[Maths Class Notes] on What are Triangles? Pdf for Exam

What are Triangles in Geometry?

Shapes and patterns have always fascinated us since the dawn of time. The rigidness in some and swirls in others have often puzzled our minds and nature housing such beautiful designs is beyond fascinating. Triangles are one of those shapes that possess a variety of features and applications in our world. A triangle is defined as a closed two-dimensional figure with 3 sides, 3 angles, and 3 vertices in Geometry. They are a type of polygon whose sum of all three angles usually equals 180°.

Types of Triangles

Since a triangle is a two-dimensional shape, the types of triangles are classified based on their sides and angle measurement.

A triangle is classified into three types based on the length of its sides,

1. Scalene Triangle – The length of all sides differs or is not equal.

2. Isosceles Triangle – The properties of an isosceles triangle involve a triangle having two sides equal in length and the third is not equal.

3. Equilateral Triangle – The properties of an equilateral triangle involve a triangle having the length of all three sides equal.

A triangle is classified based on the angles,

1. Acute Angle Triangle – The angles of a triangle are less than 90°

2. Obtuse Angle Triangle – One angle of a triangle is greater than 90°

3. Right Angle Triangle – One angle of a triangle is equal to 90°

Properties of a Triangle

Triangles follow certain properties and rules to achieve a particular state and to solve complex geometric problems. Some of the properties of triangles are listed below,

1. The presence of three sides, three angles, and three vertices.

2. All the interior angles equal to 180°.

3. The third side of a triangle is less than the sum of the other two sides. 

4. The half product of the base and the height gives the area of the triangle.

5. The sum of all the three sides of a triangle provides the perimeter of the triangle.

Properties of Median in a Triangle

The following properties are established to find the median of a triangle,

• A triangle has 3 medians, one from each vertex. 

• All medians meet at a single point.

• The Centroid of the triangle is the point where the 3 medians meet. 

• The median of a triangle divides it into two smaller triangles.

What is a Congruent Triangle?

When all three sides and three angles of a given triangle are equal it is referred to as a congruent triangle. It is based on the shape and size of the triangle. The object and its mirror image are generally referred to as congruence. Two images are congruent if they superimpose each other. In geometric figures, a similar length of line segments are congruent and so is it’s the angle of measurement.

Conditions for a Congruent Triangle are established as the property of sides of a triangle.

All three sides are equivalent concerning the second triangle. 

Any two sides and an angle are equivalent concerning the second triangle. 

Any two angles and a side are equivalent concerning property f a second triangle.

A non-included side and two angles are equal to corresponding angles and sides of another triangle.

• RHS (Right Angle-Hypotenuse-Side) 

The hypotenuse with a side of a right-angled triangle is equivalent to the second triangle’s hypotenuse and right-angled side.

Example of a Congruent Triangle

In the figure, ΔABC and ΔPQR are congruent triangles. Therefore,

Vertices:  A and P, B and Q, C and R vertices are equal

Sides:  AB=PQ, QR= BC, and AC=PR;

Angles: ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R  

State The Angle Sum Property of a Triangle

The Angle Sum Property of a given triangle is defined as the sum of all the interior angles of a triangle is equal to 180°.

Devise a theorem to calculate the angle sum property of a triangle. 

Proof: 

Let’s take a ΔABC , prove the property of the triangle by drawing a line PQ, 

Since PQ is a straight line,

∠PAB + ∠BAC + ∠QAC = 180⁰ 

PQ ll BC and AB, AC are transversals

Hence, ∠QAC = ∠ACB and ∠PAB = ∠CBA are a pair of alternate angles

Substituting ∠QAC and ∠PAB , we get

∠ACB + ∠BAC + ∠CBA = 180⁰

Hence, the sum of all interior angles is equal to  180⁰.