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 180^{0}
Given : A triangle ABC. To Prove:∠A + ∠B + ∠C= 180^{0} Construction: Draw CE such that CE || AB |
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 = 180^{0} | 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