( 2 ×20)°, i. e . 80°, 60° and 40°.
Q u e s t i o n : 5
One of the acute angles of a right triangle is 36°. find the other.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.
It is a right angle triangle. Hence, one of the angle is 90°.
? 36° + 90° +x = 180°x = 54°
Hence, the other angle measures 54°.
Q u e s t i o n : 6
The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.
S o l u t i o n :
Suppose the acute angles are (2x)
°
 and (x)
°
Sum of the angles of any triangle is 180°
? 2x+x+ 90 = 180
?
(3x) = 180-90
?
(3x) = 90
?
x = 30
So, the angles measure (2 ×30)° and 30°i. e. 60° and 30°
Q u e s t i o n : 7
One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.
S o l u t i o n :
The other two angles are equal. Let one of these angles be x°.
Sum of angles of any triangle is 180°.
? x + x+ 100 = 180
2x = 80
x = 40
Hence, the equal angles of the triangle are 40° each.
Q u e s t i o n : 8
Page 2


Q u e s t i o n : 1
In a ?ABC, if ?A = 72° and ?B = 63°, find ?C.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
? ?A + ?B + ?C = 180°72° + 63° + ?C = 180° ?C = 45° 
Hence, ?C measures 45°.
Q u e s t i o n : 2
In a ?DEF, if ?E = 105° and ?F = 40°, find ?D.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?DEF:
?D + ?E + ?F = 180° ?D +105°+40° = 180°or ?D = 180°-(105°+40°)or ?D = 35°
Q u e s t i o n : 3
In a ?XYZ, if ?X = 90° and ?Z = 48°, find ?Y.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?XYZ:
?X + ?Y + ?Z = 180°90°+ ?Y +48° = 180° = > ?Y = 180°-138° = 42°
 
Q u e s t i o n : 4
Find the angles of a triangle which are in the ratio 4 : 3 : 2.
S o l u t i o n :
Suppose the angles of the triangle are 4x
o
, 3x
o
 and 2x
o
.
Sum of the angles of any triangle is 180
o
.
? 4x + 3x + 2x = 180
9x = 180
x = 20
Therefore, the angles of the triangle are (4 ×20)°, (3 ×20)° and ( 2 ×20)°, i. e . 80°, 60° and 40°.
Q u e s t i o n : 5
One of the acute angles of a right triangle is 36°. find the other.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.
It is a right angle triangle. Hence, one of the angle is 90°.
? 36° + 90° +x = 180°x = 54°
Hence, the other angle measures 54°.
Q u e s t i o n : 6
The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.
S o l u t i o n :
Suppose the acute angles are (2x)
°
 and (x)
°
Sum of the angles of any triangle is 180°
? 2x+x+ 90 = 180
?
(3x) = 180-90
?
(3x) = 90
?
x = 30
So, the angles measure (2 ×30)° and 30°i. e. 60° and 30°
Q u e s t i o n : 7
One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.
S o l u t i o n :
The other two angles are equal. Let one of these angles be x°.
Sum of angles of any triangle is 180°.
? x + x+ 100 = 180
2x = 80
x = 40
Hence, the equal angles of the triangle are 40° each.
Q u e s t i o n : 8
Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.
S o l u t i o n :
Suppose the third angle of the isosceles triangle is x
o
.
Then, the two equal angles are (2x)
o
 and (2x)
o
.
Sum of the angles of any triangle is 180
o
.
? 2x +2x+ x= 180
5x  = 180
x = 36
Hence, the angles of the triangle are 36°, (2 ×36)° and (2 ×36)°, i. e. 36°, 72°and 72°
.
Q u e s t i o n : 9
If one angle of a triangle is equal to the sum of the other two, show that the triangle is right-angled.
S o l u t i o n :
Suppose the angles are ?A, ?B and ? C. Given: ?A = ?B + ?CAlso, ?A + ?B + ?C = 180° ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
                  (Sum of the angles of a triangle is 180
°
)
Hence, the triangle ABC is right angled at ?
