MODULE - 7 EUCLIDEAN GEOMETRY

Through investigations, produce conjectures and generalizations related to triangles, quadrilaterals and other polygons, and attempt to validate, justify, explain or prove them, using any logical method (Euclidean, co-ordinate and/or transformation). (LO 3 AS 2)

Lines and angles Adjacent supplementary angles In the diagram, angle B one + angle B two = 180 degrees

Angles round a point In the diagram, a + b + c = 360 degrees.

Vertically opposite angles Properties: Vertically opposite angles are equal

Corresponding angles If AB//CD, then the Corresponding angles are equal

Alternate angles Properties: If AB//CD, then the alternate angles are equal.

Co – interior angles Properties: If AB//CD the co – interior angles add up to 180 degrees.

TRIANGLES There are four kinds of triangles: Scalene Triangle Properties: No sides are equal in length. Isosceles Triangle Two sides are equal. Base angles are equal.

Equilateral Triangle All three sides are equal. All three interior angles are equal Right-angled triangle One interior angle is 90 degrees

Sum of the angles of a triangle Exterior angles of a triangle

The Theorem of Pythagoras

Congruency of triangles Condition 1: Two triangles are congruent if three sides of are triangle are equal in length to the three sides of the other triangle.

Condition 2 Two triangles are congruent if two sides and the included angle are equal to two sides and the included angle of the other triangle.

Condition 3 Two triangles are congruent if two angles and one side are equal to two angles and one side of the other triangle.

Condition 4 Two right-angled triangles are congruent if the hypotenuse and a side of the one triangle is equal to the hypotenuse and a side of the other triangle.

Similar Triangles If two triangles are similar (equiangular), then their corresponding sides are in the same proportion. If ,then

POLYGONS Definition: A polygon is a closed figure with three or more sides. Properties: The sum of the interior angles of a polygon of n sides is given by the formula: 180° (n - 2).

This polygon is called a hexagon. Since there are 6 sides, Example: The figure below represents a regular polygon made up of 6 equal sides and 6 equal interior angles. This polygon is called a hexagon. Since there are 6 sides, the sum of the interior angles is .

The size of the interior angle of a regular polygon (all sides and interior angle equal) is given by the formula: Example In the previous hexagon, each interior angle (x) is equal to:

The sum of the exterior angles of a convex polygon is 360°. Example The sum of the exterior angles of the hexagon is 360°

Exercise Calculate the size of the angles marked with small letters: X 490 X Y

(a) Calculate AC (b) Calculate XY.

3. Are the following pairs of triangles similar? (Give a reason for your answer).

4. The two triangles below are similar. Calculate the value of x and y.

5. Prove that using two different conditions of congruency.