The Sum Property

The smallest polygon is a triangle, which has three sides and interior angles and three edges and three vertices. According to the angle sum property, the total of a triangle’s angles equals 180º. The sum of a triangle’s interior angles is always 180o, whether it’s an acute, obtuse, or right triangle. One of the most commonly utilised properties in geometry is the triangle’s angle sum. This characteristic is primarily used to determine unknown angles. When the values of two other angles are known, the angle sum property is used to determine the constant meaning and measure an unknown inner angle.

A few of the angle sum property theorems are discussed below:

According to this property, a triangle’s internal angles add up to 180 degrees. 

Interior angles are created when any two edges of a triangle meet at the vertex. The internal angle is formed by two of a triangle’s sides. The inner angle property of a triangle is another name for this feature. According to this property, the total of a triangle’s internal angles is 180º. 

The angle sum property formula for the triangle ABC is A+B+C = 180° if the triangle is ABC.

Theorem 1:

The sum of a triangle’s internal angles is 180°, according to Angle Sum Properties of Triangle.

Proof:

Consider a triangle ABC. Draw a line to show that triangles have the above feature.

Parallel with the triangle’s BC side.

Triangle’s Angle Sum Property

Given that

PQ is a straight line; the following constant meanings can be drawn:

180° = PAB + BAC + QAC…….. (1)

PQ||BC, 

AB and AC are transversals. Hence they are transversals. 

QAC = ACB is the result (a pair of alternate angles)

PAB = CBA, as well (a pair of alternate angles)

In equation, substituting the values of QAC and PAB (1),

180º = ACB + BAC + CBA

As a result, a triangle’s interior angles add up to 180 degrees.

The Interior angle sum property theorem is another example:

A triangle’s inner angles add up to 180 degrees or two right angles (2x 90 degrees). 

Assume that you have an ABC triangle. 

To Demonstrate: A+B+C = 180° 

Construction: Draw a line PQ through point A parallel to the triangle’s BC side. 

Proof: Because PQ is a straight line, we may deduce from the linear pair that: 1 + 2 + 3 = 180°……… (1)

PQ || BC, AB, and AC are transversals since PQ || BC and AB, respectively. 

As a result, 3 = ACB is the correct answer (a pair of alternate angles), 1 = ABC, as well (a pair of alternate angles) 

Substituting 3 and 1 in equation (1), A + B + C = 180° = 2 x 90° = 2 right angles. As a result, a triangle’s internal angles add up to 180 degrees.

Triangle’s Exterior Angle Property Theorem 2:

If one of the triangle’s sides is formed, the exterior angle formed equals the sum of two interior opposite angles.

Given that: 

Consider a triangle ABC with a side BC stretched D, resulting in an exterior angle ACD.

To Demonstrate: ACD = BAC + ABC or 4 = 1 + 2

Proof: Because 3 and 4 represent adjacent angles on a straight line, they constitute a linear pair. 

As a result, 3 + 4 = 180°……….. (2)

Also, based on the interior angle sum properties of triangles, the following triangle: 

1 + 2 + 3 = 180°……….. (3)

From equations (2) and (3), 

we deduce: 4 = 1 + 2 ACD = BAC + ABC 

As a result, a triangle’s exterior angle equals the sum of its opposite inner angles.

Note: 

There are a few key points to remember when it comes to triangle angles: 

• An equilateral triangle has 60-degree angles on each side. In an isosceles triangle, the angles opposite equal sides are the same. 

• More than one right angle or more than one obtuse angle can exist in a triangle. 

• The sum of two sharp angles in a right-angled triangle equals 90 degrees. 

• The longer side’s angle is bigger, and vice versa.

• An equilateral triangle has 60 degrees for each angle.

• A right-angled triangle’s two acute angles add to 90 degrees.

Example:

48o is one of the sharp angles in a right triangle. Find the other acute angle’s measurement.

One of the sharp angles is 48°.

The triangle’s other angle is 90°.

Let’s say y is the other acute angle.

As a result, y+90°+48° and y=180°–138°=42°.

As a result, the opposite sharp angle is 42°.