A.
Q u e s t i o n : 1 0
In a ?ABC, if 2 ?A = 3 ?B = 6 ?C, calculate ?A, ?B and ?C.
S o l u t i o n :
Suppose: 2 ?A = 3 ?B = 6 ?C = x
°
Then, ?A = 
x
2
°
?B =
x
3
°
and ?C =
x
6 °
Sum of the angles of any triangle is 180°.
?A + ?B + ?C = 180°
?
x
2
+
x
3
+
x
6
= 180° ?
3x+2x+x
6
= 180° ?
6x
6
= 180° ? x = 180
 ? ?A =
180
2
°
= 90°
?B =
180
3
°
= 60
°
?C =
180
6
°
= 30°
Q u e s t i o n : 1 1
What is the measure of each angle of an equilateral triangle?
S o l u t i o n :
We know that the angles of an equilateral triangle are equal.
Let the measure of each angle of an equilateral triangle be x
°
.
? x + x + x = 180
x = 60
Hence, the measure of each angle of an equilateral triangle is 60°
.
Q u e s t i o n : 1 2
In the given figure, DE || BC. If ?A = 65° and ?B° = 55, find
i
?ADE
ii
?AED
iii
?C
S o l u t i o n :
i
DE ? BC ? ?ABC = ?ADE = 55°
                   
Correspondingangles
ii
Sum of the angles of any triangle is 180°.
? ?A + ?B + ?C = 180° ?C = 180° -(65° +55°) = 60°
( )
( ) ( )
( )
( ) ( )
Page 3


Q u e s t i o n : 1
In a ?ABC, if ?A = 72° and ?B = 63°, find ?C.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
? ?A + ?B + ?C = 180°72° + 63° + ?C = 180° ?C = 45° 
Hence, ?C measures 45°.
Q u e s t i o n : 2
In a ?DEF, if ?E = 105° and ?F = 40°, find ?D.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?DEF:
?D + ?E + ?F = 180° ?D +105°+40° = 180°or ?D = 180°-(105°+40°)or ?D = 35°
Q u e s t i o n : 3
In a ?XYZ, if ?X = 90° and ?Z = 48°, find ?Y.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?XYZ:
?X + ?Y + ?Z = 180°90°+ ?Y +48° = 180° = > ?Y = 180°-138° = 42°
 
Q u e s t i o n : 4
Find the angles of a triangle which are in the ratio 4 : 3 : 2.
S o l u t i o n :
Suppose the angles of the triangle are 4x
o
, 3x
o
 and 2x
o
.
Sum of the angles of any triangle is 180
o
.
? 4x + 3x + 2x = 180
9x = 180
x = 20
Therefore, the angles of the triangle are (4 ×20)°, (3 ×20)° and ( 2 ×20)°, i. e . 80°, 60° and 40°.
Q u e s t i o n : 5
One of the acute angles of a right triangle is 36°. find the other.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.
It is a right angle triangle. Hence, one of the angle is 90°.
? 36° + 90° +x = 180°x = 54°
Hence, the other angle measures 54°.
Q u e s t i o n : 6
The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.
S o l u t i o n :
Suppose the acute angles are (2x)
°
 and (x)
°
Sum of the angles of any triangle is 180°
? 2x+x+ 90 = 180
?
(3x) = 180-90
?
(3x) = 90
?
x = 30
So, the angles measure (2 ×30)° and 30°i. e. 60° and 30°
Q u e s t i o n : 7
One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.
S o l u t i o n :
The other two angles are equal. Let one of these angles be x°.
Sum of angles of any triangle is 180°.
? x + x+ 100 = 180
2x = 80
x = 40
Hence, the equal angles of the triangle are 40° each.
Q u e s t i o n : 8
Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.
