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Properties of Parallelograms Unit 12, Day 1 From the presentation by Mrs. Spitz, Spring 2005 http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.2%20Parallelograms.ppt You will need: • • • • • Index card Scissors 1 piece of tape Ruler Protractor Exploration 1. Mark a point somewhere along the bottom edge of your paper. 2. Draw a line from that point to the top right corner of the rectangle to form a triangle. Amy King Exploration 3. Cut along this line to remove the triangle. 4. Attach the triangle to the left side of the rectangle. 5. What shape have you created? Amy King Q R In this lesson . . . P S And the rest of the unit, you will study special quadrilaterals. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. When you mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel. For example, in the diagram above, PQ║RS and QR║SP. The symbol PQRS is read “parallelogram PQRS.” Exploration Measure the lengths of the sides of your parallelogram. What conjecture could you make regarding the lengths of the sides of a parallelogram? Amy King Theorems about parallelograms Q • 9-1—If a quadrilateral is a parallelogram, then its opposite sides are congruent. ►PQ≅RS and SP≅QR P R S Exploration Measure the angles of your parallelogram. What conjecture could you make regarding the angles of a parallelogram? Amy King Theorems about parallelograms Q R • 9-2—If a quadrilateral is a parallelogram, then its opposite angles are congruent. P ≅ R and Q ≅ S P S Theorems about parallelograms Q • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary (add up to 180°). mP +mQ = 180°, mQ +mR = 180°, mR + mS = 180°, mS + mP = 180° P R S Exploration Draw both of the diagonals of your parallelogram. Measure the distance from each corner to the point where the diagonals intersect. Amy King Exploration What conjecture could you make regarding the lengths of the diagonals of a parallelogram? Amy King Theorems about parallelograms Q • 9-3—If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM ≅ SM and PM ≅ RM R M P S Ex. 1: Using properties of Parallelograms • FGHJ is a parallelogram. Find the unknown length. a. JH b. JK 5 F G K J b. 3 H Ex. 1: Using properties of Parallelograms 5 F SOLUTION: a. JH = FG so JH = 5 G K J b. 3 H Ex. 1: Using properties of Parallelograms 5 F SOLUTION: b. JK = GK, so JK = 3 G K J b. 3 H Ex. 2: Using properties of parallelograms Q PQRS is a parallelogram. Find the angle measure. a. mR b. mQ P R 70° S Ex. 2: Using properties of parallelograms Q R a. mR = mP , so mR = 70° 70° P S Ex. 2: Using properties of parallelograms Q R b. mQ + mP = 180° mQ + 70° = 180° mQ = 110° P 70° S Ex. 3: Using Algebra with Parallelograms P PQRS is a parallelogram. Find the value of x. mS + mR = 180° 3x + 120 = 180 3x = 60 x = 20 S 3x° 120° R Q Ex. 4: Using parallelograms in real life FURNITURE DESIGN. A drafting table is made so that the legs can be joined in different ways to change the slope of the drawing surface. In the arrangement below, the legs AC and BD do not bisect each other. Is ABCD a parallelogram? C B A D Ex. 4: Using parallelograms in real life FURNITURE DESIGN. A drafting table is made so that the legs can be joined in different ways to change the slope of the drawing surface. In the arrangement below, the legs AC and BD do not bisect each other. Is ABCD a parallelogram? ANSWER: NO. If ABCD were a parallelogram, then by Theorem 6.5, AC would bisect BD and BD would bisect AC. They do not, so it cannot be a parallelogram. C B A D Homework Work Packets: Properties of Parallelograms