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Mathematics in Daily Life 9 th Grade Theorems on Parallelograms

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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. 2

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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 3

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Flowchart on Procedure to Prove a Theorem 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 1 2 Write what is to be proved using symbols 4

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

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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 BA O 6

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

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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 BA O 8

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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. StatementReasons 1)AB = DCOpposite sides of the parallelogram 2) BC = ADOpposite sides of the parallelogram 3) AC is common S.S.S. postulate 9

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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. 10

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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. 11

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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. 90˚ S PQ R 12

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Corollary 1 Proof Contd.. Proof: Hence, PQRS is a rectangle. StatementReasons 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)Opposite angles of parallelogram PQRS From the equations (1), (2) and (3) 13

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

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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) 90˚ 90˚ 90˚ 90˚ CD BA 15

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Corollary 3 Proof Contd.. Proof: Hence, diagonals of a square bisect each other StatementReasons 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 Congruent parts of congruent ∆s 16

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

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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 18

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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: 80˚ 70˚ D C BA 19

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Examples Contd... Solution: StatementReasons In ∆BDC, Opposite angles of the parallelogram ABCD. Sum of three angles of a triangle By substitution By transposing Alternate angles, AD || BC. 20

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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 B A Q P 21

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Examples Contd... Solution: StatementReasons 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) 22

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Exercises 1.In a parallelogram ABCD, =60⁰. If the bisectors of and meet at P on DC. Prove that 2.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. 3.If the diagonals PR and QS of a parallelogram PQRS are equal, prove that PQRS is a rectangle. 4.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. 5.ABCD is a parallelogram. If AB = 2 x AD and P is the mid-point of AB, prove that 23

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