Mathematics in Daily Life

1. Mathematics in Daily Life 9th Grade Theorems on Parallelograms

2. Objective After learning this chapter, you should be able to • Prove the properties of parallelograms logically. • Explain the meaning of corollary. • State the corollaries of the theorems. • Solve problems and riders based on the theorem.

3. Flowchart on Procedure to Prove a Theorem Let us recall the procedure of proving a theorem logically. Observe the following flow chart. Consider/take a statement or the Enunciation of the theorem For example, in any triangle the sum of three angles is 180˚ Draw the appropriate figure and name it. A B C Write the data using symbols. ABC is a triangle 2 1

4. Flowchart on Procedure to Prove a Theorem 2 1 Write what is to be proved using symbols Analyze the statement of the theorem and write the hypothetical construction if needed and write it symbolically Through the Vertex A draw EF || BC E A F Write the reason for construction Draw the appropriate figure and name it. Use postulates, definitions and previously proved theorems along with what is given and construction

5. Theorems on Parallelograms Theorem 1: The diagonals of a parallelogram bisect each other. Theorem 2: Each diagonal divides the parallelogram into two congruent triangles.

6. Theorem 1 Proof Theorem: The diagonals of a parallelogram bisect each other. Given: ABCD is a parallelogram. AC and BD are the diagonals intersecting at O. To Prove: AO = OC BO = OD D C O A B

7. Theorem 1 Proof Contd.. Proof: i.e., The diagonals of parallelogram bisect each other.

8. Theorem 2 Proof Theorem: Each diagonal divides the parallelogram into two congruent triangles. Given: ABCD is a parallelogram in which AC is a diagonal. AC = DC, AD = BC To Prove: D C O A B

9. Theorem 2 Proof Contd.. Proof: Diagonal AC divides the parallelogram ABCD into two congruent triangles. Similarly, we can prove that Each diagonal divides the parallelogram into two congruent triangles.

10. Corollary A corollary is a proposition that follows directly from a theorem or from accepted statements such as definitions. Corollaries of the Theorems There are four corollaries for the theorems explained in the previous slides. They are, Corollary-1: In a parallelogram, if one angle is a right angle, then it is a rectangle.

11. Corollaries of the Theorems Contd… Corollary-2: In a parallelogram, if all the sides are equal and all the angles are equal, then it is a square. Corollary-3: The diagonals of a square are equal and bisect each other perpendicularly. Corollary-4: The straight line segments joining the extremities of two equal and parallel line segments on the same side are equal and parallel.

12. Corollary 1 Proof Corollary: In a parallelogram, if one angle is a right angle, then it is a rectangle. Given: PQRS is a parallelogram. Let To Prove: PQRS is a rectangle. R S 90˚ P Q

13. Corollary 1 Proof Contd.. Proof: Hence, PQRS is a rectangle.

14. Corollary 2 Proof Corollary: In a parallelogram, if all the sides are equal and all the angles are equal, then it is a square. D C 90˚ 90˚ 90˚ 90˚ A B Activity!!! Prove this corollary logically

15. Corollary 3 Proof Corollary: The diagonals of a square are equal and bisect each other perpendicularly. Given: ABCD is a square. AB = BC = CD = DA. To Prove: 1) AC = BD 2) AO = CO, BO = DO. 3) D C 90˚ 90˚ 90˚ 90˚ A B

16. Corollary 3 Proof Contd.. Proof: Hence, diagonals of a square bisect each other

17. Corollary 3 Proof Contd.. Hence, the digonals bisect each other at right angles.

18. Corollary 4 Proof Corollary: The straight line segments joining the extremities of two equal and parallel line segments on the same side are equal and parallel. Activity!!! Prove this corollary logically Hint :- S.A.S. Postulate of congruency triangles

19. Examples Example-1: In the given figure, ABCD is a parallelogram in which Calculate the angles Given: ABCD is a parallelogram AB = DC, AD = BC AB || DC, AD || BC To Find: D C 80˚ 70˚ A B

20. Examples Contd... Solution:

21. Examples Contd… Example-2: In the figure, ABCD is a parallelogram. P is the mid point of BC. Prove that AB = BQ. Given: ABCD is a parallelogram ‘P’ is the mid point of BC. BP = PC To Prove: AB = BQ D C P Q A B

22. Examples Contd... Solution:

23. Exercises • In a parallelogram ABCD, =60⁰. If the bisectors of and meet at P on DC. Prove that • In a parallelogram ABCD, X is the mid-point of AB and Y is the mid-point of DC. Prove that BYDX is a parallelogram. • If the diagonals PR and QS of a parallelogram PQRS are equal, prove that PQRS is a rectangle. • PQRS is a parallelogram. PS is produced to M so that SM = SR and MR is produced to meet PQ produced at N. prove that QN = QR. • ABCD is a parallelogram. If AB = 2 x AD and P is the mid-point of AB, prove that