Form 2 Mathematics – CONGRUENCE OF SIMPLE POLYGON

 

 
 

The triangles above are drawn such that

CB= ZY

AC=XZ

B=YX

Corresponding sides in the triangles are those sides which are opposite to the equal angles i.e.

If the corresponding sides are equal i.e.

In general, polygons are congruent if corresponding sides and corresponding angles are equal.
The symbol for congruence is

Congruence of triangles

Case 1: Given three sides

Two triangles are congruent if the three pairs of corresponding sides are such that the sides in each pair are equal.

Consider the triangles below:

Note: SSS- is an abbreviation of side- side- side

Examples :

Solution

Construction of A is joined C

 Construction; A joined to D

  

Case 2; Given two sides and the included angle (SAS)

Two triangles are congruent if two pairs of corresponding sides are such that the sides in each pair are equal and the angles included between the given sides in each triangle are equal.

Examples

Case 3; Given two angles and a corresponding side

Two triangles are congruent if two pairs of corresponding angles are such that the angles in each triangle are equal.

Example

Solution

Case 4: Given that a right angle hypotenuse and one side (RHS)

The right angled triangles are congruent if the hypotenuse and a side of one triangle are respectively equal to the hypotenuse and side of another triangle

Example:

Use the figure below to prove that

Solution

AC= AB -right angles

Therefore

 Note:

R.H.S – Right angle hypotenuse side

Isosceles triangle theorem

The base angles of an isosceles triangle are equal

 

Construction:-

 Exercise 1.

Solution

ABCD = Common line

They are  alternate interior angle

 
 

 
 

AB=CD given

BC =AD given

 
 

SOLUTION

6. O is the center of the circle ABCD, if AC and BD and diameter of the circle and the line segments AD, AB and CB are drawn prove that

 
 

Solution

CONVERSE THE ISOSCELES TRIANGLE THEOREM

If two angles of a triangle are equal then sides opposite those angles are equal

 
 

Given that C=

Required to prove =

Construction A and D are joined such that

 
 

THEOREMS OF PARALLELOGRAMS

1)   The opposite sides of the parallelogram are equal

Given a parallelogram ABCD

Required to prove

Construction:D is formed to B

AB= CD -is interior angles AB//DC

AD= BC -is interior angles AB//DC

Therefore

2. The opposite angles of the parallelogram are equal

 
 

DB= DB

AC + DB=180Interior angle of the same side of //

AC + DB=180º interior angles on side of //

Therefore

Similarly

DB + AC=180º interior angles the same side of //

BD + AC=180º interior angles the same side of ,//

Therefore

DB + ABC= BD + AC

DB= BD

Hence opposite angles of a parallelogram are equal.

3.The diagonals of a parallelogram bisect each other

4. The diagonals of a parallelogram intersect each other

If one pair of the opposite sides of a quadrilateral are equal and parallel then the other pair of the opposite side are equal and parallel.

Example