Presentation on theme: "Mathematics in Daily Life"— Presentation transcript:
1Mathematics in Daily Life 9th GradeTheorems on Parallelograms
2Objective 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.
3Flowchart 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 theoremFor example, in any triangle the sum of three angles is 180˚Draw the appropriate figure and name it.AB CWrite the data using symbols.ABC is a triangle12
4Flowchart on Procedure to Prove a Theorem 12Write what is to be proved using symbolsAnalyze the statement of the theorem and write the hypothetical construction if needed and write it symbolicallyThrough the Vertex A drawEF || BC E A F Write thereason forconstructionDraw the appropriate figure and name it.Use postulates, definitions and previously proved theorems along with what is given and construction
5Theorems on Parallelograms The diagonals of a parallelogram bisect each other.Theorem 2:Each diagonal divides the parallelogram into two congruent triangles.
6Theorem 1 ProofTheorem: 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 = ODDCOAB
7Theorem 1 Proof Contd..Proof: i.e., The diagonals of parallelogram bisect each other.StatementReasonsProcess of AnalysisIn ∆AOB and ∆COD,AB = DCOpposite sides of the parallelogramRecognise the ∆s which contain the sides AO, BO, CO, DO.Use the data to prove the congruency of these two ∆s2)Vertically opposite angles3)Alternate anglesAB || DC and BD is a transversal.ASA PostulateCorresponding sides of congruent ∆s
8Theorem 2 ProofTheorem: 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:DCOAB
9Theorem 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.StatementReasonsAB = DCOpposite sides of the parallelogram2) BC = AD3) AC is commonS.S.S. postulate
10Corollary 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 TheoremsThere 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.
11Corollaries 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.
12Corollary 1 ProofCorollary: 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.RS90˚PQ
13Corollary 1 Proof Contd.. Proof: Hence, PQRS is a rectangle. Statement ReasonsGiven---- (1)Opposite angles of parallelogram PQRSSum of the consecutive angles of a parallelogram PQRS is equal to 180˚By substitutionBy transposing----(2)----(3)From the equations (1), (2) and (3)
14Prove this corollary logically Corollary 2 ProofCorollary: In a parallelogram, if all the sides are equal and all the angles are equal, then it is a square.DC90˚ ˚90˚ ˚ABActivity!!!Prove this corollary logically
15Corollary 3 ProofCorollary: 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)DC90˚90˚ ˚AB
16Corollary 3 Proof Contd..Proof: Hence, diagonals of a square bisect each otherStatementReasons1) Consider the ∆ABD and ∆ABCAB = BCAB is common.Sides of the square are equalAngles of the square are equalS.A.S postulateCongruent parts of congruent ∆s2) Consider the ∆AOB and ∆DOCAB = DCOpposite sides of the squareAlternate angles AB || DCA.S.A postulate
17Corollary 3 Proof Contd..Hence, the digonals bisect each other at right angles.StatementReasons3) Consider the ∆AOD and ∆CODAD = CDAO = CODO is commonSides of the square are equalThe diagonals bisect each otherS.S.S postulateCongruent parts of congruent ∆sLinear pair
18Corollary 4 ProofCorollary: 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 logicallyHint :- S.A.S. Postulate of congruency triangles
19ExamplesExample-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:DC80˚70˚AB
20Examples Contd... Solution: Statement Reasons In ∆BDC, Opposite angles of the parallelogram ABCD.Sum of three angles of a triangleBy substitutionBy transposingAlternate angles, AD || BC.
21Examples 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 = BQDCPQAB
22Examples Contd... Solution: Statement Reasons Consider the ∆BPQ and ∆CPD,BP = PC------(1)But DC = AB------(2)GivenVertically opposite anglesAlternate angles, AB || DCA.S.A. postulateC.P.C.TOpposite sides of the parallelogramFrom (1) and (2)
23ExercisesIn a parallelogram ABCD, =60⁰. If the bisectors of and meet at P on DC. Prove thatIn 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