Angle Sum Property of Triangles

In this section, we shall state and prove angle sum property of triangles. Here we will discuss some problems based on it.

Theorem 1: The sum of all the angles of a triangle is 1800
Given : A triangle ABC.
To Prove:∠A + ∠B + ∠C= 1800
Construction: Draw CE such that CE || AB
Statements
Reasons
1)BA || CE 1) By Construction
2) ∠A = ∠ACE 2) Alternate interior angle
3) ∠B = ∠DCE 3) Corresponding angles
4)∠A + ∠B = ∠ACE + ∠DCE 4) Addition property of (1) and (2)
5) ∠A + ∠B + ∠ACB = ∠ACE + ∠DCE + ∠ACB 5) Adding ∠ACB to both sides
6) ∠A + ∠B + ∠C = 1800 6) Straight line angles.

Examples :

1) Two angles of a triangle are of measures 75
0 and 35 0 . Find the measures of the third angle.

Solution :
Let ABC be a triangle such that ∠B = 75
0 and ∠C = 35 0 . Then, we have to find the measure of the third angle A.

By angle sum property of triangles,

∠A + ∠B + ∠C = 180

∠A + 75 + 35 = 180

∠A + 110 = 180

∠A = 180 -110

∠A = 70
0

2) Of the three angles of a triangle, one is twice the smallest and another is three times the smallest. Find the angles.

Solution :
Let the smallest angle be x ,

Other two angles be 2x and 3x.

By angle sum property,

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
0 , 60 0 and 90 0 .

3) If the angles of a triangle are in the ratio 2:3:4, determine the three angles.

Solution :
Let the ratio be x .

So, the angles are 2x, 3x and 4x.

By angle sum property,

2x + 3x + 4x =180

9x = 180

x = 180/9

x = 20

three angles are 2x = 2(20) = 40
0

3x = 3(20) = 60
0

4x = 4(20) = 80
0

Triangles

Introduction to Triangles
Types of Triangles on the basis of Sides
Types of Triangles on the basis of Angles
Angle Sum Property of Triangles
Exterior and Interior angles of Triangle
Triangle Inequality Property
Congruent Triangles
Postulates of Congruent Triangle
Inequality in Triangle

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