# 9th Class Mathematics Triangles Triangle

Triangle

Category : 9th Class

TRIANGLE

FUNDAMENTALS

• A closed figure formed by three intersecting lines is called a triangle

TYPES OF TRIANGLE

• Equilateral triangle:- All the three sides are equal. i.e., $\angle A=\angle B=\angle C$ and also $AB=BC=AC$

• Isosceles triangle:- Two sides are equal. i.e., $\angle B=\angle C\ne \angle A$          $AB=AC$

• Scalene triangle:- None of the three sides are equal. i.e., $AB\ne BC\ne AC$

and $\angle A\ne \angle B\ne \angle C$

• Acute angle triangle:- Ail the three angles are less then${{90}^{{}^\circ }}$. $\angle A,\angle B$ and $\angle C$are less than$90{}^\circ$. • Right angle triangle:- One Angle is equal to ${{90}^{{}^\circ }}$ Here, $\angle B={{90}^{{}^\circ }}$ • Obtuse Angle triangle:- One of the angle greater than ${{90}^{{}^\circ }}$, $\angle B>{{90}^{{}^\circ }}$ • Similar triangle:- All the angles of triangle are equal to the angles of another triangle. $\angle A=\angle X$,        $\angle B=\angle Y$,        $\angle C=\angle Z$

• Congruent triangle:- Two triangles are congruent if they are exactly same to each other in sides or angles. $AB=XY$,                  $BC=YZ$,                 $AC=XZ$

• Congruency conditions:- S - S - S (Side - Side - Side) $AB=XY,BC=YZ$ and $AC=XZ$

Then $\Delta ABC\cong \Delta XYZ$

• S - A - S (Side - Angle - Side) $AB=XY$,    $BC=YZ$ and $\angle B=\angle Y$

Then $\Delta ABC\cong \Delta XYZ$

• A - S - A (Angle - Side - Angle) $\angle B=\angle Y,\text{ }\angle C=\angle Z,\text{ }BC=YZ$

Then, $\Delta ABC\cong \Delta XYZ$

• R - H - S (Right - Hypotenuse - Side) $\angle B=\angle Y={{90}^{{}^\circ }}$       $BC=YZ$, then $\Delta ABC\cong \Delta XYZ.$

SIMILARITY OF A TRIANGLE

• Two triangles are similar if their corresponding angles are equal and also their corresponding sides are in the same ratio $\Delta ABC$ and $\Delta XYZ$ are similar if $\angle A=\angle X,\angle B=\angle Y,\text{ }\angle C=\angle Z$ and $\frac{AB}{XY}\text{=}\frac{BC}{YZ}=\frac{AC}{XZ}$

• Centroid of a triangle:- Point of intersection of all the three medians. Here, Point G is centroid of triangle ABC

In Centre:- Meeting Point of all the three angle bisectors of a triangle. I is the in-centre of triangle ABC.

• Circum-centre:- Meeting point of the perpendicular bisectors of the perpendicular bisectors of the sides of a triangle.
• Ortho-centre:- Meeting Point of the three altitudes of a triangles. Here, Point 0 is the ortho - centre of the triangle.

PROPERTIES OF TRIANGLE

• Sum of interior angles of triangle ${{180}^{{}^\circ }}$ and sum of exterior angles is${{360}^{{}^\circ }}$. $\angle 1+\angle 2+\angle 3={{180}^{{}^\circ }}$

$\angle 4+\angle 5+\angle 6={{360}^{{}^\circ }}$

• If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. $\angle 1+\angle 2=\angle 4$

• If two angles of a triangles are equal, then sides opposite to them are equal. i.e., If $\angle B=\angle C$, then $AB=AC.$

• If the two sides are unequal then, the longer side has the greater angle opposite to it. If $AC>AB,$

Then $\angle B>\angle C.$

• The sum of any two sides of mangle is greater than the third side, $AB+BC>AC$

$AB+AC>BC$

$AC+BC>AB$

• In a $\Delta ABC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at a point O. Then $\angle BOC=90+\frac{1}{2}\angle A$ • In a triangle ABC, the sides AB and AC are produced to X and Y respectively. The bisectors of $\angle XBC$ and $\angle YCB$ intersect at a point 0. Then $\angle BOC=90-\frac{1}{2}\angle A$ • In a certain triangle ABC, $\angle B$ is an obtuse angle and $AD\bot BC$, then $A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}+2BC.CD$ • In a certain triangle ABC $\angle B$ is an acute angle and $AD\bot BC$, then $A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}-2BC.CD$ $\angle B$ is an acute angle.

• The internal bisector of one base angle and the external bisector of the other is equal to one half of the vertical angle. $\angle E=\frac{1}{2}\angle A$

• The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. $\frac{Area\,of\,ABC}{Area\,of\,DEF}={{\left( \frac{AB}{DE} \right)}^{2}}{{\left( \frac{BC}{EF} \right)}^{2}}{{\left( \frac{AC}{DF} \right)}^{2}}$

#### Other Topics

##### Notes - Triangle

You need to login to perform this action.
You will be redirected in 3 sec 