Chapter - 6 Triangles

Exercise

In each of the questions 1 to 49, four options are given, out of which only one is correct. Choose the correct one.

1. The sides of a triangle have lengths (in cm) 10, 6.5 and a, where a is a

whole number. The minimum value that a can take is

1. 6 (b) 5 (c) 3 (d) 4

Solution:

Given: The sides of a triangle have lengths (in cm) 10, 6.5 and a. As we know that sum of lengths of any two sides of a triangle is greater than length of third side.

So, <math>a + 6.5 > 10</math>

<math>a > 10 - 6.5</math>

a > 3.5

According to the question, a is whole number. So, the minimum value a can take is 4. Hence, the correct option is (d).

1. Triangle DEF of Fig. 6.6 is a right triangle with <math>\angle E = 90^{\circ}</math>. What type of angles are <math>\angle D</math> and <math>\angle F</math>?
2. They are equal angles (b) They form a pair of adjacent angles
3. They are complementary angles (d) They are supplementary angles

Fig. 6.6

Solution: <math>\angle D + \angle E + \angle F = 180^{\circ}</math> [Angle sum property of a triangle] <math>\angle D + \angle F = 180^{\circ} - 90^{\circ} [\angle E = 90^{\circ} (given)]</math> <math>= 90^{\circ}</math> So, <math>\angle D</math> and <math>\angle F</math> are complementary angles. Hence, the correct option is (c).

1. In Fig. 6.7, <math>PQ = PS</math>. The value of x is
2. <math>35^{\circ}</math> (b) <math>45^{\circ}</math> <math>(c) 55^{\circ}</math>

x

110° 25° O S

Fig. 6.7

Solution:

110° <math>25\%</math> Q S

See the above figure, in triangle PQS, <math>\angle 2 + \angle 3 = 110^{\circ}</math> ...(i) [Exterior angle property of a triangle]

<math>\angle 2 + \angle 3 + \angle 4 = 180^{\circ}</math>

[Angle sum property of a triangle]

[Using equation (i)]

<math>\angle 4 = 180^{\circ} = 110^{\circ}</math>

So, <math>\angle 4 = 70^{\circ}</math>

Now, <math>PQ = PS</math>

[Given] ...(ii)

<math>\angle 2 = \angle 4 = 70^{\circ}</math>

Now, in <math>\triangle PRS</math>:

<math>\angle 2 = x + 25^{\circ}</math>

[Exterior angle property of a triangle]

<math>x = 70^{\circ} - 25^{\circ}</math>

<math>x = 45^{\circ}</math>

[Using equation (ii)]

Hence, the correct option is (b).

1. In a right-angled triangle, the angles other than the right angle are
2. obtuse (b) right (c) acute (d) straight

Solution:

As we know that the sum of angles other than right angle in a right-angled triangle is 90° So, both angles other than the right angle must be acute.

Hence, the correct option is (c).

1. In an isosceles triangle, one angle is <math>70^{\circ}</math>. The other two angles are of
2. <math>55^{\circ}</math> and <math>55^{\circ}</math> (ii) <math>70^{\circ}</math> and <math>40^{\circ}</math> (iii) any measure

In the given option(s) which of the above statement(s) are true?