Properties of Triangles (Exercise 15.3) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

 Page 1


 
1. In Fig. 35, ?CBX is an exterior angle of ?ABC at B. Name
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ?CBX
Also, name the interior opposite angles to an exterior angle at A.
Solution: 
(i) The interior adjacent angle is ?ABC
(ii) The interior opposite angles to exterior ?CBX is ?BAC and ?ACB 
Also the interior angles opposite to exterior ?BAY are ?ABC and ?ACB
2. In the fig. 36, two of the angles are indicated. What are the measures of ?ACX and
?ACB?
Solution: 
Given that in ?ABC, ?A = 50
o
 and ?B = 55
o
We know that the sum of angles in a triangle is 180
o
Page 2


 
1. In Fig. 35, ?CBX is an exterior angle of ?ABC at B. Name
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ?CBX
Also, name the interior opposite angles to an exterior angle at A.
Solution: 
(i) The interior adjacent angle is ?ABC
(ii) The interior opposite angles to exterior ?CBX is ?BAC and ?ACB 
Also the interior angles opposite to exterior ?BAY are ?ABC and ?ACB
2. In the fig. 36, two of the angles are indicated. What are the measures of ?ACX and
?ACB?
Solution: 
Given that in ?ABC, ?A = 50
o
 and ?B = 55
o
We know that the sum of angles in a triangle is 180
o
 
 
 
 
 
 
 
Therefore we have 
?A + ?B + ?C = 180
o 
50
o
+ 55
o
+ ?C = 180
o 
?C = 75
o 
?ACB = 75
o 
?ACX = 180
o
- ?ACB = 180
o 
- 75
o
 = 105
o
 
 
3. In a triangle, an exterior angle at a vertex is 95
o
 and its one of the interior opposite 
angles is 55
o
. Find all the angles of the triangle. 
 
Solution: 
 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Hence, for the given triangle, we can say that: 
?ABC+ ?BAC = ?BCO 
55
o
 + ?BAC = 95
o 
?BAC= 95
o
– 55
o 
?BAC = 40
o 
We also know that the sum of all angles of a triangle is 180
o
. 
Hence, for the given ?ABC, we can say that: 
?ABC + ?BAC + ?BCA = 180
o 
55
o
 + 40
o
 + ?BCA = 180
o 
?BCA = 180
o
 –95
o 
?BCA = 85
o 
 
4. One of the exterior angles of a triangle is 80
o
, and the interior opposite angles are 
equal to each other. What is the measure of each of these two angles? 
 
Solution: 
Let us assume that A and B are the two interior opposite angles. 
We know that ?A is equal to ?B. 
Page 3


 
1. In Fig. 35, ?CBX is an exterior angle of ?ABC at B. Name
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ?CBX
Also, name the interior opposite angles to an exterior angle at A.
Solution: 
(i) The interior adjacent angle is ?ABC
(ii) The interior opposite angles to exterior ?CBX is ?BAC and ?ACB 
Also the interior angles opposite to exterior ?BAY are ?ABC and ?ACB
2. In the fig. 36, two of the angles are indicated. What are the measures of ?ACX and
?ACB?
Solution: 
Given that in ?ABC, ?A = 50
o
 and ?B = 55
o
We know that the sum of angles in a triangle is 180
o
 
 
 
 
 
 
 
Therefore we have 
?A + ?B + ?C = 180
o 
50
o
+ 55
o
+ ?C = 180
o 
?C = 75
o 
?ACB = 75
o 
?ACX = 180
o
- ?ACB = 180
o 
- 75
o
 = 105
o
 
 
3. In a triangle, an exterior angle at a vertex is 95
o
 and its one of the interior opposite 
angles is 55
o
. Find all the angles of the triangle. 
 
Solution: 
 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Hence, for the given triangle, we can say that: 
?ABC+ ?BAC = ?BCO 
55
o
 + ?BAC = 95
o 
?BAC= 95
o
– 55
o 
?BAC = 40
o 
We also know that the sum of all angles of a triangle is 180
o
. 
Hence, for the given ?ABC, we can say that: 
?ABC + ?BAC + ?BCA = 180
o 
55
o
 + 40
o
 + ?BCA = 180
o 
?BCA = 180
o
 –95
o 
?BCA = 85
o 
 
4. One of the exterior angles of a triangle is 80
o
, and the interior opposite angles are 
equal to each other. What is the measure of each of these two angles? 
 
Solution: 
Let us assume that A and B are the two interior opposite angles. 
We know that ?A is equal to ?B. 
 
 
 
 
 
 
 
We also know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore from the figure we have,  
?A + ?B = 80
o 
?A + ?A = 80
o
 (because ?A = ?B) 
2 ?A = 80
o 
?A = 80/2 =40
o 
?A= ?B = 40
o 
Thus, each of the required angles is of 40
o
. 
 
5. The exterior angles, obtained on producing the base of a triangle both ways are 
104
o
 and 136
o
. Find all the angles of the triangle. 
 
