Properties of Triangles (Exercise 15.2) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

 Page 1


 
1. Two angles of a triangle are of measures 105
o
 and 30
o
. Find the measure of the third 
angle.
Solution: 
Given two angles of a triangle are of measures 10 5
o
 and 30
o
 
Let the required third angle be x 
We know that sum of all the angles of a triangle = 180
o
10 5
o
 + 30
o
 + x = 180
o
135
o
 + x = 180
o
x = 180
o
 – 135
o
x = 45
o
Therefore the third angle is 45
o
 
2. One of the angles of a triangle is 130
o
, and the other two angles are equal. What is 
the measure of each of these equal angles?
Solution: 
Given one of the angles of a triangle is 130
o
 
Also given that remaining two angles are equal 
So let the second and third angle be x 
We know that sum of all the angles of a triangle = 180
o
130
o
 + x + x = 180
o
130
o
 + 2x = 180
o
2x = 180
o 
– 130
o
2x = 50
o
x = 50/2 
x = 25
o
Therefore the two other angles are 25
o
 each 
3. The three angles of a triangle are equal to one another. What is the measure of 
each of the angles?
Solution: 
Given that three angles of a triangle are equal to one another 
Page 2


 
1. Two angles of a triangle are of measures 105
o
 and 30
o
. Find the measure of the third 
angle.
Solution: 
Given two angles of a triangle are of measures 10 5
o
 and 30
o
 
Let the required third angle be x 
We know that sum of all the angles of a triangle = 180
o
10 5
o
 + 30
o
 + x = 180
o
135
o
 + x = 180
o
x = 180
o
 – 135
o
x = 45
o
Therefore the third angle is 45
o
 
2. One of the angles of a triangle is 130
o
, and the other two angles are equal. What is 
the measure of each of these equal angles?
Solution: 
Given one of the angles of a triangle is 130
o
 
Also given that remaining two angles are equal 
So let the second and third angle be x 
We know that sum of all the angles of a triangle = 180
o
130
o
 + x + x = 180
o
130
o
 + 2x = 180
o
2x = 180
o 
– 130
o
2x = 50
o
x = 50/2 
x = 25
o
Therefore the two other angles are 25
o
 each 
3. The three angles of a triangle are equal to one another. What is the measure of 
each of the angles?
Solution: 
Given that three angles of a triangle are equal to one another 
 
 
 
 
 
 
 
So let the each angle be x 
We know that sum of all the angles of a triangle = 180
o 
x + x + x = 180
o 
3x = 180
o 
x = 180/3 
x = 60
o 
Therefore angle is 60
o
 each 
 
4. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles. 
 
Solution: 
Given angles of the triangle are in the ratio 1: 2: 3  
So take first angle as x, second angle as 2x and third angle as 3x 
We know that sum of all the angles of a triangle = 180
o 
x + 2x + 3x = 180
o 
6x = 180
o 
x = 180/6 
x = 30
o 
2x = 30
o
 × 2 = 60
o 
3x = 30
o
 × 3 = 90
o 
Therefore the first angle is 30
o
, second angle is 60
o
 and third angle is 90
o
. 
 
5. The angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. Find the value of x. 
 
Solution: 
Given the angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. 
We know that sum of all the angles of a triangle = 180
o 
(x - 40)
o
 + (x - 20)
o
 + (1/2 - 10)
o
 = 180
o 
x + x + (1/2) – 40
o
 – 20
o 
– 10
o
 = 180
o 
x + x + (1/2) – 70
o
 = 180
o
 
(5x/2) = 180
o 
+ 70
o
 
(5x/2) = 250
o
 
x = (2/5) × 250
o
 
x = 100
o 
Hence the value of x is 100
o
 
 
6. The angles of a triangle are arranged in ascending order of magnitude. If the 
Page 3


 
1. Two angles of a triangle are of measures 105
o
 and 30
o
. Find the measure of the third 
angle.
Solution: 
Given two angles of a triangle are of measures 10 5
o
 and 30
o
 
