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
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 THEO
REM
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

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