MATHEMATICS Standard -8

By

Shelin Elizabeth VargheseB.Ed Mathematics

Reg No:13304013

CONTENTS

CONGRUENT TRIANGLESCONGRUENCE OF TRIANGLESSSS CONGRUENCESAS CONGRUENCEASA CONGRUENCEAAS CONGRUENCERHS CONGRUENCE ISOSCELES TRIANGLEBISECTORSAPPLICATION AND EXAMPLES

CONGRUENT TRIANGLES1.1 CONCEPT OF CONGRUENCE

In our daily life you observe various figures and objects. These figures or objects can be categorised in terms of their shapes and sizes in the following manner.(i) Figures, which have different shapes and sizes as shown in Fig. 11.1

(ii) Objects, which have same shapes but different sizes as shown in Fig. 11.2

(iii) Two one-rupee coins.

.

Two figures, which have the same shape and same size are called congruent figures and this property is called congruence.

What is "Congruent" ... ?Equal in size and shape. Two objects are congruent if they

have the same dimensions and shape.

If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent.

Rotation Turn!

Reflection Flip!

Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.Examples:These shapes are all Congruent:

RotatedReflected and

Moved Reflected and

Rotated

1.2   CONGRUENCE  OF TRIANGLES

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

If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:

Refresh your mind

1.3 SSS CONGRUENCESSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

For example:

is congruent to:

and

because they all have exactly the same sides.But:

is NOT congruent to:

because the two triangles do not have exactly the same sides.

1.4 SAS CONGRUENCE

SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

For example:

is congruent to:

1.5 ASA CONGRUENCEASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

For example:

is congruent to:

1.6 AAS CONGRUENCE

AAS (Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent.

For example:

is congruent to:

1.7 RHS CONGRUENCE

RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

For example:

is congruent to:

Refresh your mind!

Caution ! Don't Use "AAA" !

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

CHECK YOUR PROGRESSProve that the points of intersection of the two diagonals of a parallelogram is the midpoint of both the diagonals.

ISOSCELES TRIANGLE  An isosceles triangle is a triangle that has two sides of equal length. The angles opposite the equal sides are also equal.

Theorem 1 If a triangle has two sides of equal length, then the angles opposite to these sides are congruent. 

Proof Let ABC be a triangle with sides AC and BC of equal length (Figure 1).                                              We need to prove that angles BAC and ABC are congruent. 

Consider the triangle BAC.The side AC of the triangle ABC corresponds to the side BC of the triangle BAC.The side BC of the triangle ABC corresponds to the side AC of the triangle BAC.The side AB is common to the triangles ABC and BAC.Since all three of the corresponding sides of the triangles ABC and BAC are of equal length, these two triangles are congruent, in accordance to the postulate 3 (SSS) of the triangle congruency.Hence, the corresponding angles BAC and ABC are congruent. 

Figure 1. To the Theorem 1

SSA DOESN’T WORK

AB is the same length as PQ, BC is the same length as QR, and the angle A is the same measure as P. And yet the triangles are clearly not congruent - they have a different shape and size.

The proof is completed.Thus we proved that in isosceles triangle two angles

opposite to the equal sides are congruent. 

Theorem 2 If in a triangle two angles are congruent, then the sides opposite to these angles are of equal length. 

Proof Let ABC be a triangle with congruent angles BAC and ABC (Figure 2).                                                We need to prove that angles BAC and ABC are congruent. 

Consider the triangle BAC.The angle BAC of the triangle ABC corresponds to the angle ABC of the triangle BAC.The angle ABC of the triangle ABC corresponds to the angle BAC of the triangle BAC.The side AB is common to the triangles ABC and BAC.Since two angles and the included sides of the triangles ABC and BAC are congruent, these two triangles are congruent, in accordance to the postulate 2 (ASA) of the triangle congruency .Hence, the corresponding sides AC and BC are of equal length. 

The proof is completed.

Figure 2. To the Theorem 2

Thus we proved that in isosceles triangle two sides opposite to the congruent angles are of equal length.

BISECTORS

A line which cuts an angle into two equal halves. For every angle, there exists a line that divides the angle into two equal parts. This line is known as the angle bisector. In a triangle, there are three such lines. Three angle bisectors of a triangle meet at a point called the incenter of the triangle. 

The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.

Point G is the circumcenter of ABC.

APPLICATION AND EXAMPLES

Example 1: Let ABCD be a parallelogram and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer. 

Solution to Example 1:

In a parallelogram, opposite sides are congruent. Hence sides 

BC and AD are congruent, and also sides AB and CD are congruent. 

In a parallelogram opposite angles are congruent. Hence angles 

ABC and CDA are congruent. 

Two sides and an included angle of triangle ABC are congruent to two corresponding sides and an included angle in triangle CDA. According to the above postulate the two triangles ABC and CDA are congruent.

Example 2: Let ABCD be a square and AC be one of its diagonals. What can you say about triangles ABC andCDA? Explain your answer

Solution to Example 2:  In a square, all four sides are congruent. Hence sides AB and CD are

congruent, and also sides BC and DA are congruent

The two triangles also have a common side: AC. Triangles ABC has three sides congruent to the corresponding three sides in triangle CDA. According to the above postulate the two triangles are congruent. The triangles are also right triangles and isosceles. 

Example 3: ABC is an isosceles triangle. BB' is the angle bisector. Show that triangles ABB' and CBB' are congruent. 

Solution to Example 3:

Since ABC is an isosceles triangle its sides AB and BC are congruent and also its angles BAB' and BCB' are congruent. Since BB' is an angle bisector, angles ABB' and CBB' are congruent. 

Two angles and an included side in triangles ABB' are congruent to two corresponding angles and one included side in triangle CBB'. According to the above postulate triangles ABB' and CBB' are congruent.

Example 4: What can you say about triangles ABC and QPR shown below. 

Solution to Example 4:

In triangle ABC, the third angle ABC may be calculated using the theorem that the sum of all three angles in a triangle is equal to 180 derees. Hence 

angle ABC = 180 - (25 + 125) = 30 degrees 

The two triangles have two congruent corresponding angles and one congruent side. 

angles ABC and QPR are congruent. Also angles BAC and PQR are congruent. Sides BC and PR are congruent. 

Two angles and one side in triangle ABC are congruent to two corresponding angles and one side in triangle PQR. According to the above theorem they are congruent.

Example 5: Show that the two right triangles shown below are congruent. 

Solution to Example 5:

We first use Pythagora's theorem to find the length of side AB in triangle ABC. 

length of AB = sqrt [5 2 - 3 2] = 4 

One side and the hypotenuse in triangle ABC are congruent to a corresponding side and hypotenuse in the right triangle A'B'C'. According to the above theorem, triangles ABC and B'A'C' are congruent.

TERMINAL EXERCISE

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