Angle sum property of a triangle‌

A triangle is a three-sided convex polygon. It has three interior angles on each of its vertices. Triangles are classified on the basis of

Interior angles as an acute-angled triangle, obtuse-angled triangle and right-angled triangle.
Length of sides as an equilateral triangle, isosceles triangle and scalene triangle.

A common property of all kinds of triangles is the angle sum property. The angle sum property of triangles is 180°. This means that the sum of all the interior angles of a triangle is equal to 180°. This property is useful in calculating the missing angle in a triangle or to verify whether the given shape is a triangle or not. It is also frequently used to calculate the exterior angles of a triangle when interior angles are given. For example,

In a given triangle ABC,

∠ABC + ∠ACB + ∠CAB = 180°

Theorems of angle sum property of triangle

Theorem 1: The angle sum property of a triangle states that the sum of interior angles of a triangle is 180°.

To prove the above theorem, consider PQR.

Step 1: Construct a line XY such that XY passes through the point P and is parallel to QR.

Step 2: Since XY is parallel to QR, ∠XPQ = ∠ PQR. Also, ∠YPR = ∠PRQ. This is because these angles are pairs of alternate interior angles and alternate interior angles are equal to each other.

Step 3: We know that XY is a straight line passing through point P. Therefore, ∠ XPQ + ∠QPR + ∠YPR = 180°. This is the property of a straight line.

Step 4: From step 2, we know that alternate angles are equal and substituting the values from step 2 in step 3 we get,

∠PQR + ∠QPR + ∠PRQ = 180°

which is the angle sum property of a triangle.

Theorem 2: The exterior angle of a triangle formed by extending any side of the triangle is equal to the sum of angles corresponding to the other sides of the triangle.

To prove the above statement, consider a triangle XYZ whose side YZ is extended to meet a point A on the line.

Step 1: Extend the side YZ to a point A on the extended line.

Step 2: Now AZ is a straight line.

Step 3: By the property of straight line, ∠AYX + ∠XYZ = 180°.

Step 4: By the angle sum property of triangle, ∠XYZ + ∠YXZ +∠XZY = 180°.

Step 5: Using equation obtained in step 3 and step 4, on equating we get,

∠AYX + ∠XYZ = ∠XYZ + ∠YXZ +∠XZY

∠AYX = ∠YXZ +∠XZY

which implies that the exterior angle is equal to the sum of the opposite interior angles of the triangle.