S o l u t i o n :
Suppose the third angle of the isosceles triangle is x
o
.
Then, the two equal angles are (2x)
o
 and (2x)
o
.
Sum of the angles of any triangle is 180
o
.
? 2x +2x+ x= 180
5x  = 180
x = 36
Hence, the angles of the triangle are 36°, (2 ×36)° and (2 ×36)°, i. e. 36°, 72°and 72°
.
Q u e s t i o n : 9
If one angle of a triangle is equal to the sum of the other two, show that the triangle is right-angled.
S o l u t i o n :
Suppose the angles are ?A, ?B and ? C. Given: ?A = ?B + ?CAlso, ?A + ?B + ?C = 180° ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
                  (Sum of the angles of a triangle is 180
°
)
Hence, the triangle ABC is right angled at ?
A.
Q u e s t i o n : 1 0
In a ?ABC, if 2 ?A = 3 ?B = 6 ?C, calculate ?A, ?B and ?C.
S o l u t i o n :
Suppose: 2 ?A = 3 ?B = 6 ?C = x
°
Then, ?A = 
x
2
°
?B =
x
3
°
and ?C =
x
6 °
Sum of the angles of any triangle is 180°.
?A + ?B + ?C = 180°
?
x
2
+
x
3
+
x
6
= 180° ?
3x+2x+x
6
= 180° ?
6x
6
= 180° ? x = 180
 ? ?A =
180
2
°
= 90°
?B =
180
3
°
= 60
°
?C =
180
6
°
= 30°
Q u e s t i o n : 1 1
What is the measure of each angle of an equilateral triangle?
S o l u t i o n :
We know that the angles of an equilateral triangle are equal.
Let the measure of each angle of an equilateral triangle be x
°
.
? x + x + x = 180
x = 60
Hence, the measure of each angle of an equilateral triangle is 60°
.
Q u e s t i o n : 1 2
In the given figure, DE || BC. If ?A = 65° and ?B° = 55, find
i
?ADE
ii
?AED
iii
?C
S o l u t i o n :
i
DE ? BC ? ?ABC = ?ADE = 55°
                   
Correspondingangles
ii
Sum of the angles of any triangle is 180°.
? ?A + ?B + ?C = 180° ?C = 180° -(65° +55°) = 60°
( )
( ) ( )
( )
( ) ( )
 DE || BC
? ?AED = ?ACB = 60°
                correspondingangles
iii
  We have found in point ii
that ?C is equal to 60°.
Q u e s t i o n : 1 3
Can a triangle have
i
two right angles?
ii
two obtuse angles?
iii
two acute angles?
iv
all angles more than 60°?
v
all angles less than 60°?
vi
al angles equal to 60°?
S o l u t i o n :
i
No. This is because the sum of all the angles is 180°.
ii
No. This is because a triangle can only have one obtuse angle.
iii
Yes
iv
No. This is because the sum of the angles cannot be more than 180°.
v
No. This is because one angle has to be more than 60° as the sum of all angles is always 180°.
vi
Yes, it will be an equilateral triangle.
Q u e s t i o n : 1 4
Answer the following in 'Yes' or 'No'.
i
Can an isosceles triangle be a right triangle?
ii
Can a right triangle be a scalene triangle?
iii
Can a right triangle be an equilateral triangle?
iv
Can an obtuse triangle be an isosceles triangle?
S o l u t i o n :
i
Yes, it will be an isosceles right triangle.
ii
Yes, a right triangle can have all sides of different measures. For example, 3, 4 and 5 are the sides of a scalene right triangle.
iii
No, it cannot be an equilateral triangle since the hypotenuse square will be the sum of the square of the other two sides.
iii
Yes, if an obtuse triangle has an obtuse angle of 120° and the other two angles of 30° each, then it will be an isosceles triangle.
