UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 10: Proving Theorems About Parallelograms Instruction

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1 Prerequisite Skills This lesson requires the use of the following skills: applying angle relationships in parallel lines intersected by a transversal applying triangle congruence postulates applying triangle similarity postulates setting up and solving linear equations writing proofs Introduction What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner of that room and looked directly across the room, you would be looking at the opposite corner. If you looked to your right, that corner would be a consecutive corner. If you looked to your left, that corner would also be a consecutive corner. The walls of the room could also be described similarly. If you were to stand with your back at the center of one wall, the wall straight across from you would be the opposite wall. The walls next to you would be consecutive walls. There are two pairs of opposite walls in a rectangular room, and there are two pairs of opposite angles. efore looking at the properties of parallelograms, it is important to understand what the terms opposite and consecutive mean. Key oncepts quadrilateral is a polygon with four sides. convex polygon is a polygon with no interior angle greater than 180º and all diagonals lie inside the polygon. diagonal of a polygon is a line that connects nonconsecutive vertices. onvex polygon U1-671

2 onvex polygons are contrasted with concave polygons. concave polygon is a polygon with at least one interior angle greater than 180º and at least one diagonal that does not lie entirely inside the polygon. > 180º oncave polygon parallelogram is a special type of quadrilateral with two pairs of opposite sides that are parallel. y definition, if a quadrilateral has two pairs of opposite sides that are parallel, then the quadrilateral is a parallelogram. Parallelograms are denoted by the symbol. Parallelogram If a polygon is a parallelogram, there are five theorems associated with it. U1-67

3 In a parallelogram, both pairs of opposite sides are congruent. Theorem If a quadrilateral is a parallelogram, opposite sides are congruent. The converse is also true. If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. U1-673

4 Parallelograms also have two pairs of opposite angles that are congruent. Theorem If a quadrilateral is a parallelogram, opposite angles are congruent. The converse is also true. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. U1-674

5 onsecutive angles are angles that lie on the same side of a figure. In a parallelogram, consecutive angles are supplementary; that is, they sum to 180º. Theorem If a quadrilateral is a parallelogram, then consecutive angles are supplementary. m + m = 180 m + m = 180 m + m = 180 m + m = 180 U1-675

6 The diagonals of a parallelogram have a relationship. They bisect each other. Theorem The diagonals of a parallelogram bisect each other. P P P P P The converse is also true. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. U1-676

7 Notice that each diagonal divides the parallelogram into two triangles. Those two triangles are congruent. Theorem The diagonal of a parallelogram forms two congruent triangles. ommon Errors/Misconceptions thinking that all angles in a parallelogram are congruent even if the parallelogram isn t a rectangle or square misidentifying opposite pairs of sides misidentifying opposite pairs of angles and consecutive angles U1-677

8 Guided Practice Example 1 Quadrilateral has the following vertices: ( 4, 4), (, 8), (3, 4), and ( 3, 0). etermine whether the quadrilateral is a parallelogram. Verify your answer using slope and distance to prove or disprove that opposite sides are parallel and opposite sides are congruent. 1. Graph the figure. y 10 8 (, 8) 4 ( 4, 4) (3, 4) 6 ( 3, 0) x U1-678

9 . etermine whether opposite pairs of lines are parallel. alculate the slope of each line segment. is opposite ; is opposite. y (8 4) 4 y (4 m = = = = = = 8) x [ ( 4)] 6 3 = 4 m = 4 x (3 ) 1 y (4 0) 4 y (0 4) m = = = = = = = 4 m = 4 x [3 ( 3)] 6 3 x [ 3 ( 4)] 1 alculating the slopes, we can see that the opposite sides are parallel because the slopes of the opposite sides are equal. y the definition of a parallelogram, quadrilateral is a parallelogram. 3. Verify that the opposite sides are congruent. alculate the distance of each segment using the distance formula. d= ( x x ) + ( y y ) 1 1 = [ ( 4)] + (8 4) = (6) + (4) = = 5 = 13 = [3 ( 3)] + (4 0) = (6) + (4) = = 5 = 13 = (3 ) + (4 8) = (1) + ( 4) = = 17 = [ 3 ( 4)] + (0 4) = (1) + ( 4) = = 17 From the distance formula, we can see that opposite sides are congruent. ecause of the definition of congruence and since = and =, then and. U1-679

10 Example Use the parallelogram from Example 1 to verify that the opposite angles in a parallelogram are congruent and consecutive angles are supplementary given that and. 10 y 8 (, 8) 4 ( 4, 4) (3, 4) 6 ( 3, 0) x U1-680

11 1. Extend the lines in the parallelogram to show two pairs of intersecting lines and label the angles with numbers. y x Prove 4 9. and Given 4 13 lternate Interior ngles Theorem Vertical ngles Theorem 16 9 lternate Interior ngles Theorem 4 9 Transitive Property We have proven that one pair of opposite angles in a parallelogram is congruent. U1-681

12 3. Prove and Given 7 10 lternate Interior ngles Theorem Vertical ngles Theorem lternate Interior ngles Theorem 7 14 Transitive Property We have proven that both pairs of opposite angles in a parallelogram are congruent. 4. Prove that consecutive angles of a parallelogram are supplementary. and Given 4 and 14 are supplementary. Same-Side Interior ngles Theorem 14 and 9 are supplementary. Same-Side Interior ngles Theorem 9 and 7 are supplementary. Same-Side Interior ngles Theorem 7 and 4 are supplementary. Same-Side Interior ngles Theorem We have proven consecutive angles in a parallelogram are supplementary using the Same-Side Interior ngles Theorem of a set of parallel lines intersected by a transversal. U1-68

13 Example 3 Use the parallelogram from Example 1 to prove that diagonals of a parallelogram bisect each other. 10 y x Find the midpoint of, where M stands for midpoint. y definition, the midpoint is the point on a segment that divides the segment into two congruent parts. + + = 1, 1 M x x y y Midpoint formula M = 4+ 3, , 8 1 = =,4 Substitute values for x 1, x, y 1, and y, then solve. U1-683

14 . Find the midpoint of. + + = 1, 1 M x x y y Midpoint formula M = 3+, , 8 1 = =,4 Substitute values for x 1, x, y 1, and y, then solve. 3. Mark the midpoint of each segment on the graph. Notice that the midpoint of and the midpoint of are the same point. 10 y 8 6 M x Write statements that prove the diagonals bisect each other. Since M is the midpoint of, M M. M is also a point on. Therefore, is the bisector of. The midpoint of is M. This means that M. Since M is a point on, is the bisector of. The diagonals bisect each other. U1-684

15 Example 4 Use the parallelogram from Example 1 and the diagonal to prove that a diagonal of a parallelogram separates the parallelogram into two congruent triangles. 10 y x U1-685

16 1. Use theorems about parallelograms to mark congruent sides. Opposite sides of a parallelogram are congruent, as proven in Example 1. and Opposite sides of a parallelogram are congruent. So far, we know that the triangles each have two sides that are congruent to the corresponding sides of the other triangle. To prove triangles congruent, we could use S, SS, or SSS. From the information we have, we could either try to find the third side congruent or the included angles congruent. 10 y x U1-686

17 . Use the Reflexive Property to identify a third side of the triangle that is congruent. by the Reflexive Property. 10 y x Now, all three sides of the triangles are congruent. 3. State the congruent triangles. Using SSS, we verified that. Therefore, the diagonal splits the parallelogram into two congruent triangles. U1-687