Introduction
Angle Sum Triangle is a term used in geometry that refers to a figure formed by three line segments that intersect at three angles. The sum of the angles in a triangle is always 180 degrees. The Angle Sum Triangle theorem states that the sum of the angles in any triangle is 180 degrees. This theorem can be proven using basic geometry principles. Angles in a triangle can be classified as either acute, right, or obtuse.
The Angle Sum Triangle theorem is an important theorem in geometry that can be used to solve problems involving triangles. It can be used to determine the size of the angles in a triangle or to determine whether a triangle is acute, right, or obtuse. The Angle Sum Triangle theorem can also be used to find the length of the sides of a triangle, given the size of the angles.
Triangle Sum Property
In geometry, one of the most used shapes is a triangle. A triangle has three sides and three angles. These sides and angles are the elements of the triangle. All the polygons have two types of angles which are interior angles and exterior angles. As the triangle is the smallest polygon, it has three interior angles and six exterior angles. A triangle with vertices A, B, C is denoted by ∆ABC. There are various kinds of triangles with different angles and edges, but all of them follow the triangle sum properties. The two most important properties are the angle sum property of a triangle and the exterior angle property of a triangle.
Angle Sum Property of a Triangle
This property states the sum of the interior angles of a triangle is 180 degrees. Interior angles are formed at the vertex where any two edges of a triangle join. The angle between two sides of a triangle is called the interior angle. It is also known as the interior angle property of a triangle. This property states that the sum of all the interior angles of a triangle is 180°. If the triangle is ∆ABC, the angle sum property formula is ∠A+∠B+∠C = 180°.
Some other Important Angle Properties of a Triangle
Besides the angle sum property and the exterior angle property of a triangle, there are some other essential properties of the angles of a triangle, and they are as follows.
The value of each angle of an equilateral triangle is 60°.
The sum of the two acute angles of a right-angled triangle is 90°.
The angle opposite to the smallest side is the smallest, and the largest angle is the opposite to the largest side.
The two angles of a triangle opposite to the two equal sides are equal.
A triangle has a maximum of one right angle or one obtuse angle.
Solved Examples
1. Find Out the Angle ∠ABC of the Triangle ∆ABC. The Exterior ∠ACD = 125° and the Other Interior Angle ∠BAC = 61°.
Ans: BC a side of ∆ABC is extended up to D, and the exterior angle is 125°. So, the two opposite angles are ∠ABC and ∠BAC. The sum of the two angles is equal to the value of ∠ACD = 125°.
Therefore, ∠ABC = ∠ACD – ∠BAC
= 125° – 61°
= 64°
2. The Ratio of the Three Angles of a Triangle is 1:2:3. Determine the Largest Angle of the Triangle and the Type of the Triangle.
Ans: According to the angle sum property,
x + 2x + 3x = 180°
3x = 90°
Therefore, the largest angle is 90°, and it is a right-angled triangle.
Conclusion:
The Angle Sum Triangle Theorem states that the sum of the angles in any triangle is 180 degrees. This theorem can be proven using basic geometry principles. Angles in a triangle can be classified as either acute, right, or obtuse. The Angle Sum Triangle theorem is an important theorem in geometry that can be used to solve problems involving triangles. It can be used to determine the size of the angles in a triangle or to determine whether a triangle is acute, right, or obtuse. The Angle Sum Triangle theorem can also be used to find the length of the sides of a triangle, given the size of the angles.
FAQs on Introduction to Angle Sum Triangle
1. How to Find Exterior Angles of a Triangle?
An exterior angle is an angle between one side of the triangle and the extended adjacent side. A triangle has six exterior angles. Each edge of a triangle can form two exterior angles with two of its extended adjacent sides. If you want to get an exterior angle of a triangle, you have to extend the straight line of one side of the triangle. The straight line can be extended to any of the two sides. After the extension, the extended part and the adjacent side of the triangle form an angle. Thus, the exterior angle of a triangle is formed.
2. What is the Right Angle Property of a Triangle?
The right-angle property of a triangle states that a triangle has one ninety-degree angle and two smaller angles. It can be expressed as A triangle that has one angle of 90 degrees, called the "right angle," and some of the other two smaller angles is 90 degrees. The opposite side of the 90-degree angle is the hypotenuse.
3. How to Identify Acute, Right, and Obtuse Angles?
There are three types of angles in a triangle: acute angles, right angles, and obtuse angles. Acute angles are angles that are less than ninety degrees. Right angles are angles that are equal to ninety degrees. Obtuse angles are angles that are greater than ninety degrees and less than 180 degrees.
4. What are the Principal Properties of the Angles of a Triangle?
The angles of a triangle are of two types, which are interior angles and exterior angles. The main two properties for these two angles of the triangle are angle sum property and exterior angle property. Apart from these two, there are many properties of the angle of a triangle. The equality of angles of an equilateral triangle comes under interior angle properties. The exterior angle property is that the exterior angle adjacent to the largest exterior angle measures the smallest. The angles of a triangle opposite to the equal sides are equal. A triangle cannot have two or more right angles.