Q u e s t i o n : 1 5
Fill in the blanks:
i
A right triangle cannot have an ...... angle.
ii
The acute angles of a right triangle are ...... .
iii
Each acute angle of an isosceles right triangle measures ...... .
iv
Each angle of an equilateral triangle measures ...... .
v
The side opposite the right angle of a right triangle is called ...... .
vi
The sum of the lengths of the sides of a triangle is called its...... .
S o l u t i o n :
i
obtuse (since the sum of the other two angles of the right triangle is 90
o
)
ii
equal to the sum of 90
o
iii
45
o
 (since their sum is equal to 90
o
)
iv
60
o
v
a hypotenuse
vi
perimeter
Page 4


Q u e s t i o n : 1
In a ?ABC, if ?A = 72° and ?B = 63°, find ?C.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
? ?A + ?B + ?C = 180°72° + 63° + ?C = 180° ?C = 45° 
Hence, ?C measures 45°.
Q u e s t i o n : 2
In a ?DEF, if ?E = 105° and ?F = 40°, find ?D.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?DEF:
?D + ?E + ?F = 180° ?D +105°+40° = 180°or ?D = 180°-(105°+40°)or ?D = 35°
Q u e s t i o n : 3
In a ?XYZ, if ?X = 90° and ?Z = 48°, find ?Y.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?XYZ:
?X + ?Y + ?Z = 180°90°+ ?Y +48° = 180° = > ?Y = 180°-138° = 42°
 
Q u e s t i o n : 4
Find the angles of a triangle which are in the ratio 4 : 3 : 2.
S o l u t i o n :
Suppose the angles of the triangle are 4x
o
, 3x
o
 and 2x
o
.
Sum of the angles of any triangle is 180
o
.
? 4x + 3x + 2x = 180
9x = 180
x = 20
Therefore, the angles of the triangle are (4 ×20)°, (3 ×20)° and ( 2 ×20)°, i. e . 80°, 60° and 40°.
Q u e s t i o n : 5
One of the acute angles of a right triangle is 36°. find the other.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.
It is a right angle triangle. Hence, one of the angle is 90°.
? 36° + 90° +x = 180°x = 54°
Hence, the other angle measures 54°.
Q u e s t i o n : 6
The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.
S o l u t i o n :
Suppose the acute angles are (2x)
°
 and (x)
°
Sum of the angles of any triangle is 180°
? 2x+x+ 90 = 180
?
(3x) = 180-90
?
(3x) = 90
?
x = 30
So, the angles measure (2 ×30)° and 30°i. e. 60° and 30°
Q u e s t i o n : 7
One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.
S o l u t i o n :
The other two angles are equal. Let one of these angles be x°.
Sum of angles of any triangle is 180°.
? x + x+ 100 = 180
2x = 80
x = 40
Hence, the equal angles of the triangle are 40° each.
Q u e s t i o n : 8
Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.
S o l u t i o n :
Suppose the third angle of the isosceles triangle is x
o
.
Then, the two equal angles are (2x)
o
 and (2x)
o
.
Sum of the angles of any triangle is 180
o
.
? 2x +2x+ x= 180
5x  = 180
x = 36
Hence, the angles of the triangle are 36°, (2 ×36)° and (2 ×36)°, i. e. 36°, 72°and 72°
.
Q u e s t i o n : 9
If one angle of a triangle is equal to the sum of the other two, show that the triangle is right-angled.
S o l u t i o n :
Suppose the angles are ?A, ?B and ? C. Given: ?A = ?B + ?CAlso, ?A + ?B + ?C = 180° ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
                  (Sum of the angles of a triangle is 180
°
)
Hence, the triangle ABC is right angled at ?
A.
Q u e s t i o n : 1 0
In a ?ABC, if 2 ?A = 3 ?B = 6 ?C, calculate ?A, ?B and ?C.
S o l u t i o n :
Suppose: 2 ?A = 3 ?B = 6 ?C = x
°
Then, ?A = 
x
2
°
?B =
x
3
°
and ?C =
x
6 °
Sum of the angles of any triangle is 180°.
