In this particular article, the topic of the Angle Sum Triangle has been explained in great detail using simple language and examples by the subject experts at Vedantu. Students are advised to download the free PDF of the notes and can also watch the video lectures on the topic to get a complete understanding and excel in their performance in any examination. Show What is a Triangle? Properties of Triangle Interior and Exterior Angle of a Triangle Angle sum property Exterior Angle property Solved examples What is a Triangle?In our daily life, we all see a lot of things that are of different shapes and structures. Things like Traffic signs, pyramids, truss bridges, sailboards, and roofing a house are a few things that look like a triangle. So what is a triangle? A triangle is a closed polygon that is formed by only three line segments. Triangles can be classified based on their sides and angles. Based on their sides, triangles can be equilateral, isosceles, and scalene. And based on their angle, a triangle can be an acute triangle, right triangle, an obtuse triangle. Properties of TrianglesThe properties of a triangle are listed below
Interior and Exterior Angle of a TriangleThere are two important attributes of a triangle i.e., angle sum property and exterior angle property. We may have questions in our minds about what is an angle sum property of a triangle? And how can we prove angle sum property? So let's clear our heads with these questions. We know already that a triangle has an interior as well as an exterior angle. The Interior angle is an angle between the adjacent sides of a triangle and an exterior angle is an angle between the side of a triangle and an adjacent side extending outward. Angle Sum PropertyTheorem: Prove that the sum of all the three angles of a triangle is 180 degrees or 2 right angles. Proof: ∠1 = ∠B and ∠3 = ∠C………….(i) Alternative angle = PQ||BC ∠1 +∠2 +∠3 = 180 ∠B +∠2 + ∠C = 180 ∠B +∠CAB + ∠C = 180 = 2 right angles Proved Theorem 2: In a triangle, if one side is formed then the exterior angle formed will be equal to the sum of two interior opposite angles. ∠4 = ∠1 + ∠2 Proof: ∠3 = 180 - (∠1 +∠2)..............(i) ∠3 + ∠4 = 180 Or ∠3 = 180 - ∠4…………(ii) By (i) and (ii) 180 - (∠1 + ∠2) = 180 - ∠4 ∠1 + ∠2 = ∠4 Proved Exterior Angle PropertyProof of exterior angle property The exterior angle theorem asserts that if a triangle’s side gets extended, then the resultant exterior angle will be equal to the total of the two opposite interior angles of the triangle. According to the Exterior Angle Theorem, the sum of measures of ∠ABC and ∠CAB will be equal to the exterior angle ∠ACD. The general proof of this theorem is explained below: Proof: Consider ∆ABC as given below such that the side BC of ∆ABC is extended. A parallel line to the side AB is drawn.
Thus, from the above statements, we can see that the exterior angle ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC. Solved ExamplesExample 1) In the following triangle, find the value of x. Solution) x + 24 + 32 = 180 x + 56 = 180 x = 180 - 56 x = 124 Example 2) In the triangle ABC given below, find the area of a triangle inscribed inside a square of 20 cm. Solution) Area of a triangle = \[\frac{1}{2} \times base \times height\] = \[\frac{1}{2} (20) (20) \] = \[200 cm^{2}\] Key Learnings from the Chapter -
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