Solution: 
 
 
In the given figure, ?ABE and ?ABC form a linear pair. 
?ABE + ?ABC =180
o 
?ABC = 180
o
– 136
o 
?ABC = 44
o
  
We can also see that ?ACD and ?ACB form a linear pair. 
?ACD + ?ACB = 180
o 
?ACB = 180
o
– 104
o 
?ACB = 76
o 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore, we can write as 
?BAC + ?ABC = 104
o 
?BAC = 104
o
 – 44
o
 = 60
o 
Thus, 
?ACE = 76
o
 and ?BAC = 60
o
 
 
Page 4


 
1. In Fig. 35, ?CBX is an exterior angle of ?ABC at B. Name
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ?CBX
Also, name the interior opposite angles to an exterior angle at A.
Solution: 
(i) The interior adjacent angle is ?ABC
(ii) The interior opposite angles to exterior ?CBX is ?BAC and ?ACB 
Also the interior angles opposite to exterior ?BAY are ?ABC and ?ACB
2. In the fig. 36, two of the angles are indicated. What are the measures of ?ACX and
?ACB?
Solution: 
Given that in ?ABC, ?A = 50
o
 and ?B = 55
o
We know that the sum of angles in a triangle is 180
o
 
 
 
 
 
 
 
Therefore we have 
?A + ?B + ?C = 180
o 
50
o
+ 55
o
+ ?C = 180
o 
?C = 75
o 
?ACB = 75
o 
?ACX = 180
o
- ?ACB = 180
o 
- 75
o
 = 105
o
 
 
3. In a triangle, an exterior angle at a vertex is 95
o
 and its one of the interior opposite 
angles is 55
o
. Find all the angles of the triangle. 
 
Solution: 
 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Hence, for the given triangle, we can say that: 
?ABC+ ?BAC = ?BCO 
55
o
 + ?BAC = 95
o 
?BAC= 95
o
– 55
o 
?BAC = 40
o 
We also know that the sum of all angles of a triangle is 180
o
. 
Hence, for the given ?ABC, we can say that: 
?ABC + ?BAC + ?BCA = 180
o 
55
o
 + 40
o
 + ?BCA = 180
o 
?BCA = 180
o
 –95
o 
?BCA = 85
o 
 
4. One of the exterior angles of a triangle is 80
o
, and the interior opposite angles are 
equal to each other. What is the measure of each of these two angles? 
 
Solution: 
Let us assume that A and B are the two interior opposite angles. 
We know that ?A is equal to ?B. 
 
 
 
 
 
 
 
We also know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore from the figure we have,  
?A + ?B = 80
o 
?A + ?A = 80
o
 (because ?A = ?B) 
2 ?A = 80
o 
?A = 80/2 =40
o 
?A= ?B = 40
o 
Thus, each of the required angles is of 40
o
. 
 
5. The exterior angles, obtained on producing the base of a triangle both ways are 
104
o
 and 136
o
. Find all the angles of the triangle. 
 
Solution: 
 
 
In the given figure, ?ABE and ?ABC form a linear pair. 
?ABE + ?ABC =180
o 
?ABC = 180
o
– 136
o 
?ABC = 44
o
  
We can also see that ?ACD and ?ACB form a linear pair. 
?ACD + ?ACB = 180
o 
?ACB = 180
o
– 104
o 
?ACB = 76
o 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore, we can write as 
?BAC + ?ABC = 104
o 
?BAC = 104
o
 – 44
o
 = 60
o 
Thus, 
?ACE = 76
o
 and ?BAC = 60
o
 
 
 
 
 
 
 
 
 
6. In Fig. 37, the sides BC, CA and BA of a ?ABC have been produced to D, E and F 
respectively. If ?ACD = 105
o
 and ?EAF = 45
o
; find all the angles of the ?ABC. 
 
Solution: 
In a ?ABC, ?BAC and ?EAF are vertically opposite angles. 
Hence, we can write as 
?BAC = ?EAF = 45
o 
Considering the exterior angle property, we have 
?BAC + ?ABC = ?ACD = 105
o 
On rearranging we get 
?ABC = 105
o
– 45
o
 = 60
o
 
We know that the sum of angles in a triangle is 180
o
 
?ABC + ?ACB + ?BAC = 180° 
?ACB = 75
o 
Therefore, the angles are 45
o
, 60
o
 and 75
o
. 
 
7. In Fig. 38, AC perpendicular to CE and C ?A: ?B: ?C= 3: 2: 1. Find the value of ?ECD. 
 
Solution: 
In the given triangle, the angles are in the ratio 3: 2: 1. 
Let the angles of the triangle be 3x, 2x and x. 
We know that sum of angles in a triangle is 180
o 
Page 5


 
1. In Fig. 35, ?CBX is an exterior angle of ?ABC at B. Name
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ?CBX
Also, name the interior opposite angles to an exterior angle at A.
Solution: 
(i) The interior adjacent angle is ?ABC
(ii) The interior opposite angles to exterior ?CBX is ?BAC and ?ACB 
Also the interior angles opposite to exterior ?BAY are ?ABC and ?ACB
2. In the fig. 36, two of the angles are indicated. What are the measures of ?ACX and
?ACB?
Solution: 
Given that in ?ABC, ?A = 50
o
 and ?B = 55
o
We know that the sum of angles in a triangle is 180
o
 
 
 
 
 
 
 
Therefore we have 
?A + ?B + ?C = 180
o 
50
o
+ 55
o
+ ?C = 180
o 
?C = 75
o 
?ACB = 75
o 
?ACX = 180
o
- ?ACB = 180
o 
- 75
o
 = 105
o
 
 
3. In a triangle, an exterior angle at a vertex is 95
o
 and its one of the interior opposite 
angles is 55
o
. Find all the angles of the triangle. 
 
