Prove that the sum of the angle of a traingle is 180°
Answers
Answer:
consider a triangle abc
angle a measures 60° angle b measures 60°
angle c measures 60°
hence sum of the angles =60+60+60=180°
Angle Sum Property
A triangle has three sides and three angles, one at each vertex, bounded by a pair of adjacent sides. In a Euclidean space, the sum of angles of a triangle equals 180 degrees. Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be 180º. Thus, the angle sum property states that the sum of the angles of a triangle is equal to 180º.
The angle sum property of a triangle is one of the most frequently carried out steps in geometry. However, in order to do so, you need to have your basic geometrical concepts in place, which includes certain properties and theorems of mathematics. The angle sum property of triangles is in great use while calculating the unknown angles of a polygon.
What is Angle Sum Property?
A triangle is a closed figure formed by three line segments, consisting of interior as well as exterior angles. An interior angle is an angle formed between two adjacent sides of a triangle, whereas an exterior angle is an angle formed between a side of the triangle and an adjacent side extending outward. As per the angle sum property, the sum of all three angles(interior) of a triangle is 180 degrees, and the exterior angle of a triangle measures the same as the sum of its two opposite interior angles. The triangle angle sum theorem is useful for finding the measure of an unknown angle when the values of the other two angles are known.
Definition
The angle sum property states that the sum of the angles of a triangle is equal to 180°. Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be 180°. This can be represented by the formula:
In a triangle ABC, ∠B + ∠A + ∠C = 180°
Angle Sum Property of a Triangle
Exterior Angles of a Triangle
The exterior angle of a triangle is formed if any side of a triangle is extended and the exterior angle thus formed is equal to the sum of the two opposite interior angles of the triangle. This is referred to as the exterior angle property of a triangle.
Exterior Angles of a Triangle
In the above image of triangle ABC, the interior angles are a, b, c and the exterior angles are d, e, f. According to the Exterior Angle Property of a triangle, the exterior angle is equal to the sum of the two remote interior angles. i.e., in this case,
∠d = ∠b + ∠a
∠e = ∠c + ∠a
∠f = ∠b + ∠c
Proof of the Angle Sum Property
Let's have a look at the proof of the angle sum property of the triangle.
Angle Sum Property Proof
The steps for proving the angle sum property of a triangle are listed below:
Step 1: Draw a line PQ that passes through the vertex A and is parallel to side BC of the triangle ABC.
Step 2: We know that the sum of the angles on a straight line is equal to 180°. In other words, ∠PAB + ∠BAC + ∠CAQ = 180°, which gives, Equation 1: ∠PAB + ∠BAC + ∠CAQ = 180°
Step 3: Now, since line PQ is parallel to BC. ∠PAB = ∠ABC and ∠CAQ = ∠ACB. (Interior alternate angles), which gives, Equation 2: ∠PAB = ∠ABC, and Equation 3: ∠CAQ = ∠ACB
Step 4: Substitute ∠PAB and ∠CAQ with ∠ABC and ∠ACB respectively, in Equation 1 as shown below.
Equation 1: ∠PAB + ∠BAC + ∠CAQ = 180°. Thus we get, ∠ABC + ∠BAC + ∠ACB = 180°
Hence proved, in triangle ABC, ∠B + ∠A + ∠C = 180°. Thus, the sum of all the angles of a triangle is equal to 180°.
Step-by-step explanation:
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