MATHEMATICS O LEVEL(FORM TWO) NOTES – CONGRUENCE OF SIMPLE POLYGON

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: A is joined to 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 a right angle, hypotenuse, and one side (RHS)

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 = A – 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 angles.

AB = CD given

BC = AD given

SOLUTION

6. O is the center of the circle ABCD. If AC and BD are diameters 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 the 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 joined to B.

AB = CD – interior angles, AB // DC

AD = BC – interior angles, AB // DC

Therefore

2. The opposite angles of the parallelogram are equal.

DB = DB

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

Therefore

Similarly,

DB + A = 180º interior angles on the same side of

Therefore

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 sides are equal and parallel.

Example