Properties of Parallelograms Lesson 6-2 Lesson Quiz Use parallelogram ABCD for Exercises 1–5 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 + 100 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. 7 50 72 46 x = 10, y = 20 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Check Skills You’ll Need (For help, go to Lessons 1-8 and 3-7.) 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. Check Skills You’ll Need 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Check Skills You’ll Need Solutions 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. x1 + x2 2 y1 + y2 2 1 + 4 2 2 + 1 2 5 2 3 , = , = , . x1 + x2 2 y1 + y2 2 1 + 4 2 0 + 3 2 5 2 3 2 , = , = , 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Check Skills You’ll Need Solutions (continued) 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. y2 – y1 x2 – x1 1 – 0 4 – 1 = 1 3 m = 3 – 2 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes Statements Reasons 1. 2. 3. 4. 5. 6. 7. Given Reflexive POC SSS CPCTC Defn of AIA Converse of AIA Thm WXYZ is a  Defn of  6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Notes 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Additional Examples Finding Values for Parallelograms 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. 10x – 24 = 8x + 12 Diagonals of parallelograms 2y – 80 = y + 9 bisect each other. 2x – 24 = 12 y – 80 = 9 Collect the variable terms on one side. x = 18 2x = 36 y = 89 Solve. If x = 18 and y = 89, then ABCD is a parallelogram. Quick Check 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Additional Examples Is the Quadrilateral a Parallelogram? Determine whether the quadrilateral is a parallelogram. Explain. a. All you know about the quadrilateral is that only one pair of opposite sides is congruent. a. Therefore, you cannot conclude that the quadrilateral is a parallelogram. b. 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. If x represents the measure of the unmarked angle, x + 75 + 105 + 75 = 360, so x = 105. Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-6. Quick Check 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Additional Examples 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 chart’s compass. Quick Check 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Additional Examples Real-World Connection 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 chart’s 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 chart’s compass is congruent to the angle the boat should travel, which is 36° north of due east. Quick Check 6-3

Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Lesson Quiz Find the values of the variables for which GHIJ must be a parallelogram. 1. 2. x = 6, y = 0.75 a = 34, b = 26 Determine whether the quadrilateral must be a parallelogram. Explain. 3. 4. 5. Yes; the diagonals bisect each other. Yes; one pair of opposite sides is both congruent and parallel. No; both pairs of opposite sides are not necessarily congruent. 6-3