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5.3 Parallelograms and Rhombuses parallelogram ______________________________________________________________________ 1 In Chapter 3, we used the idea of symmetry to show that the diagonal of a parallelogram divided the figure into two congruent triangles. How can we prove that using what we know about parallel lines? 2 3 4 This leads us to the next theorem… Theorem 5.14 A diagonal of a parallelogram forms two congruent triangles. B B C A C A D If ABCD is a parallelogram, then D +ABD ≅ +CDB and +ABC ≅ +CDA …and from this theorem, we can state the following corollaries. Ú Corollary 5.15 (This gives us TWO sets of conditions that are true IF THE QUADRILATERAL IS A PARALLELOGRAM) In a parallelogram, both pairs of opposite sides and opposite angles are congruent. B B C A D A C D If ABCD is a parallelogram, then ∠A ≅ ∠C and ∠B ≅ ∠D and if ABCD is a parallelogram, then AB ≅ DC and AD ≅ BC Corollary 5.16 Parallel lines are everywhere equidistant. a A C If a & b , then AB=CD . b B D Proof: Given a & b , let AB and CD be two segments both perpendicular to line b. Because corresponding angles are congruent, AB & CD and therefore ABCD is a parallelogram by the definition of parallelogram. Since opposite sides of a parallelogram are congruent, AB=CD so the distance between the two lines is equal. The following theorems are converses two conditionals contained in Corollary 5.15. Corollary 5.17 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B C Key Steps For Proof + + Draw diagonal AC and prove ABD – CDB by SSS. pBCA –pDAC by CPCTC which means that BC & AD because Alt. Int. angles are congruent. A Repeat this process after drawing diagonal BD to show pCBD –pADB by CPCTC which will show AB & DC thereby proving ABCD is a parallelogram by definition. D Corollary 5.18 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B C A Key Steps For Proof We will use the total number of degrees in a quadrilateral (360E) and since pB –pD and pA –pC we have 2mpA + 2mpB = 360. Simplify this to mpA + mpB = 180 making them supplementary. Then since same-side interior angles are supplementary, one pair of sides is parallel. Repeat to get the second pair of sides parallel which will prove ABCD is a parallelogram by definition. D Theorem 5.19 If a quadrilateral has one pair of opposite sides both parallel and congruent, then the quadrilateral is a parallelogram. B C Proof is in the book on Page 260. It uses congruent triangles again. A D Theorem 5.20 Diagonals of a parallelogram bisect each other. (If it’s a parallelogram, then the diagonals bisect each other.) B C Remember: If it’s a parallelogram, then both pairs of opposite sides are congruent. Then you can prove either + BMC – + DMA or + BMA – + DMC which will M give you BM ≅ MD and AM ≅ MC by CPCTC. A D The converse is also true! Theorem 5.21 If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram. What is a rhombus? ____________________________________________________________________ Theorem 5.22 Every rhombus is a parallelogram. B A So everything that is true for parallelograms is true for rhombuses. 1. Diagonals form 2 congruent triangles 2. Both pairs of opposite angles (and sides) are congruent. 3. Diagonals bisect each other. D C Theorem 5.23 The diagonals of a rhombus are perpendicular to each other. B A So everything that is true for parallelograms is true for rhombuses. 1. Diagonals form 2 congruent triangles 2. Both pairs of opposite angles (and sides) are congruent. 3. Diagonals bisect each other. D C And the converse… Theorem 5.24 If the diagonals of a quadrilateral are perpendicular, then it is a rhombus. The next theorem combines two conditionals together (we can also do that with Thm. 5.23 and Thm. 5.24) Theorem 5.25 A parallelogram is a rhombus if and only if the diagonals bisect the opposite angles. Examples from the book: 15) 2 1 3 55E 115E 17) The measures of consecutive angles in a parallelogram are in the ratio 2:3. Find the measures of the angles of the parallelogram. Hint for #46 - Read the Applied Problem and its solution. the angle of incidence = the angle of reflection