?A + ?B + ?C = 180°
?
x
2
+
x
3
+
x
6
= 180° ?
3x+2x+x
6
= 180° ?
6x
6
= 180° ? x = 180
 ? ?A =
180
2
°
= 90°
?B =
180
3
°
= 60
°
?C =
180
6
°
= 30°
Q u e s t i o n : 1 1
What is the measure of each angle of an equilateral triangle?
S o l u t i o n :
We know that the angles of an equilateral triangle are equal.
Let the measure of each angle of an equilateral triangle be x
°
.
? x + x + x = 180
x = 60
Hence, the measure of each angle of an equilateral triangle is 60°
.
Q u e s t i o n : 1 2
In the given figure, DE || BC. If ?A = 65° and ?B° = 55, find
i
?ADE
ii
?AED
iii
?C
S o l u t i o n :
i
DE ? BC ? ?ABC = ?ADE = 55°
                   
Correspondingangles
ii
Sum of the angles of any triangle is 180°.
? ?A + ?B + ?C = 180° ?C = 180° -(65° +55°) = 60°
( )
( ) ( )
( )
( ) ( )
 DE || BC
? ?AED = ?ACB = 60°
                correspondingangles
iii
  We have found in point ii
that ?C is equal to 60°.
Q u e s t i o n : 1 3
Can a triangle have
i
two right angles?
ii
two obtuse angles?
iii
two acute angles?
iv
all angles more than 60°?
v
all angles less than 60°?
vi
al angles equal to 60°?
S o l u t i o n :
i
No. This is because the sum of all the angles is 180°.
ii
No. This is because a triangle can only have one obtuse angle.
iii
Yes
iv
No. This is because the sum of the angles cannot be more than 180°.
v
No. This is because one angle has to be more than 60° as the sum of all angles is always 180°.
vi
Yes, it will be an equilateral triangle.
Q u e s t i o n : 1 4
Answer the following in 'Yes' or 'No'.
i
Can an isosceles triangle be a right triangle?
ii
Can a right triangle be a scalene triangle?
iii
Can a right triangle be an equilateral triangle?
iv
Can an obtuse triangle be an isosceles triangle?
S o l u t i o n :
i
Yes, it will be an isosceles right triangle.
ii
Yes, a right triangle can have all sides of different measures. For example, 3, 4 and 5 are the sides of a scalene right triangle.
iii
No, it cannot be an equilateral triangle since the hypotenuse square will be the sum of the square of the other two sides.
iii
Yes, if an obtuse triangle has an obtuse angle of 120° and the other two angles of 30° each, then it will be an isosceles triangle.
Q u e s t i o n : 1 5
Fill in the blanks:
i
A right triangle cannot have an ...... angle.
ii
The acute angles of a right triangle are ...... .
iii
Each acute angle of an isosceles right triangle measures ...... .
iv
Each angle of an equilateral triangle measures ...... .
v
The side opposite the right angle of a right triangle is called ...... .
vi
The sum of the lengths of the sides of a triangle is called its...... .
S o l u t i o n :
i
obtuse (since the sum of the other two angles of the right triangle is 90
o
)
ii
equal to the sum of 90
o
iii
45
o
 (since their sum is equal to 90
o
)
iv
60
o
v
a hypotenuse
vi
perimeter
Q u e s t i o n : 1 6
In the figure given alongside, find the measure of ?ACD.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
?ACD = ?CAB + ?CBA ?ACD = 75°+45° = 120°
Q u e s t i o n : 1 7
In the figure given alongside, find the values of x and y.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? ?BAC + ?ABC = ?ACDx +68 = 130x = 62
Sum of the angles in any triangle is 180
o
.
