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Application of Angle Sum Property of Triangle
We are familiar with the angle sum property of a triangle which can be stated as the sum of all angles in a triangle is 180°. We can verify this by doing the following activity.
Draw any triangle and colour the angles. Check the property as shown below:
From the above we have verified that the sum of three angles of any triangle is 180°
From the above activity, we get the result as, the sum of three angles of any triangle is 180°.
Now we prove the result in a formal method.
Given: A triangle ABC, where ∠A = x , ∠B = y and ∠C = z .
Now let us prove, x + y + z = 180°.
To show this we need to extend BC to D and draw a line CE, parallel to AB.
Now CE makes two angles, ∠ACE and ∠ECD .
Let it be u and v respectively.
Now z, u, v are the angles formed at a point on a straight line.
Therefore z + u + v = 180° . ... (1)
Since AB and CE are parallel and DB is a transversal,
v = y (corresponding angles).
Again AB and CE are parallel lines and AC is a transversal,
u = x (alternate angles). Also z + u + v = 180° [by (1)]
Hence, by replacing u as x and v as y, we get x + y + z = 180°.
Hence the sum of all three angles in a triangle is 180°.
Can the following angles form a triangle?
(i) 80°, 70°, 50°
(ii) 56°, 64°, 60°
(i) Given angles 80°, 70°, 50°
Sum of the angles = 80°+70°+ 50° = 200° ≠ 180°
The given angles cannot form a triangle.
(ii) Given angles 56°, 64°, 60°
Sum of the angles = 56°+ 64°+ 60° = 180°
The given angles can form a triangle.
Find the measure of the missing angle in the given triangle ABC.
Let ∠A = x
We know that,
∠A+ ∠B+∠C=180° (angle sum property)
x + 44° + 31° = 180°
x + 75° = 180°
x = 180°– 75°
x = 105°
In ∆STU, if SU = UT, ∠SUT = 70°, ∠STU = x, find the value of x.
Given, ∠SUT = 70°
∠UST = ∠STU = x [Angles opposite to equal sides]
∠SUT + ∠UST +∠STU = 180°
70° + x + x = 180°
70° + 2x = 180°
2x = 180°– 70°
2x = 110°
x = 110º / 2 = 55°
If two angles of a triangle
having measures 65° and 35°, find the measure of the third angle.
Given angles are 65° and 35°.
Let the third angle be x
65° + 35°+ x = 180°
100° + x = 180°
x = 80°
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