Construction of Triangles: Properties and Types of Constructing Triangles

The Congruence Rule of Triangles is a very important chapter of class 9 mathematics book which helps in constructing triangles. Generally, a polygonal shape can be formed by joining three line segments and the final structure thus formed is also called a triangle. Constructing triangles involves using given information to accurately draw a triangle.

• In geometry, any three non-collinear points provide a distinct triangle and, separately, a distinct plane (known as two-dimensional Euclidean space).
• The three main parts of a triangle are its sides, angles, and vertices.
• Some of the real-life examples of triangles are slice of pizza, roof of house, hills, pyramids etc. 
• Construction of triangles with specific angles, we will need three tools – protractor, compass, ruler and scale.

What are Triangles?

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A triangle is a three-sided polygon with three vertices and three edges. The sum of any triangle's three interior angles is 180 degrees. Triangles can be classified according to their sides, angles, and other characteristics. It is significant to have the following dimensions for construction of triangles. 

• All the three sides of a triangle
• Two sides and one angle
• Two angles and one side
• An additional triangular side and
• Hypotenuse of a right-angled triangle.

Types of Triangle

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Construction of triangle is simple when we are provided measurements based on various characteristics such as SSS, SAS, and ASA congruency rule of triangle. Based on their sides and angles, triangles are classified as: 

Based on Sides 

There are three types of triangles based on sides. They are:

• Equilateral Triangles: An equilateral triangle is one that has three equal sides and three equal angles.
• Isosceles Triangles: Isosceles triangles are any triangles with two equal sides.
• Scalene Triangles: A scalene triangle is a triangle having uneven sides on all sides.

Triangles based on sides

Based on Angles

The types of triangles based on angles are as follows: 

• Acute-angled triangles: All the angles less than 90 degrees.
• Right-angled triangles: A triangle with a 90 degree angle.
• Obtuse-angled triangles: One angle more than 90 degree. 

Triangles based on angles

Properties of Triangle

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The properties of triangles are as follows: 

• The sum of all the internal angles of a triangle equals 180 degrees. 
• The exterior angle of the opposing side is equal to the sum of two adjacent internal angles.
• Any two triangle sides added together have lengths that are bigger than the triangle's third side.

Pythagoras Theorem: A right-angled triangle's hypotenuse (the side that faces the right angle) has a square length that is equal to the sum of the squares of the other two sides.

a² + b² = c²

Pythagoras Theorem

Types of Construction of Triangles

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They can be divided into three groups based on the construction dimensions:

• SSS - If all three sides are provided
• SAS - If only one angle and two sides are provided
• ASA - If two angles and one side are provided
• RHS - If the base and hypotenuse are provided

Construction of SSS Triangle 

According to the SSS (Side-Side-Side) Triangle Congruence theorem, two triangles are congruent if their three sides are the same length. When a triangle's three sides are known, the following instructions can be utilized to construct a SSS triangle:

SSS Triangle

• Draw a line segment that is the same length as the triangle's longest side.
• Draw an arc after measuring the second side's length with a ruler.
• Once the previous arc has been cut and the point has been marked, measure the third side.
• The needed triangle is constructed by connecting the line segment's endpoints to the intersections of the two arcs.

Construction of SAS Triangle 

According to the side-angle-side (SAS) triangle congruence rule, two triangles are congruent if their included angle and two sides match those of the other triangle. The SAS triangle can be created as follows when two sides and an internal angle of a triangle are given: 

SAS Triangle

• Using a ruler and pencil, draw a line segment that is the same length as the triangle's longest side.
• Measure the angle by placing the protractor's center on one end of a line segment. Join the points together and build the ray so that it is closer to the line segment.
• Using a ruler and a compass, measure the triangle's other provided side.
• After that, cut the ray at a different place and position the compass at one end.
• Connect the point to the opposite end of the line segment now.

Read more: Law of Cosines

Construction of ASA Triangle 

According to the Angle-Side-Angle (ASA) Triangle theorem, two triangles are congruent if their included sides and two angles match those of another triangle. An ASA triangle can be created using the following formula when two angles and a side are provided:

ASA Triangle

• Using a ruler and the specified triangle side, draw a line segment that is the same length.
• Segments measure one of the specified angles at one end of a line and draw a ray.
• Draw another ray such that it intersects the first one at a point at the other end of the line segment.
• Use a protractor to measure the other angle.
• The necessary triangle is obtained by connecting the previous point with both of the line segment's endpoints.

Construction of Right-Angled Triangle 

A triangle that has one angle that is exactly 90 degrees is known as a right-angled triangle. A right-angled triangle can be created as follows when the hypotenuse of a triangle is given together with its two other sides:

Right angled triangle

• Draw a line segment with a length equal to the hypotenuse side.
• Measure the angle to be 90 degrees at one of the line segment's endpoints and create a ray.
• Then calculate the length of a second side, draw an arc to stop the ray at a point, and give the point a name.
• To create the necessary right-angled triangle, connect the point to the opposite side of the line segment.

Things to remember

• The sum of all the internal angles of a triangle equals 180 degrees. 
• A triangle is a three-sided polygon with three vertices and three edges.
• One angle more than 90 degrees is referred to as an obtuse-angled triangle. 
• A scalene triangle is a triangle having uneven sides on all sides.
• An equilateral triangle is one that has three equal sides and three equal angles.
• ASA construction if two angles and one side are provided
• All the angles less than 90 degrees is regarded as acute-angled triangles