Congruency of triangles

Congruency of Triangles

What is a Triangle?

A triangle is a two-dimensional plane figure bounded by three line segments. It is a convex polygon.

Elements of a Triangle -

• SIDES- They are the line segments forming a two-dimensional polygon. A triangle has 3 sides.

• VERTICES- Each point, where two points intersect is called vertex (plural- vertices). A triangle has 3 vertices.

• ANGLES- A triangle has 3 interior angles and 3 exterior angles.

Interior Angles

In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle if a point within the angle is in the interior of the polygon. The sum of all the 3 interior angles of a triangle is equal to 180°.

Exterior Angles

The Exterior Angle is the angle between any side of a shape, and a line extended from the next side. In triangles the sum of two interior angles in equal to the value of the opposite angles. The sum of all the 3 exterior angles of a triangle is equal to 360°.

Types of Triangles on the basis of sides -

• Equilateral Triangles

• Isosceles Triangles

• Scalene Triangles

Equilateral Triangles

In geometry, an equilateral triangle is a triangle in which all three sides have the same length, an equilateral triangle is also equiangular; that is, all three internal angles are also equal to each other and are each 60°.

Isosceles Triangles

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Also both the angles opposite to the equal sides measure the same.

Scalene Triangles

A scalene triangle is a triangle in which all the three sides are in different lengths, and all the three angles are of various measures. However, the sum of all the interior angles is always equal to 180°.

Types of Triangles on the basis of angles -

• Acute-angled Triangle

• Right-angled Triangle

• Obtuse-angled Triangle

Acute-angled Triangle

An acute-angled triangle is a triangle which has three acute angles i.e. the all angles measure less than 90 degrees.

Right-angled Triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle and the others are acute angles.The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called legs.

Obtuse-angled Triangle

An obtuse-angled triangle is a triangle with one obtuse angle (angle greater than 90°) and two acute angles.

Medians of a Triangle

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid.

Here, CO:OP is 2:1;

AO:ON is 2:1;

BO:OM is also 2:1.

Altitudes of a Triangle

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). Every triangle has exactly three altitudes, one from each vertex, and they all intersect each other at the point known as the orthocenter of the circle.

What are Congruent Figures?

In geometry, two figures or objects are congruent if they have the same shape and size. The '≅' symbol is used to mark to things congruent.

Congruent Triangles

Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.

Axioms for Congruency of Triangles

• SSS- Side Side Side

• SAS- Side Angle Side

• AAS- Angle Angle Side

• ASA- Angle Side Angle

• RHS- Right Hypotenuse Side

SSS-axiom

The Side-Side-Side rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

SAS-axiom

The Side-Angle-Side rule states that

If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

AAS-axiom

The Angle-Side-Angle rule states that if two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

ASA-axiom

The Angle-Side-Angle rule states that if two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

RHS-axiom

The Right-Hypotenuse-Side rule states that in two right-angled triangles, if the length of the hypotenuse and one side of one triangle, is equal to the length of the hypotenuse and corresponding side of the other triangle, then the two triangles are congruent.