? ?BAC + ?ABC + ?ACB = 180°62 + 68 + y = 180y = 50
Q u e s t i o n : 1 8
In the figure given alongside, find the values of x and y.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? ?BAC + ?CBA = ?ACD32 + x = 65x = 33
Also, sum of the angles in any triangle is 180°
.
? ?BAC + ?CBA + ?ACB = 180°32 +33 + y = 180y = 115
? x= 33
    y =115
Q u e s t i o n : 1 9
An exterior angle of a triangle measures 110° and its interior opposite angles are in the ratio 2 : 3. Find the angles of the triangle.
S o l u t i o n :
Suppose the interior opposite angles are (2x)° and (3x)°.
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? 3x +2x= 110
x = 22
The interior opposite angles are (2 ×22)° and (3 ×22)°, i. e.
44° and 66°.
Suppose the third angle of the triangle is y°.
Now, sum of the angles in any triangle is 180°.
? 44 + 66 + y = 180
y = 70
Hence, the angles of the triangle are 44°, 66° and 70°.
Q u e s t i o n : 2 0
An exterior angle of a triangle is 100° and its interior opposite angles are equal to each other. Find the measure of each angle of the triangle.
S o l u t i o n :
Suppose the interior opposite angles of an exterior angle 100
o 
are x
o
 and x
o
.
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? x + x = 100
2x= 100
x= 50
Page 5


Q u e s t i o n : 1
In a ?ABC, if ?A = 72° and ?B = 63°, find ?C.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
? ?A + ?B + ?C = 180°72° + 63° + ?C = 180° ?C = 45° 
Hence, ?C measures 45°.
Q u e s t i o n : 2
In a ?DEF, if ?E = 105° and ?F = 40°, find ?D.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?DEF:
?D + ?E + ?F = 180° ?D +105°+40° = 180°or ?D = 180°-(105°+40°)or ?D = 35°
Q u e s t i o n : 3
In a ?XYZ, if ?X = 90° and ?Z = 48°, find ?Y.
S o l u t i o n :
Sum of the angles of any triangle is 180°.
In ?XYZ:
?X + ?Y + ?Z = 180°90°+ ?Y +48° = 180° = > ?Y = 180°-138° = 42°
 
Q u e s t i o n : 4
Find the angles of a triangle which are in the ratio 4 : 3 : 2.
S o l u t i o n :
Suppose the angles of the triangle are 4x
o
, 3x
o
 and 2x
o
.
Sum of the angles of any triangle is 180
o
.
? 4x + 3x + 2x = 180
9x = 180
x = 20
Therefore, the angles of the triangle are (4 ×20)°, (3 ×20)° and ( 2 ×20)°, i. e . 80°, 60° and 40°.
Q u e s t i o n : 5
One of the acute angles of a right triangle is 36°. find the other.
S o l u t i o n :
Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.
It is a right angle triangle. Hence, one of the angle is 90°.
? 36° + 90° +x = 180°x = 54°
Hence, the other angle measures 54°.
Q u e s t i o n : 6
The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.
S o l u t i o n :
Suppose the acute angles are (2x)
°
 and (x)
°
Sum of the angles of any triangle is 180°
? 2x+x+ 90 = 180
?
(3x) = 180-90
?
(3x) = 90
?
x = 30
So, the angles measure (2 ×30)° and 30°i. e. 60° and 30°
Q u e s t i o n : 7
One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.
S o l u t i o n :
The other two angles are equal. Let one of these angles be x°.
Sum of angles of any triangle is 180°.
? x + x+ 100 = 180
2x = 80
x = 40
Hence, the equal angles of the triangle are 40° each.
Q u e s t i o n : 8
Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.
S o l u t i o n :
Suppose the third angle of the isosceles triangle is x
o
.
Then, the two equal angles are (2x)
o
 and (2x)
o
.
Sum of the angles of any triangle is 180
o
.
? 2x +2x+ x= 180
5x  = 180
x = 36
Hence, the angles of the triangle are 36°, (2 ×36)° and (2 ×36)°, i. e. 36°, 72°and 72°
.
