# Angle Sum Property

**Theorem 8**

The sum of the three angles of a triangle is equal to 180Â°.

**Given: **A triangle ABC.

**To Prove: âˆ **ABC +âˆ ACB + âˆ BAC = 180Â° or âˆ 1 + âˆ 2 + âˆ 3 = 180Â°

**Construction: **Through A, draw a line MN parallel to BC.

**Proof: ** Since BC || MN,

âˆ´ âˆ 5 = âˆ 1 and âˆ 4 = âˆ 2 â€¦.(Alternate interior angles)

âˆ´ âˆ 5 + âˆ 4 = âˆ 1 + âˆ 2

Adding âˆ 3 on both sides, we get

Ãž âˆ 5 + âˆ 4 + âˆ 3 = âˆ 1 + âˆ 2 + âˆ 3

But âˆ 5 + âˆ 3 + âˆ 4 = 180Â° â€¦ (Linear pair axiom)

âˆ´ âˆ 1 + âˆ 2 + âˆ 3 = 180Â°

Thus the sum of the three angles of a triangle is 180Â°.

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If the ratio of three angles of a triangle is 2 : 3 : 4, find the angles.

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**Example:**

**Solution : ** Let the measure of three angles of the triangle ABC be 2k, 3k and 4k

âˆ´ 2k + 3k + 4k = 180Â°

âˆ´ Â Â Â Â Â Â Â Â Â Â Â Â Â Â 9k = 180Â°

âˆ´ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â k = 20

Hence the angles of the triangle are 40Â°, 60Â° and 80Â°.

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