Let the required third angle be x 
We know that sum of all the angles of a triangle = 180
o
10 5
o
 + 30
o
 + x = 180
o
135
o
 + x = 180
o
x = 180
o
 – 135
o
x = 45
o
Therefore the third angle is 45
o
 
2. One of the angles of a triangle is 130
o
, and the other two angles are equal. What is 
the measure of each of these equal angles?
Solution: 
Given one of the angles of a triangle is 130
o
 
Also given that remaining two angles are equal 
So let the second and third angle be x 
We know that sum of all the angles of a triangle = 180
o
130
o
 + x + x = 180
o
130
o
 + 2x = 180
o
2x = 180
o 
– 130
o
2x = 50
o
x = 50/2 
x = 25
o
Therefore the two other angles are 25
o
 each 
3. The three angles of a triangle are equal to one another. What is the measure of 
each of the angles?
Solution: 
Given that three angles of a triangle are equal to one another 
 
 
 
 
 
 
 
So let the each angle be x 
We know that sum of all the angles of a triangle = 180
o 
x + x + x = 180
o 
3x = 180
o 
x = 180/3 
x = 60
o 
Therefore angle is 60
o
 each 
 
4. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles. 
 
Solution: 
Given angles of the triangle are in the ratio 1: 2: 3  
So take first angle as x, second angle as 2x and third angle as 3x 
We know that sum of all the angles of a triangle = 180
o 
x + 2x + 3x = 180
o 
6x = 180
o 
x = 180/6 
x = 30
o 
2x = 30
o
 × 2 = 60
o 
3x = 30
o
 × 3 = 90
o 
Therefore the first angle is 30
o
, second angle is 60
o
 and third angle is 90
o
. 
 
5. The angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. Find the value of x. 
 
Solution: 
Given the angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. 
We know that sum of all the angles of a triangle = 180
o 
(x - 40)
o
 + (x - 20)
o
 + (1/2 - 10)
o
 = 180
o 
x + x + (1/2) – 40
o
 – 20
o 
– 10
o
 = 180
o 
x + x + (1/2) – 70
o
 = 180
o
 
(5x/2) = 180
o 
+ 70
o
 
(5x/2) = 250
o
 
x = (2/5) × 250
o
 
x = 100
o 
Hence the value of x is 100
o
 
 
6. The angles of a triangle are arranged in ascending order of magnitude. If the 
 
 
 
 
 
 
 
difference between two consecutive angles is 10
o
. Find the three angles. 
 
Solution: 
Given that angles of a triangle are arranged in ascending order of magnitude 
Also given that difference between two consecutive angles is 10
o
 
Let the first angle be x 
Second angle be x + 10
o 
Third angle be x + 10
o
 + 10
o 
We know that sum of all the angles of a triangle = 180
o 
x + x + 10
o
 + x + 10
o
 +10
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
First angle is 50
o 
Second angle x + 10
o
 = 50 + 10 = 60
o 
Third angle x + 10
o
 +10
o
 = 50 + 10 + 10 = 70
o
 
 
7. Two angles of a triangle are equal and the third angle is greater than each of those 
angles by 30
o
. Determine all the angles of the triangle 
 
Solution: 
Given that two angles of a triangle are equal 
Let the first and second angle be x 
Also given that third angle is greater than each of those angles by 30
o
 
Therefore the third angle is greater than the first and second by 30
o
 = x + 30
o 
The first and the second angles are equal 
We know that sum of all the angles of a triangle = 180
o 
x + x + x + 30
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
Third angle = x + 30
o
 = 50
o
 + 30
o
 = 80
o 
The first and the second angle is 50
o
 and the third angle is 80
o
. 
Page 4


 
1. Two angles of a triangle are of measures 105
o
 and 30
o
. Find the measure of the third 
angle.
Solution: 
Given two angles of a triangle are of measures 10 5
o
 and 30
o
 