Q u e s t i o n : 9
If one angle of a triangle is equal to the sum of the other two, show that the triangle is right-angled.
S o l u t i o n :
Suppose the angles are ?A, ?B and ? C. Given: ?A = ?B + ?CAlso, ?A + ?B + ?C = 180° ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
                  (Sum of the angles of a triangle is 180
°
)
Hence, the triangle ABC is right angled at ?
A.
Q u e s t i o n : 1 0
In a ?ABC, if 2 ?A = 3 ?B = 6 ?C, calculate ?A, ?B and ?C.
S o l u t i o n :
Suppose: 2 ?A = 3 ?B = 6 ?C = x
°
Then, ?A = 
x
2
°
?B =
x
3
°
and ?C =
x
6 °
Sum of the angles of any triangle is 180°.
?A + ?B + ?C = 180°
?
x
2
+
x
3
+
x
6
= 180° ?
3x+2x+x
6
= 180° ?
6x
6
= 180° ? x = 180
 ? ?A =
180
2
°
= 90°
?B =
180
3
°
= 60
°
?C =
180
6
°
= 30°
Q u e s t i o n : 1 1
What is the measure of each angle of an equilateral triangle?
S o l u t i o n :
We know that the angles of an equilateral triangle are equal.
Let the measure of each angle of an equilateral triangle be x
°
.
? x + x + x = 180
x = 60
Hence, the measure of each angle of an equilateral triangle is 60°
.
Q u e s t i o n : 1 2
In the given figure, DE || BC. If ?A = 65° and ?B° = 55, find
i
?ADE
ii
?AED
iii
?C
S o l u t i o n :
i
DE ? BC ? ?ABC = ?ADE = 55°
                   
Correspondingangles
ii
Sum of the angles of any triangle is 180°.
? ?A + ?B + ?C = 180° ?C = 180° -(65° +55°) = 60°
( )
( ) ( )
( )
( ) ( )
 DE || BC
? ?AED = ?ACB = 60°
                correspondingangles
iii
  We have found in point ii
that ?C is equal to 60°.
Q u e s t i o n : 1 3
Can a triangle have
i
two right angles?
ii
two obtuse angles?
iii
two acute angles?
iv
all angles more than 60°?
v
all angles less than 60°?
vi
al angles equal to 60°?
S o l u t i o n :
i
No. This is because the sum of all the angles is 180°.
ii
No. This is because a triangle can only have one obtuse angle.
iii
Yes
iv
No. This is because the sum of the angles cannot be more than 180°.
v
No. This is because one angle has to be more than 60° as the sum of all angles is always 180°.
vi
Yes, it will be an equilateral triangle.
Q u e s t i o n : 1 4
Answer the following in 'Yes' or 'No'.
i
Can an isosceles triangle be a right triangle?
ii
Can a right triangle be a scalene triangle?
iii
Can a right triangle be an equilateral triangle?
iv
Can an obtuse triangle be an isosceles triangle?
S o l u t i o n :
i
Yes, it will be an isosceles right triangle.
ii
Yes, a right triangle can have all sides of different measures. For example, 3, 4 and 5 are the sides of a scalene right triangle.
iii
No, it cannot be an equilateral triangle since the hypotenuse square will be the sum of the square of the other two sides.
iii
Yes, if an obtuse triangle has an obtuse angle of 120° and the other two angles of 30° each, then it will be an isosceles triangle.
Q u e s t i o n : 1 5
Fill in the blanks:
i
A right triangle cannot have an ...... angle.
ii
The acute angles of a right triangle are ...... .
iii
Each acute angle of an isosceles right triangle measures ...... .
iv
Each angle of an equilateral triangle measures ...... .
v
The side opposite the right angle of a right triangle is called ...... .
vi
The sum of the lengths of the sides of a triangle is called its...... .
