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Q1. Find the value of x in the figure below:    

Solution

Here, exterior angle = (3x + 6)^o  Interior angles are x^o and 30^o  We know that,  Exterior angle = Sum of interior opposite angles  or (3x + 6) = x + 30^o  3x + 6^o - x = 30^o (subtract x from both sides)  (3x - x) + 6^o = 30^o (combining like terms)  2x + 6^o = 30^o (subtract 6^o from both sides)  2x = 24^o (divide both sides by 2)  Thus, x = 12^o      

Q2. PQ is parallel to BC. In the following diagram, the measure of angle A is:

• 1) 70o
• 2) 60o
• 3) 80o
• 4) 100o

Solution

PQ is parallel to BC, APQ =  B = 70°  .....(Corresponding Angles) Now in triangle APQ, A + P + Q = 180°  .....(Sum of all angles of a triangle is 180°) A + 70° + 30° = 180° A = 180° - 100° A = 80°

Q3. Raju takes a rectangular piece of plywood that is 12 cm long and uses a table saw to make a 20 cm cut from corner to opposite corner. What was the width of the piece of plywood?

Solution

Draw a diagram. Let a be the width of the plywood. Using the Pythagorean theorem, with b = 12 and h = 20. p² + b² = h² a² + 12² = 20² a² + 144 = 400 (subtract 144 from both sides) a² = 256 a² = 16² a = 16 The width of a piece of plywood is 16 cm.

Q4. The exterior angle of a triangle is 80⁰. If one of the interior opposite angles is 35⁰, the measure of the other two angles is

• 1) 80⁰; 20⁰
• 2) 40⁰; 60⁰
• 3) 100⁰; 45⁰
• 4) 50⁰; 50⁰

Solution

In the figure, a + 80⁰ = 180⁰     ......(linear pair) Thus, a = 100⁰ b + 35⁰ = 80⁰      ........(Since exterior angle = sum of interior opposite angles) Thus, b = 45⁰

Q5. The length of x is such that 2 < x < 8. What is the length y of the given triangle?

Solution

In a triangle sum of lengths of either two sides is always greater than the third side and also difference of lengths of either two sides is always lesser than the third side. So, 5 - y < x < 5 + y Also, it is given that 2 < x < 8 So, comparing the above two inequalities, we get 5 + y = 8 i.e. y = 8 - 5 = 3

Q6. In a triangle PQR, PQ = 2 cm, QR = 3 cm and RP = 4 cm. The angle with larger measure is:

• 1) Angle Q
• 2) Angle P
• 3) None
• 4) Angle R

Solution

Larger side = RP = 4 cm Thus, the angle with larger measure is Angle Q as it is opposite to the largest side RP.

Q7. What is the value of a?

Solution

Use the Pythagorean theorem, with p = 8 and h = 10. h² = p² + b² 10² = 8² + a² 8² + a² = 10² (transposing both the sides) 64 + a² = 100 (subtract 64 from both sides) a² = 36 a² = 6² a = 6 The length of the missing leg is 6 m.

Q8.   In the given figure, the measures of angle a and b is

• 1) a = 55⁰; b = 55⁰
• 2) a = 122⁰; b = 58⁰
• 3) a = 58⁰; b = 122⁰
• 4) a = 58⁰; b = 52⁰

Solution

Q9. Consider the following figure, Choose the correct one.

• 1) AB + MC > AM
• 2) AB+AC >2BM
• 3) AB + AC < 2BM
• 4) AB + AC = 2BM

Solution

In a triangle ABC, AB + AC > BC But M is the mid-point of BC, therefore AB + AC > 2BM

Q10. Which of the following can be the sides of a right angled triangle?

• 1) 8, 10, 5
• 2) 8, 17, 15
• 3) 7, 8, 9
• 4) 1, 3, 6

Solution

We have 17² = 289 and 8² + 15² = 64 + 225 = 289 This give 17² = 8² + 15² Therefore, 8, 17, 15 are the sides of a right angled triangle.

Q11. Find the value of x in the figure given below:      

Solution

In triangle KML,   Since all the equal angle x's are vertically opposite to angles KLM, MKL and KML, we have  KLM = MKL = KML = x^o   Using angle sum property of triangles we have,   KLM + MKL + KML = 180^o   i.e. x^o + x^o + x^o = 180^o    3x^o = 180^o(divide both sides by 3)   x^o = 60^o   Thus, all interior angles are of equal measure 60^o      

Q12. The perimeter of the following triangle is

• 1) 5.5 cm
• 2) 2.5 cm
• 3) 6.5 cm
• 4) 6 cm

Solution

Here,   Therefore, perimeter = 1.5 + 2 + 2.5 = 6 cm

Q13. If the sum of the sides of a right triangle is 49 inches and the hypotenuse is 41 inches, find the two sides.

Solution

Let "a" and "b" be the lengths of the two shorter sides. The sum is: a + b = 49 so, a = 49 - b Using Pythagorean Theorem: perpendicular² + base² = hypotenuse² (49 - b)² + b² = 41² (by substitution) 2401 - 98b + b² + b² = 1681 2b² - 98b + 720 = 0 (take 2 common from L.H.S) b² - 49b + 360 = 0 (b - 9)(b - 40) = 0 b = 9 or b = 40 In this case, either solution will do. If b = 9, then a = 49 - b = 49 - 9 = 40. Or if b = 40, then a = 49 - b = 49 - 40 = 9. Thus, one side is 40 inches long, and the other side is 9 inches long.

Q14. Between what two measures should the length of the side DB fall?

Solution

In a triangle sum of lengths of either two sides is always greater than the third side and also difference of lengths of either two sides is always lesser than the third side. Here third side AB will be lesser than 7 + 18 = 25 and also it will be greater than 18 - 7 = 11. That is, 11 < AB < 25 Given AD = 6 i.e. 11 < AD + DB < 25 i.e. 11 < 5 + DB < 25 Thus, 5 < DB < 19

Q15. Answer what kind of triangle it is: (a) A triangle has angle measurements of 90^o, 20^o and 70^o? (b) A triangle has angle measurements of 14^o, 30^o and 136^o? (c) A triangle has angle measurements of 76^o, 68^o and 36^o?

Solution

We know that In an acute-angled triangle, all three angles are less than 90°. In a right-angled triangle, one angle is exactly 90^o. In an obtuse-angled triangle, one angle is greater than 90^o. (a) This triangle is a right-angled triangle because it has a 90^o angle which is a right angle. (b) This triangle is an obtuse-angled triangle because it has a 136^o angle which is an obtuse angle. (c) This triangle is an acute-angled triangle because it has all three angles acute, that is less than 90^o.