SUM OF ANGLE MEASURES IN A TRIANGLE WORKSHEET

1.  Can a triangle have two right angles ? Explain.

2.  Describe the relationship between the two acute angles in a right triangle. Explain your reasoning.

3.  Can 30°, 60° and 90° be the angles of a triangle ?

4.  Can 35°, 55° and 95° be the angles of a triangle ?

5.  In a triangle, if the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle. 

6. Calculate the size of x in the diagram.

7. Jacob has measured the three angles in a triangle. Two of his measurements are 45°and 70° What is the third measurement?

8. James says that a triangle is right angled. Olivia says that the same triangle is isosceles. They are both correct. Explain how.

9. The ratio of three angles in a triangle are 1:2:3. Work out the size of each angle.

10. The angles of a triangle are (x+21)° , x° and (2x-45)°.Find the value of x.

11. Of the three angles of a triangle ,one is three times the smallest and the other is five times the smallest. Find the angles.

12. In a right triangle the two acute angles are in the ratio 7:11 .Find the acute angles.

13. One angle of a triangle is 65°.Find the remaining two angles if their difference is 20°.

14. The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.

15. One of the exterior angle of a triangle is 80° and the interior opposite angles are in the ratio 3:5. Find the angles of the triangle.

1. Answer : 

No

The sum of the measures of two right angles is 180°. That means the measure of the third angle would be

180° - 180°  =  0°

which is impossible.

2. Answer :

Two acute angles of a right triangle are complimentary angles.

A right triangle is triangle that contains one right angle and two acute angles which all add up to 180°. In a right triangle, one of the angles is a right angle which is 90°. 

Therefore, the sum of the other two acute angles in the right triangle must be equal to 90°. 

So, the relationship between the two acute angles in a right triangle complementary.  

3. Answer :

Let us add all the three given angles and check whether the sum is equal to 180°.

 30° +  60° + 90°  =  180°

Because the sum of the angles is equal 180°, the given three angles can be the angles of a triangle. 

4. Answer :

Let us add all the three given angles and check whether the sum is equal to 180°.

 35° +  55° + 95°  =  185°

Since the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle. 

5. Answer :

Let x be the first angle.

The second angle  =  x + 5

The third angle  =  x + 5 + 5  =  x + 10

We know that,

the sum of the three angles of a triangle  =  180°

x + (x + 5) + (x + 10)  =  180°

3x + 15  =  180°

3x  =  165°

x  =  55°

• The first angle  =  55°
• The second angle  =  55 + 5  =  60°
• The third angle  =  60 + 5  =  65°

So, the three angles of a triangle are 55°, 60° and 65°. 

6. Answer :

x + 60 + 80 = 180

x + 140 = 180

x = 180 - 140

x = 40

7. Answer :

Let x be the third angle.

<1 + <2 + <3 = 180

45 + 70 + <3 = 180

115 + x = 180

x = 180 - 115

x = 65

8. Answer :

Since it is right angle, one of the angle measure will be 90 degree.

In isosceles triangle, two angles will be equal, the measure of that two angles measures will be 45 degree.

9. Answer :

Since the given angle measures are in the ratio 1 : 2 : 3, then those three angle measures will be x, 2x and 3x.

x + 2x + 3x = 180

6x = 180

x = 180/6

x = 30

2x = 2(30) ==> 60

3x = 3(30) ==> 90

So, the three angles are 30, 60 and 90.

10. Answer :

Sum of interior angles of triangle = 180

x + 21 + x + 2x - 45 = 180

4x - 24 = 180

4x = 180 + 24

4x = 204

x = 204/4

x = 51

So, the value of x is 51.

11. Answer :

Let x be the smallest angle.

One angle = 3x

Other angle = 5x

x + 3x + 5x = 180

9x = 180

x = 180/9

x = 20

3x = 3(20) ==> 60

5x = 5(20) ==> 100

So, the required angle measures are 20, 60 and 100.

12. Answer :

7x and 11 are the two angles.

7x + 11x = 180

18x = 180

x = 180/18

x = 10

7x = 7(10) ==> 70

11x = 11(10) ==> 110

So, 70 is the acute angle.

13. Answer :

Sum of interior angle of triangle = 180

One of the angle = 65

65 + sum of other two angles = 180

sum of other two angles = 180 - 65

= 115

Since the difference between the two angles = 20

Let the angles be x and y.

x + y = 115  -----(1)

x - y = 20 -----(2)

(1) + (2)

2x = 135

x = 135/2

x = 67.5

y = 47.5

14. Answer :

Let x be the sum of the two angles. 

Let z be the third angle.

Sum of interior angles = 180

x + z = 180

z = 180 - x

15. Answer :

In a triangle,

exterior angle = sum of interior angles

80 = 3x + 5x

80 = 8x

x = 80/8

x = 10

3x = 30

5x = 50

So, the required angles are 30 and 50.

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