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Mathematics in Daily Life

9th 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.

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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

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Flowchart on Procedure to Prove a Theorem

1 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

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Theorems on Parallelograms

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

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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 O A B

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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

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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 O A B

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

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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.

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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.

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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. R S 90˚ P Q

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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)

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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

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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) D C 90˚ 90˚ ˚ A B

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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

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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

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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

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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: D C 80˚ 70˚ A B

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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.

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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 P Q A B

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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)

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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

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