Solution: 
 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Hence, for the given triangle, we can say that: 
?ABC+ ?BAC = ?BCO 
55
o
 + ?BAC = 95
o 
?BAC= 95
o
– 55
o 
?BAC = 40
o 
We also know that the sum of all angles of a triangle is 180
o
. 
Hence, for the given ?ABC, we can say that: 
?ABC + ?BAC + ?BCA = 180
o 
55
o
 + 40
o
 + ?BCA = 180
o 
?BCA = 180
o
 –95
o 
?BCA = 85
o 
 
4. One of the exterior angles of a triangle is 80
o
, and the interior opposite angles are 
equal to each other. What is the measure of each of these two angles? 
 
Solution: 
Let us assume that A and B are the two interior opposite angles. 
We know that ?A is equal to ?B. 
 
 
 
 
 
 
 
We also know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore from the figure we have,  
?A + ?B = 80
o 
?A + ?A = 80
o
 (because ?A = ?B) 
2 ?A = 80
o 
?A = 80/2 =40
o 
?A= ?B = 40
o 
Thus, each of the required angles is of 40
o
. 
 
5. The exterior angles, obtained on producing the base of a triangle both ways are 
104
o
 and 136
o
. Find all the angles of the triangle. 
 
Solution: 
 
 
In the given figure, ?ABE and ?ABC form a linear pair. 
?ABE + ?ABC =180
o 
?ABC = 180
o
– 136
o 
?ABC = 44
o
  
We can also see that ?ACD and ?ACB form a linear pair. 
?ACD + ?ACB = 180
o 
?ACB = 180
o
– 104
o 
?ACB = 76
o 
We know that the sum of interior opposite angles is equal to the exterior angle. 
Therefore, we can write as 
?BAC + ?ABC = 104
o 
?BAC = 104
o
 – 44
o
 = 60
o 
Thus, 
?ACE = 76
o
 and ?BAC = 60
o
 
 
 
 
 
 
 
 
 
6. In Fig. 37, the sides BC, CA and BA of a ?ABC have been produced to D, E and F 
respectively. If ?ACD = 105
o
 and ?EAF = 45
o
; find all the angles of the ?ABC. 
 
Solution: 
In a ?ABC, ?BAC and ?EAF are vertically opposite angles. 
Hence, we can write as 
?BAC = ?EAF = 45
o 
Considering the exterior angle property, we have 
?BAC + ?ABC = ?ACD = 105
o 
On rearranging we get 
?ABC = 105
o
– 45
o
 = 60
o
 
We know that the sum of angles in a triangle is 180
o
 
?ABC + ?ACB + ?BAC = 180° 
?ACB = 75
o 
Therefore, the angles are 45
o
, 60
o
 and 75
o
. 
 
7. In Fig. 38, AC perpendicular to CE and C ?A: ?B: ?C= 3: 2: 1. Find the value of ?ECD. 
 
Solution: 
In the given triangle, the angles are in the ratio 3: 2: 1. 
Let the angles of the triangle be 3x, 2x and x. 
We know that sum of angles in a triangle is 180
o 
 
 
 
 
 
 
 
3x + 2x + x = 180
o 
6x = 180
o 
x = 30
o 
Also, ?ACB + ?ACE + ?ECD = 180
o 
x + 90
o
 + ?ECD = 180
o
 ( ?ACE = 90
o
) 
We know that x = 30
o
 
Therefore  
?ECD = 60
o
 
 
8. A student when asked to measure two exterior angles of ?ABC observed that the 
exterior angles at A and B are of 103
o
 and 74
o
 respectively. Is this possible? Why or 
why not? 
Solution: 
We know that sum of internal and external angle is equal to 180
o
 
Internal angle at A + External angle at A = 180
o 
Internal angle at A + 103
o
 =180
o 
Internal angle at A = 77
o 
Internal angle at B + External angle at B = 180
o 
Internal angle at B + 74
o
 = 180
o 
Internal angle at B = 106
o 
Sum of internal angles at A and B = 77
o
 + 106
o
 = 183
o 
It means that the sum of internal angles at A and B is greater than 180
o
, which cannot be 
possible. 
 
9. In Fig.39, AD and CF are respectively perpendiculars to sides BC and AB of ?ABC. If 
?FCD = 50
o
, find ?BAD 
 
Solution: 
We know that the sum of all angles of a triangle is 180
o 
Therefore, for the given ?FCB, we have