1. Grade 8 Lesson 10 The Parallelogram Sunday, June 12, 2016

2. A parallelgram is a quadrilateral whose opposite sides are parallel 1.The sides and the angles of parallelogram Let ABCD is a parallelogram. Consider the triangles ABC and ADC. We have [AC] common side; The two triangles are equal by A.S.A, so and similarly We can prove In a parallelogram, both pairs of opposite sides and opposite angles are equal I. Definitions and properties

3. Sunday, June 12, 2016 2. The diagonals of a parallelogram: In a parallelogram, the diagonals bisect each other and their point of intersection is a center of symmetry. Let ABCD be a parallelogram We can prove that the two triangles AOC and COD are Congruent by A.S.A and we conclude that OA=OC and OB=OD So, O is the midpoint of AC and BD and therefore, it’s a Center of symmetry, we have

4. Sunday, June 12, 2016 1.Starting from the two sides II. When is a quadrilateral a parallelogram Let ABCD be a quadrilateral in which the sides AB and CD are parallel and equal. C is the translate of D by the translation taking A to B. Consequently, BC is the translate of AD and the two segments are parallel, so ABCD is a parallelogram. A quadrilateral, where two opposite sides are equal and parallel is a parallelgram.

5. Sunday, June 12, 2016 2. Starting from the foir sides A quadrilateral whose opposite sides are equal is a parallelogram. Let ABCD be a quadrilatral where AB = CD & AD=BC. Triangles ABC and CDA are congruent by S.S.S. Consequently, So, AB and CD are parallel because they form Two equal alternate interior angles. Similarly AD and BC.Therefore, ABCD is a parallelogram since two of its sides are equal and parallel.

6. Sunday, June 12, 2016 3. Starting from diagonals A quadrilateral whose diagonals bisect each other is a parallelogram and also we can say : A parallelogram that admits a center of symmetry is a parallelgram. Let ABCD be a quadrilateral whose diagonals AC and BD bisect each other at O. In this case, O is a center of symmetry, and then AB = CD and AD = BC by symmetry

7. 2. Starting from the angles Let ABCD be a quadrilateral where the opposite angles are equal. Since the sum of the angles is 360°, the sum of two adjacent angles is 180° and, consequently But, therefore. Also AD and BC are parallel since they make two equal corresponding angles with DC. Similarly, AB and DC are parallel, since they form two equal alternate interior angles with BC. So ABCD has opposite parallel sides. Consequently: A quadrilateral where both pairs of opposite angles are equal is a parallelogram Sunday, June 12, 2016

8. III. The parallelogram and its descendants We have seen that in a square, in a rectangle, and in a rhombus the opposite sides are parallel; thus: The square, the rectangle and the rhombus are particular parallelograms A parallelogram having two adjacent sides equal is a rhombus A parallelogram having a right angle is a rectangle A square is a rectangle and a rhombus at the same time Sunday, June 12, 2016

9. The End Sunday, June 12, 2016