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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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§ 8.1 Quadrilaterals § 8.4 Rectangles, Rhombi, and Squares § 8.3 Tests for Parallelograms § 8.2 Parallelograms § 8.5 Trapezoids.

§ 8.1 Quadrilaterals § 8.4 Rectangles, Rhombi, and Squares § 8.3 Tests for Parallelograms § 8.2 Parallelograms § 8.5 Trapezoids.

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