Let the required third angle be x 
We know that sum of all the angles of a triangle = 180
o
10 5
o
 + 30
o
 + x = 180
o
135
o
 + x = 180
o
x = 180
o
 – 135
o
x = 45
o
Therefore the third angle is 45
o
 
2. One of the angles of a triangle is 130
o
, and the other two angles are equal. What is 
the measure of each of these equal angles?
Solution: 
Given one of the angles of a triangle is 130
o
 
Also given that remaining two angles are equal 
So let the second and third angle be x 
We know that sum of all the angles of a triangle = 180
o
130
o
 + x + x = 180
o
130
o
 + 2x = 180
o
2x = 180
o 
– 130
o
2x = 50
o
x = 50/2 
x = 25
o
Therefore the two other angles are 25
o
 each 
3. The three angles of a triangle are equal to one another. What is the measure of 
each of the angles?
Solution: 
Given that three angles of a triangle are equal to one another 
 
 
 
 
 
 
 
So let the each angle be x 
We know that sum of all the angles of a triangle = 180
o 
x + x + x = 180
o 
3x = 180
o 
x = 180/3 
x = 60
o 
Therefore angle is 60
o
 each 
 
4. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles. 
 
Solution: 
Given angles of the triangle are in the ratio 1: 2: 3  
So take first angle as x, second angle as 2x and third angle as 3x 
We know that sum of all the angles of a triangle = 180
o 
x + 2x + 3x = 180
o 
6x = 180
o 
x = 180/6 
x = 30
o 
2x = 30
o
 × 2 = 60
o 
3x = 30
o
 × 3 = 90
o 
Therefore the first angle is 30
o
, second angle is 60
o
 and third angle is 90
o
. 
 
5. The angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. Find the value of x. 
 
Solution: 
Given the angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. 
We know that sum of all the angles of a triangle = 180
o 
(x - 40)
o
 + (x - 20)
o
 + (1/2 - 10)
o
 = 180
o 
x + x + (1/2) – 40
o
 – 20
o 
– 10
o
 = 180
o 
x + x + (1/2) – 70
o
 = 180
o
 
(5x/2) = 180
o 
+ 70
o
 
(5x/2) = 250
o
 
x = (2/5) × 250
o
 
x = 100
o 
Hence the value of x is 100
o
 
 
6. The angles of a triangle are arranged in ascending order of magnitude. If the 
 
 
 
 
 
 
 
difference between two consecutive angles is 10
o
. Find the three angles. 
 
Solution: 
Given that angles of a triangle are arranged in ascending order of magnitude 
Also given that difference between two consecutive angles is 10
o
 
Let the first angle be x 
Second angle be x + 10
o 
Third angle be x + 10
o
 + 10
o 
We know that sum of all the angles of a triangle = 180
o 
x + x + 10
o
 + x + 10
o
 +10
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
First angle is 50
o 
Second angle x + 10
o
 = 50 + 10 = 60
o 
Third angle x + 10
o
 +10
o
 = 50 + 10 + 10 = 70
o
 
 
7. Two angles of a triangle are equal and the third angle is greater than each of those 
angles by 30
o
. Determine all the angles of the triangle 
 
Solution: 
Given that two angles of a triangle are equal 
Let the first and second angle be x 
Also given that third angle is greater than each of those angles by 30
o
 
Therefore the third angle is greater than the first and second by 30
o
 = x + 30
o 
The first and the second angles are equal 
We know that sum of all the angles of a triangle = 180
o 
x + x + x + 30
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
Third angle = x + 30
o
 = 50
o
 + 30
o
 = 80
o 
The first and the second angle is 50
o
 and the third angle is 80
o
. 
 
