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FeatureLesson Geometry Lesson Main 1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x. 2. If m BAD = y and m ADC = 4y – 70, find y. 3. If m ABC = 2x and m ADC = 6x + 84, find m BCD. 4. If m BCD = 80 and m CAD = 34, find m ACD. 5. If AP = 3x, BP = y, CP = x + y, and DP = 6x – 40, find x and y. x = 10, y = 20 Use parallelogram ABCD for Exercises 1– Properties of Parallelograms Lesson 6-2 Lesson Quiz 6-3

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main (For help, go to Lessons 1-8 and 3-7.) Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Check Skills Youll Need 1.Find the coordinates of the midpoints of AC and BD. What is the relationship between AC and BD? 2. Find the slopes of BC and AD. How do they compare? 3. Are AB and DC parallel? Explain. 4. What type of figure is ABCD? Use the figure at the right for Exercises 1–4. 6-3

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FeatureLesson Geometry Lesson Main 1. For A(1, 2) and C(4, 1), or (2.5, 1.5). For B(1, 0) and D(4, 3), or (2.5, 1.5). They bisect one another x 1 + x 2 2 y 1 + y 2 2,=,,=, =,=, x 1 + x 2 2 y 1 + y ,=,,=,=,=, Solutions Proving That a Quadrilateral is a Parallelogram Lesson 6-3. Check Skills Youll Need 6-3

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Solutions (continued) Check Skills Youll Need 2. For BC, the endpoints are B(1, 0) and C(4, 1); For AD, the endpoints are A(1, 2) and D(4, 3); The slopes of BC and AD are equal. 3. Yes; they are vertical lines. 4. From Exercise 2, the slopes of BC and AD are the same, so the lines are parallel. From Exercise 3, AB and DC are parallel. Thus ABCD is a parallelogram. y 2 – y 1 x 2 – x 1 1 – 0 4 – 1 == 1313 m = y 2 – y 1 x 2 – x 1 3 – 2 4 – 1 == 1313 m = 6-3

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3 StatementsReasons Given Reflexive POC SSS CPCTC Defn of AIA Converse of AIA Thm Defn of WXYZ is a

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson Notes

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

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FeatureLesson Geometry Lesson Main Find values of x and y for which ABCD must be a parallelogram. If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other. If x = 18 and y = 89, then ABCD is a parallelogram. 10x – 24 = 8x + 12 Diagonals of parallelograms2y – 80 = y + 9 bisect each other. Proving That a Quadrilateral is a Parallelogram Lesson 6-3 x = 18 2x = 36 y = 89 Solve. 2x – 24 = 12 y – 80 = 9 Collect the variable terms on one side. Quick Check Additional Examples 6-3 Finding Values for Parallelograms

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FeatureLesson Geometry Lesson Main Determine whether the quadrilateral is a parallelogram. Explain. a. b. a.All you know about the quadrilateral is that only one pair of opposite sides is congruent. b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360. Therefore, you cannot conclude that the quadrilateral is a parallelogram. If x represents the measure of the unmarked angle, x = 360, so x = 105. Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-6. Quick Check Additional Examples 6-3 Is the Quadrilateral a Parallelogram?

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FeatureLesson Geometry Lesson Main Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Quick Check Additional Examples 6-3 The crossbars and the sections of the rulers are congruent no matter how they are positioned. So, ABCD is always a parallelogram. Since ABCD is a parallelogram, the rulers are parallel. Therefore, the direction the ship should travel is the same as the direction shown on the charts compass.

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FeatureLesson Geometry Lesson Main The captain of a fishing boat plots a course toward a school of bluefish. One side of a parallel rule connects the boat with the school of bluefish. The other side makes a 36° angle north of due east on the charts compass. Explain how the captain knows in which direction to sail to reach the bluefish. Because both sections of the rulers and the crossbars are congruent, the rulers and crossbars form a parallelogram. Therefore, the angle shown on the charts compass is congruent to the angle the boat should travel, which is 36° north of due east. Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Quick Check Additional Examples 6-3 Real-World Connection

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FeatureLesson Geometry Lesson Main Find the values of the variables for which GHIJ must be a parallelogram x = 6, y = 0.75 a = 34, b = 26 Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Determine whether the quadrilateral must be a parallelogram. Explain No; both pairs of opposite sides are not necessarily congruent. Yes; the diagonals bisect each other. Yes; one pair of opposite sides is both congruent and parallel. Lesson Quiz 6-3

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