FINDING ANGLE MEASURES OF A TRIANGLE USING RATIOS

Problem 1 :

The ratio of the angle measures of a triangle is 1: 1: 1. Find the angle measures. Then classify the triangle by its angle measures.

Solution :

Let the common ratio between the angles be x.

The sum of the interior angles of a triangle = 180˚

1x + 1x + 1x = 180˚

3x = 180˚

x = (180˚)/3

x = 60˚

The required angles are,

60˚ ; 1(60˚) = 60˚ ; 1(60˚) = 60˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Problem 2 :

The ratio of the angle measures in a triangle is 6 : 5 : 4. Find the angle measures. Then classify the triangle by its angle measures.

Solution :

Let the common ratio between the angles be x.

The sum of interior angles of a triangle = 180˚

6x + 5x + 4x = 180˚

15x = 180˚

x = (180˚)/15

x = 12˚

The required angles are,

6(12˚) = 72˚ ; 5(12˚) = 60˚ ; 4(12˚) = 48˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Find the value of x. Then classify the triangle by its angle measures.

Problem 3 :

Solution :

The sum of the interior angles of a triangle = 180˚

60˚ + 2x˚ + 2x˚ = 180˚

60˚ + 4x˚ = 180˚

4x˚ = 180˚ – 60˚

4x˚ = 120˚

x˚ = (120˚)/4

x = 30˚

The required angles are,

60˚ ; 2(30˚) = 60˚ ;  2(30˚) = 60˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Problem 4 :

Solution :

The sum of the interior angles of a triangle = 180˚

80˚ + 4x˚ + x˚ = 180˚

80˚ + 5x˚ = 180˚

5x˚ = 180˚ – 80˚

5x˚ = 100˚

x = (100˚)/5

x = 20˚

The required angles are,

80˚ ; 4(20˚) = 80˚ ;  20˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Problem 5 :

Solution :

The sum of the interior angles of a triangle = 180˚

 x˚ + (x + 25)˚ + 25˚ = 1800˚

x˚ + x˚ + 25˚ + 25˚ = 180˚

2x˚ + 50˚ = 180˚

2x˚ = 180˚ – 50˚

2x˚ = 130˚

x = 65˚

The required angles are,

65˚ ; (65˚ + 25˚) ; 25˚

65˚ ; 90˚ ; 25˚

Since one of the angles is 90˚, it is a Right Angle Triangle.

Problem 6 :

Solution :

42˚ + 2x˚ + x˚ = 180˚

42˚ + 3x˚ = 180˚

3x˚ = 180˚ – 42˚

3x˚ = 138˚

x = 46˚

The required angles are,

42˚ ; 2(46˚) = 92˚ ;  46˚

Since one of the angles is greater than 90˚, it is an Obtuse Angle Triangle.

Problem 7 :

Solution :

 4x˚ + 5x˚ + 45˚ = 180˚

60˚ + 4x˚ = 180˚

4x˚ = 180˚ – 60˚

4x˚ = 120˚

x = 30˚

The required angles are,

60˚ ; 2(30˚) = 60˚ ;  2(30˚) = 60˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Problem 8 :

Solution :

The sum of the interior angles of a triangle = 180˚

49˚ + (2x – 8)˚ + 41˚ = 180˚

49˚ + 2x˚ - 8˚ + 41˚ = 180˚

  82˚ + 2x˚ = 180˚

2x˚ = 180˚ – 82˚

x = (98˚)/2

x = 49˚

The required angles are,

49˚ ; (2(49)˚ - 8˚) ; 41˚

49˚ ; 90˚ ; 41˚

Since one of the angles is 90˚, it is a Right Angle Triangle.

Problem 9 :

Solution :

(3x + 51)˚ + x˚ + 37˚ = 180˚

3x˚ + 51˚ + x˚ + 37˚ = 180˚

4x˚ + 51˚ +37˚ = 180˚

4x˚ + 88˚ = 180˚

4x˚ = 180˚ – 88˚

x˚ = 992˚)/4

x = 23˚

The required angles are,

(3(23)˚ + 51˚) ; 23˚ ; 37˚

120˚ ; 23˚ ; 37˚

Since one of the angles is greater than 90˚, it is an Obtuse Angle Triangle.

Problem 10 :

Solution :

2x˚ + x˚ +(3x – 18)˚ = 180˚

2x˚ + x˚ + 3x˚ – 18˚ = 180˚

3x˚ + 3x˚ - 18˚ = 180˚

6x˚ - 18˚ = 180˚

6x˚ = 180˚ + 18˚

x˚ = (198˚)/6

x = 33˚

The required angles are,

2(33˚) ; 33˚ + (3(33˚) – 18˚)

66˚ ; 33˚ ; 81˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.

Problem 11 :

Solution :

(x + 15)˚ + x˚ + 39˚ = 180˚

(x˚ + 15˚) + x˚ + 39˚ = 180˚

2x˚ + 15˚ + 39˚ = 180˚

2x˚ + 54˚ = 180˚

2x˚ = 180˚ – 54˚

x˚ = (126˚)/2

x = 63˚

The required angles are,

(63˚ + 15˚) ; 63˚ ; 39˚

78˚ ; 63˚ ; 39˚

Since all the three angles are less than 90˚, it is an Acute Angle Triangle.