 
 
 
 
 
 
 
8. If one angle of a triangle is equal to the sum of the other two, show that the triangle 
is a right triangle. 
 
Solution: 
Given that one angle of a triangle is equal to the sum of the other two 
Let the measure of angles be x, y, z 
Therefore we can write above statement as x = y + z 
x + y + z = 180
o 
Substituting the above value we get 
x + x = 180
o 
2x = 180
o 
x = 180/2 
x = 90
o 
If one angle is 90
o
 then the given triangle is a right angled triangle 
 
9. If each angle of a triangle is less than the sum of the other two, show that the 
triangle is acute angled. 
 
Solution: 
Given that each angle of a triangle is less than the sum of the other two 
Let the measure of angles be x, y and z 
From the above statement we can write as 
x > y + z 
y < x + z 
z < x + y 
Therefore triangle is an acute triangle 
 
10. In each of the following, the measures of three angles are given. State in which 
cases the angles can possibly be those of a triangle: 
(i) 63
o
, 37
o
, 80
o 
(ii) 45
o
, 61
o
, 73
o 
(iii) 59
o
, 72
o
, 61
o 
(iv) 45
o
, 45
o
, 90
o 
(v) 30
o
, 20
o
, 125
o
 
 
Solution: 
Page 5


 
1. Two angles of a triangle are of measures 105
o
 and 30
o
. Find the measure of the third 
angle.
Solution: 
Given two angles of a triangle are of measures 10 5
o
 and 30
o
 
Let the required third angle be x 
We know that sum of all the angles of a triangle = 180
o
10 5
o
 + 30
o
 + x = 180
o
135
o
 + x = 180
o
x = 180
o
 – 135
o
x = 45
o
Therefore the third angle is 45
o
 
2. One of the angles of a triangle is 130
o
, and the other two angles are equal. What is 
the measure of each of these equal angles?
Solution: 
Given one of the angles of a triangle is 130
o
 
Also given that remaining two angles are equal 
So let the second and third angle be x 
We know that sum of all the angles of a triangle = 180
o
130
o
 + x + x = 180
o
130
o
 + 2x = 180
o
2x = 180
o 
– 130
o
2x = 50
o
x = 50/2 
x = 25
o
Therefore the two other angles are 25
o
 each 
3. The three angles of a triangle are equal to one another. What is the measure of 
each of the angles?
Solution: 
Given that three angles of a triangle are equal to one another 
 
 
 
 
 
 
 
So let the each angle be x 
We know that sum of all the angles of a triangle = 180
o 
x + x + x = 180
o 
3x = 180
o 
x = 180/3 
x = 60
o 
Therefore angle is 60
o
 each 
 
4. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles. 
 
Solution: 
Given angles of the triangle are in the ratio 1: 2: 3  
So take first angle as x, second angle as 2x and third angle as 3x 
We know that sum of all the angles of a triangle = 180
o 
x + 2x + 3x = 180
o 
6x = 180
o 
x = 180/6 
x = 30
o 
2x = 30
o
 × 2 = 60
o 
3x = 30
o
 × 3 = 90
o 
Therefore the first angle is 30
o
, second angle is 60
o
 and third angle is 90
o
. 
 
5. The angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. Find the value of x. 
 
Solution: 
Given the angles of a triangle are (x - 40)
o
, (x - 20)
o
 and (1/2 - 10)
o
. 
We know that sum of all the angles of a triangle = 180
o 
(x - 40)
o
 + (x - 20)
o
 + (1/2 - 10)
o
 = 180
o 
x + x + (1/2) – 40
o
 – 20
o 
– 10
o
 = 180
o 
x + x + (1/2) – 70
o
 = 180
o
 
(5x/2) = 180
o 
+ 70
o
 
(5x/2) = 250
o
 
x = (2/5) × 250
o
 
x = 100
o 
Hence the value of x is 100
o
 
 
6. The angles of a triangle are arranged in ascending order of magnitude. If the 
 
 
 
 
 
 
 
difference between two consecutive angles is 10
o
. Find the three angles. 
 
