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.
|1.||What is Angle Sum Property?|
|2.||Exterior Angle of a Triangle|
|3.||Proof of the Angle Sum Property|
|4.||FAQs on Angle Sum Property|
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.
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°
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.
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.
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°.
- The sum of the angles of a triangle is always 180°.
- If any side of a triangle is extended, then the exterior angle thus formed is equal to the sum of the two interior opposite angles.
- A triangle cannot have more than one right angle.
- In a right-angled triangle, the sum of two acute angles is equal to 90°.
Example 1: Sam needs to find the measure of the third angle of a triangle ABC in which ∠ABC = 45° and ∠ACB = 55°. Can you help him?
We know that ∠ABC =45° and ∠ACB =55°. Using the Angle Sum Property of the triangle, ∠A + ∠B + ∠C = 180, ∠A + 45 + 55° = 180°, ∠A + 100° = 180°, ∠A = 180° -100°, ∠A = 80°. Therefore, the third angle: ∠A = 80°
Example 2: If the angles of a triangle are in the ratio 3:4:5, determine the value of the three angles.
Let the angles be 3x, 4x and 5x. According to the Angle Sum Property of the triangle, 3x + 4x + 5x = 180°, 12x = 180, x= 15. Thus, the three angles will be: 3x = 3 x 15 = 45°, 4x = 4 x 15 = 60°, 5x = 5 x 15 = 75°. Therefore, the three angles are: 45°, 60°, 75°.
FAQs on Angle Sum Property
How do you Solve for Exterior Angles?
The exterior angle of a polygon is the angle formed between any side of a polygon and a line that is extended from the next side. It should be noted that the corresponding interior and exterior angles are supplementary and the exterior angles of a regular polygon are equal in measure.
What is the Angle Sum Property of a Polygon?
We know that the sum of interior angles in a triangle sums up to 180°. In order to find the sum of interior angles of a polygon, we should multiply the number of triangles in the polygon by 180°. Thus, the sum of interior angles of a polygon can be calculated with the formula: S = ( n − 2) × 180°. It should also be noted that the sum of exterior angles of a polygon is 360°
What is the Sum of All Angles in a Triangle?
The sum of the angles of any triangle is equal to 180°
What is the Angle Sum Property of a Hexagon?
Each interior angle of a regular hexagon is 120°. Since the sum of these angles will always be 360°, each exterior angle would be 60° (360° ÷ 6 = 60°).
What is the Angle Sum Property of a Quadrilateral?
According to the angle sum property of a quadrilateral, the sum of all the four interior angles is 360°.
What is the Exterior Angle Property?
The exterior angle theorem says that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.
What is the Formula of Angle Sum Property?
The formula for the angle sum property is S = ( n − 2) × 180°.
How do you Find the Sum of an Angle in a Triangle?
The sum of the angles of a triangle is always 180°. You can add the two known angles and subtract their sum from 180 to get the measure of third angle.