Class 9 Maths

# Congurency In Triangles

## Exercise 7.2

Question 1: In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠ C intersect each other at O. Join A to O. Show that: (a) OB = OC (b) AO bisects ∠ A.

∠OBC = ∠ OAC (They are halves of angless B and C)
So, OB = OC (Sides opposite to equal angles)

In ΔAOB and ΔAOC
AB = AC (given)
OB = OC (proven earlier)
∠ABO = ∠ACO (they are halves of angles B and C)
So, ΔAOB ≅ ΔAOC(SAS rule)
So, ∠BAO=∠CAO
It means that AO bisects ∠A

Question 2: In Δ ABC, AD is the perpendicular bisector of BC. Show that Δ ABC is an isosceles triangle in which AB = AC.

AD = AD (common side)
BD = CD (given)
∠ADB=∠ADC (right angle)
So, ΔABD≅ ΔACD
So, AB = AC
This proves that ΔABC is isosceles triangle.

Question 3: ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively. Show that these altitudes are equal.

AB = AC (given)
∠BAE=∠CAF (common to both triangles)
∠CFA=∠BEQ (right angles)
So, ΔABE≅ ΔACF (ASA rule)
So, BE=CF

Question 4: ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal. Show that

1. ΔABE≅ ΔACF
2. AB = AC, i.e., ABC is an isosceles triangle.

Answer: This can be solved like previous question.

Question 5: ABC and DBC are two isosceles triangles on the same base BC. Show that ∠ABD = ∠ACD.

Answer: ∠ABC=∠ACB
∠DBC=∠DCB
So, ∠ABC+∠DBC=∠ACB+∠DCB
Or, ∠ABD=∠ACD

Question 6: ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB. Show that ∠BCD is a right angle.

AD=AB
AC=AC
∠ACB=∠ABC
∠ACD=∠ACD
In ΔABC, ∠ACB+∠ABC+∠CAB=180°
Or, ∠CAB=180°-2∠ACB ----- (1)

∠DAC=180°-2∠ACD ------ (2)
As BD is a straight line, so ∠CAB+∠DAC=180°
So, adding equations (1) and (2) we get
180°=360°-2∠ACB-2∠ACD
Or, 180°=360°-2(∠ACB+∠ACD)
Or, 2(∠ACB+∠ACD)=180°
Or, ∠ACB+∠ACD=∠BCD=90°

Question 7: ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

Answer: If AB = AC then angles opposite to these sides will be equal. As you know the sum of all angles of a triangle is equal to 180°.
So, ∠A+∠B+∠C=190°
Or, 90°+∠B+∠C=180°
Or, ∠B+∠C=180°-90°=90°
Or, ∠B=∠C=45°

Question 8: Show that the angles of an equilateral triangle are 60° each.

Answer: As angles opposite to equal sides of a triangle are always equal. So, in case of equilateral triangle all angles will be equal. So they will measure one third of 180°, which is equal to 60°

## Number System

Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers.

## Polynomials

Algebraic expression with many terms having variables and coefficients is called polynomial.

## Coordinate Geometry

The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

## Linear Equations in Two Variables

A linear equation in two variables has infinite number of solutions.

## Euclid's Geometry

The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.

## Lines & Angles

An angle is formed by two rays originating from the same end point. The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

## Triangles

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

Sum of the angles of a quadrilateral is 360°. A diagonal of a parallelogram divides it into two congruent triangles.

## Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, and diagonals bisect each other

## Circles

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

## Heron's Formula

Heron's formula is generally used for calculating area of scalene triangle.

## Surface Area

How to calculate surface area of cylinder, cone and sphere?

## Volume

How to calculate volume of cylinder, cone and sphere?

## Statistics

Learn to make bar chart and histogram from given data.

## Probability

Probability tells about likelihood of an event to occur.