S o l u t i o n :
i
obtuse (since the sum of the other two angles of the right triangle is 90
o
)
ii
equal to the sum of 90
o
iii
45
o
 (since their sum is equal to 90
o
)
iv
60
o
v
a hypotenuse
vi
perimeter
Q u e s t i o n : 1 6
In the figure given alongside, find the measure of ?ACD.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
?ACD = ?CAB + ?CBA ?ACD = 75°+45° = 120°
Q u e s t i o n : 1 7
In the figure given alongside, find the values of x and y.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? ?BAC + ?ABC = ?ACDx +68 = 130x = 62
Sum of the angles in any triangle is 180
o
.
? ?BAC + ?ABC + ?ACB = 180°62 + 68 + y = 180y = 50
Q u e s t i o n : 1 8
In the figure given alongside, find the values of x and y.
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? ?BAC + ?CBA = ?ACD32 + x = 65x = 33
Also, sum of the angles in any triangle is 180°
.
? ?BAC + ?CBA + ?ACB = 180°32 +33 + y = 180y = 115
? x= 33
    y =115
Q u e s t i o n : 1 9
An exterior angle of a triangle measures 110° and its interior opposite angles are in the ratio 2 : 3. Find the angles of the triangle.
S o l u t i o n :
Suppose the interior opposite angles are (2x)° and (3x)°.
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? 3x +2x= 110
x = 22
The interior opposite angles are (2 ×22)° and (3 ×22)°, i. e.
44° and 66°.
Suppose the third angle of the triangle is y°.
Now, sum of the angles in any triangle is 180°.
? 44 + 66 + y = 180
y = 70
Hence, the angles of the triangle are 44°, 66° and 70°.
Q u e s t i o n : 2 0
An exterior angle of a triangle is 100° and its interior opposite angles are equal to each other. Find the measure of each angle of the triangle.
S o l u t i o n :
Suppose the interior opposite angles of an exterior angle 100
o 
are x
o
 and x
o
.
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? x + x = 100
2x= 100
x= 50
Also, sum of the angles of any triangle is  180°.
Let the measure of the third angle be y°.
? x + x + y = 180
50  + 50 + y= 180
y  = 80
Hence, the angles are of the measures 50°, 50° and 80°.
Q u e s t i o n : 2 1
In the figure given alongside, find:
i
?ACD
ii
?AED
S o l u t i o n :
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
In ?
ABC:
?ACD = ?BAC + ?ABC = 25°+45° ?ACD = 70°(ii) In ? ECD: ?AED = ?ECD + ?EDC = 70°+40° = > ?AED = 110°
Q u e s t i o n : 2 2
In the figure given alongside, find:
i
?ACD
ii
?ADC
iii
?DAE
S o l u t i o n :
Sum of the angles of a triangle is 180°
.
In ? ABC: ?BAC + ?CBA + ?ACB = 180° ?BAC = 180°-(40°+100°) = > ?BAC = 40°
  
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
 
?ACD = ?BAC + ?CBA = 40°+40° = 80°(i) ?ACD = 80°(ii) In ? ACD: ?CAD + ?ACD + ?ADC = 180° = > ?ADC = 180°-(50°+80°) = > ?ADC = 50° ? ?ADC = 50°(iii) ?DAB + ?DAE
Q u e s t i o n : 2 3
In the figure given alongside, x : y = 2 : 3 and ?ACD = 130°.
Find the values of x, y and z.
S o l u t i o n :
x
y
=
2
3
? 3x = 2y ? x =
2
3
y
We know that the exterior angle of a triangle is equal to the sum of the interior opposite angles.
? ?A + ?B = ?ACD
x°
+ y°
= 130°
?
2y
3
+y = 130 ? 5y = 130 ×3 ? 5y = 390 ? y = 78 ? x =
2
3
×78 ? x = 52
Also, sum of the angles in any triangle is 180°