Solution: 
Given that angles of a triangle are arranged in ascending order of magnitude 
Also given that difference between two consecutive angles is 10
o
 
Let the first angle be x 
Second angle be x + 10
o 
Third angle be x + 10
o
 + 10
o 
We know that sum of all the angles of a triangle = 180
o 
x + x + 10
o
 + x + 10
o
 +10
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
First angle is 50
o 
Second angle x + 10
o
 = 50 + 10 = 60
o 
Third angle x + 10
o
 +10
o
 = 50 + 10 + 10 = 70
o
 
 
7. Two angles of a triangle are equal and the third angle is greater than each of those 
angles by 30
o
. Determine all the angles of the triangle 
 
Solution: 
Given that two angles of a triangle are equal 
Let the first and second angle be x 
Also given that third angle is greater than each of those angles by 30
o
 
Therefore the third angle is greater than the first and second by 30
o
 = x + 30
o 
The first and the second angles are equal 
We know that sum of all the angles of a triangle = 180
o 
x + x + x + 30
o
 = 180
o 
3x + 30 = 180 
3x = 180 - 30 
3x = 150 
x = 150/3 
x = 50
o 
Third angle = x + 30
o
 = 50
o
 + 30
o
 = 80
o 
The first and the second angle is 50
o
 and the third angle is 80
o
. 
 
 
 
 
 
 
 
 
8. If one angle of a triangle is equal to the sum of the other two, show that the triangle 
is a right triangle. 
 
Solution: 
Given that one angle of a triangle is equal to the sum of the other two 
Let the measure of angles be x, y, z 
Therefore we can write above statement as x = y + z 
x + y + z = 180
o 
Substituting the above value we get 
x + x = 180
o 
2x = 180
o 
x = 180/2 
x = 90
o 
If one angle is 90
o
 then the given triangle is a right angled triangle 
 
9. If each angle of a triangle is less than the sum of the other two, show that the 
triangle is acute angled. 
 
Solution: 
Given that each angle of a triangle is less than the sum of the other two 
Let the measure of angles be x, y and z 
From the above statement we can write as 
x > y + z 
y < x + z 
z < x + y 
Therefore triangle is an acute triangle 
 
10. In each of the following, the measures of three angles are given. State in which 
cases the angles can possibly be those of a triangle: 
(i) 63
o
, 37
o
, 80
o 
(ii) 45
o
, 61
o
, 73
o 
(iii) 59
o
, 72
o
, 61
o 
(iv) 45
o
, 45
o
, 90
o 
(v) 30
o
, 20
o
, 125
o
 
 
Solution: 
 
 
 
 
 
 
 
(i) 63
o
 + 37
o 
+ 80
o
 = 180
o 
Angles form a triangle 
 
(ii) 45
o
, 61
o
, 73
o
 is not equal to 180
o 
Therefore not a triangle 
 
(iii) 59
o
, 72
o
, 61
o
 is not equal to 180
0 
Therefore not a triangle 
 
(iv) 45
o
+ 45
o
+ 90
o
 = 180
o 
Angles form a triangle 
 
(v) 30
o
, 20
o
, 125
o
 is not equal to 180
o 
Therefore not a triangle 
 
11. The angles of a triangle are in the ratio 3: 4: 5. Find the smallest angle 
 
Solution: 
Given that angles of a triangle are in the ratio: 3: 4: 5 
Therefore let the measure of the angles be 3x, 4x, 5x 
We know that sum of the angles of a triangle =180
o 
3x + 4x + 5x = 180
o 
12x = 180
o 
x = 180/12 
x = 15
o 
Smallest angle = 3x 
= 3 × 15
o 
= 45
o 
Therefore smallest angle = 45
o
 
 
12. Two acute angles of a right triangle are equal. Find the two angles. 
 
Solution: 
Given that acute angles of a right angled triangle are equal 
We know that Right triangle: whose one of the angle is a right angle 
Let the measure of angle be x, x, 90
o 
x + x + 90
o
= 180
o