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 75⁰ and 35⁰. Find the measures of the third angle.

Solution :
Let ABC be a triangle such that ∠B = 75⁰ and ∠C = 35⁰. 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 = 70⁰

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 30⁰, 60⁰ and 90⁰.

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) = 40⁰

3x = 3(20) = 60⁰

4x = 4(20) = 80⁰

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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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