Ex-15.2, (Part - 1), Properties Of Triangles, Class 7, Math RD Sharma Solutions | RD Sharma Solutions for Class 7 Mathematics PDF Download

Q1. Two angles of a triangle are of measures 150^∘ and 30^∘. Find the measure of the third angle.

Solution.

Let the third angle be x

Sum of all the angles of a triangle = 180^∘

105^∘+ 30^∘+x = 180^∘

135^∘+ x = 180^∘

x = 180^∘ – 135^∘

x = 45^∘

Therefore the third angle is 45^∘

Q2. One of the angles of a triangle is 130^∘, and the other two angles are equal What is the measure of each of these equal angles?

Solution.

Let the second and third angle be x

Sum of all the angles of a triangle = 180^∘

130∘+ x+x = 180^∘

130^∘+2x = 180^∘

2x = 180^∘– 130^∘

2x = 50^∘

x = 50/2

x = 25^∘

Therefore the two other angles are 25^∘ each

Q3. The three angles of a triangle are equal to one another. What is the measure of each of the angles?

Solution.

Let the each angle be x

Sum of all the angles of a triangle = 180^∘

x+x+x = 180^∘

3x = 180^∘

x = 1803

x = 60^∘

Therefore angle is 60^∘ each

Q4. If the angles of a triangle are in the ratio 1 : 2 : 3, determine three angles.

Solution.

If angles of the triangle are in the ratio 1:2:3 then take first angle as ‘x’, second angle as ‘2x’ and third angle as ‘3x’

Sum of all the angles of a triangle = 180^∘

x+2x+3x = 180^∘

6x = 180^∘

x = 180/6

x=30^∘

2x = 30^∘×2 = 60^∘

3x = 30^∘×3 = 90^∘

Therefore the first angle is 30^∘, second angle is 60^∘ and third angle is 90^∘

Q5. The angles of a triangle are   Find the value of x.

Solution.

Sum of all the angles of a triangle = 180^∘

Hence we can conclude that x is equal to 100^∘

Q6. The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10^∘. Find the three angles.

Solution.

Let the first angle be x

Second angle be x+10^∘

Third angle be x+10^∘+ 10^∘

Sum of all the angles of a triangle = 180^∘

x+ x+10^∘+ x+10^∘+ 10^∘ = 180^∘

3x+30 = 180

3x = 180-30

3x = 150

x = 150/3

x = 50^∘

First angle is 50

Second angle x+10^∘ = 50+10 = 60^∘

Third angle x+10^∘+ 10^∘ = 50+10+10 = 70^∘

Q7. Two angles of a triangle are equal and the third angle is greater than each of those angles by 30^∘. Determine all the angles of the triangle

Solution.

Let the first and second angle be x

The third angle is greater than the first and second by 30^∘ = x+30^∘

The first and the second angles are equal

Sum of all the angles of a triangle=180∘

x+x+x+30^∘ = 180^∘

3x+30 = 180

3x=180-30

3x = 150

x = 150/3

x = 50^∘

Third angle = x+30^∘ = 50^∘+ 30^∘ = 80^∘

The first and the second angle is 50^∘ and the third angle is 80^∘

Q8. If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.

Solution.

One angle of a triangle is equal to the sum of the other two

x = y+z

Let the measure of angles be x,y,z

x+y+z = 180^∘

x+x = 180^∘

2x = 180^∘

x = 180^∘/2

x = 90^∘

If one angle is 90^∘ then the given triangle is a right angled triangle

Q9. If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.

Solution.

Each angle of a triangle is less than the sum of the other two

Measure of angles be x,y and z

x>y+z

y<x+z

z<x+y

Therefore triangle is an acute triangle

Q10. 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^∘, 37^∘, 80^∘

(ii) 45^∘, 61^∘, 73^∘

(iii) 59^∘, 72^∘, 61^∘

(iv) 45^∘, 45^∘, 90^∘

(v) 30^∘, 20^∘, 125^∘

Solution.

(i) 63^∘, 37^∘, 80^∘ = 180^∘

Angles form a triangle

(ii) 45^∘, 61^∘, 73^∘ is not equal to 180^∘

Therefore not a triangle

(iii) 59^∘, 72^∘, 61^∘ is not equal to 180^∘

Therefore not a triangle

(iv) 45^∘, 45^∘, 90^∘ = 180

Angles form a triangle

(v) 30^∘, 20^∘, 125^∘ is not equal to 180^∘

Therefore not a triangle

Q11. 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

Measure of the angles be 3x, 4x, 5x

Sum of  the angles of a triangle = 180^∘

3x+4x+5x = 180^∘

12x = 180^∘

x = 180^∘/12

x = 15^∘

Smallest angle = 3x

= 3 x 15^∘

= 45^∘

Q12. Two acute angles of a right triangle are equal. Find the two angles.

Solution.

Given acute angles of a right angled triangle are equal

Right triangle: whose one of the angle is a right angle

Measured angle be x,x,90^∘

x+x+180^∘ = 180^∘

2x = 90^∘

x = 90^∘/2

x = 45^∘

The two angles are 45^∘  and 45^∘

Q13. One angle of a triangle is greater than the sum of the other two. What can you say about the measure of this angle? What type of a triangle is this?

Solution.

Angle of a triangle is greater than the sum of the other two

Measure of the angles be x,y,z

x>y+z   or

y>x+z  or

z>x+y

x or y or z>90^∘ which is obtuse

Therefore triangle is an obtuse angle

Q14. AC, AD and AE are joined. Find ∠FAB+∠ABC+∠BCD+∠CDE+∠DEF+∠EFA

Solution.

∠FAB+∠ABC+∠BCD+∠CDE+∠DEF+∠EFA

We know that sum of the angles of a triangle is 180^∘

Therefore in ΔABC,we have

∠CAB+∠ABC+∠BCA = 180^∘—(i)

In ΔACD,we have

∠DAC+∠ACD+∠CDA = 180^∘—(ii)

In ΔADE,we have

∠EAD+∠ADE+∠DEA = 180^∘—(iii)

In ΔAEF,we have

∠FAE+∠AEF+∠EFA = 180^∘—(iv)

Adding (i),(ii),(iii),(iv) we get

∠CAB+∠ABC+∠BCA+∠DAC+∠ACD+∠CDA+∠EAD+∠ADE+∠DEA+∠FAE+∠AEF+∠EFA = 720^∘

Therefore 

∠FAB+∠ABC+∠BCD+∠CDE+∠DEF+∠EFA = 720^∘