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Triangles CBSE Class 9

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Question 1/20

Q 1. In an isosceles triangle, if the vertex angle is twice of the sum of the base angles, then the measure of the vertex angle of the triangle is 

Solution

Let the base angle be x.

According to the question,

2(x + x) + x + x = 180° 

4x + 2x = 180° 

6x = 180° 

x = 30° 

Vertex angle = 180° - 2x = 180° - 60° = 120° 

Q 2. Two right-angled triangles are congruent to each other if one of their sides and their hypotenuses are equal. This statement is called

Solution

If the hypotenuses and one of the sides of two right-angled triangles are equal, then the triangles are congruent by the RHS criterion.

Q 3. In ABC, AB = AC, AD = AE, BAD = CAE, then   

Solution

In ABD and ACE,

AB = AC 

ABD = ACE Opposite angles of equal sides

BAD = CAE

∆ABD ∆ACE ASA

Ar(ABD) = Ar(ACE)

Ar(ABD) + ar(ADE) = Ar(ACE) + ar(ADE)

Ar(ABE) = ar(ACD)

Q 4. BD is the angle bisector of ABC. If PM is perpendicular to BA and PN is perpendicular to BC, then PMB PNB by   

Solution

In PMB and PNB,

PMB = PNB = 90° 

PB = PB Common side

PM = PN  Any point on the bisector

is equidistant from both arms

PMB PNB RHS

Q 5. Which of the following is required so that two triangles are congruent by the ASA criterion?    

Solution

Two triangles are congruent when a pair of corresponding angles and the included side are equal to each other. 

Q 6. All similar triangles are 

Solution

All similar triangles are not congruent. 

Q 7. If ABC and PQR are right-angled at B and Q, respectively, then which of the following is required so that the triangles are congruent?

Solution

If the hypotenuses and one of the sides of two right-angled triangles are equal, then the triangles are congruent by the RHS criterion. 

Q 8. If DEF PQR and E = a + 10, P = a - 10, R = a, then find the greatest angle of a triangle. 

Solution

According to the question, 

Q = E = a +10, P = a - 10, R = a

In PQR,

P + Q + R = 180° 

a + a + 10 + a - 10 = 180° 

3a = 180° 

a = 60° 

The greatest angle is a + 10 = 60˚ + 10˚ = 70˚. 

Q 9. Two sides of the triangle are of lengths 4 cm and 3.5 cm. The length of the third side of the triangle cannot be

Solution

4 + 3.5 < 8

Hence, the length of the third side of the triangle cannot be 8 cm.

Q 10. If the sides of a triangle are a, b and c, then

Solution

Using triangle inequality,

a + b > c

Q 11. If one of the angles of an isosceles triangle is 70°, then find the other angles. 

Solution

Let the remaining angles each be x as it is an isosceles triangle.

x + x + 70° = 180° 

2x + 70° = 180° 

2x = 110° 

x = 55° 

Q 12. PQR is an isosceles triangle. If Q = R and PM is the median to base QR, find QPM.   

Solution

In PQR, 

PQR = PRQ = 55° 

In PQR, 

P + Q + R = 180° 

∴ ∠P + 55° + 55° = 180° 

∴ ∠P = 70° 

QPM = RPM =  

Median is the angle bisector in an isosceles triangle. 

Q 13. If the sides of a triangle area, a and b, then

Solution

Using triangle inequalities, 2a > b.

Q 14. If angles of an isosceles triangle are in the ratio 1:1:3, then find the vertex angle. 

Solution

Let the angles be x, x and 3x.

x + x + 3x = 180° 

5x = 180° 

x = 36° 

Vertex angle is 36° × 3 = 108°. 

Q 15. If ABC ACB, then ABC is isosceles with 

Solution

If ABC ACB, then ABC is isosceles with AB = AC. 

Q 16. If one of the angles of an isosceles triangle is 100°, then find the other angles. 

Solution

Let the remaining angles each be x as it is an isosceles triangle.

2x + 100° = 180° 

2x = 80° 

x = 40° 

Q 17. In ABC, if A = 74° and B = 66°, then

Solution

According to the question,

A + B + C = 180° 

74° + 66°+ C = 180° 

C = 40° 

Hence, C < B < A

AB < AC < BC

Q 18. Which of the following sides do not form a triangle?

Solution

Consider the sides of the triangle as 7, 9, 16.

7 + 9 = 16

Hence, these sides do not form a triangle

Q 19. AB = CD and AD = BC, then ∆ADC CBA by the

Solution

  

 

Given: AB = CD and AD = BC

To prove: ADC CBA

Proof: In ADC and CBA,

AB = CD

AD = BC

AC = AC Common side

ADC CBA SSS 

Q 20. D is a point on side BC of ∆ABC such that AD = AC, then   

Solution

In DAC,

AD = AC

ADC = ACD  Angle opposite to equal sides

ADC is an exterior angle for ABD.

ADC > ABD

ACD > ABD

ACB > ABC

AB > AC

AB > AD

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Triangles Notes

Revision Notes

Read CBSE Class 9 Mathematics revision notes for Triangles

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