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 1 2

4. Flowchart on Procedure to Prove a Theorem1 2
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
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.
Statement
Reasons
Process of Analysis
In ∆AOB and ∆COD,
AB = DC
Opposite sides of the parallelogram
Recognise the ∆s which contain the sides AO, BO, CO, DO.
Use the data to prove the congruency of these two ∆s 2)
Vertically opposite angles 3)
Alternate angles
AB || DC and BD is a transversal.
ASA Postulate
Corresponding sides of congruent ∆s

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.
Statement
Reasons
AB = DC
Opposite sides of the parallelogram
2) BC = AD
3) AC is common
S.S.S. postulate

10. Corollary Corollaries of the Theorems
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. Statement
Reasons
Given
---- (1)
Opposite angles of parallelogram PQRS
Sum of the consecutive angles of a parallelogram PQRS is equal to 180˚
By substitution
By transposing
----(2)
----(3)
From the equations (1), (2) and (3)

14. Prove this corollary logically
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˚ ˚ 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˚ ˚ A B

16. Corollary 3 Proof Contd..
Proof: Hence, diagonals of a square bisect each other
Statement
Reasons
1) Consider the ∆ABD and ∆ABC
AB = BC
AB is common.
Sides of the square are equal
Angles of the square are equal
S.A.S postulate
Congruent parts of congruent ∆s
2) Consider the ∆AOB and ∆DOC
AB = DC
Opposite sides of the square
Alternate angles AB || DC
A.S.A postulate

17. Corollary 3 Proof Contd..
Hence, the digonals bisect each other at right angles.
Statement
Reasons
3) Consider the ∆AOD and ∆COD
AD = CD
AO = CO
DO is common
Sides of the square are equal
The diagonals bisect each other
S.S.S postulate
Congruent parts of congruent ∆s
Linear pair

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: Statement Reasons In ∆BDC,
Opposite angles of the parallelogram ABCD.
Sum of three angles of a triangle
By substitution
By transposing
Alternate angles, AD || BC.

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: Statement Reasons
Consider the ∆BPQ and ∆CPD,
BP = PC
------(1)
But DC = AB
------(2)
Given
Vertically opposite angles
Alternate angles, AB || DC
A.S.A. postulate
C.P.C.T
Opposite sides of the parallelogram
From (1) and (2)

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