Theorem 5-1 Opposite sides of a parallelogram are congruent. Theorem 5-2 Opposite angles of a parallelogram are congruent.

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1 Section 1: Properties of Parallelograms Definition A parallelogram ( ) is a quadrilateral with both pairs of opposite sides parallel. Theorem 5-1 Opposite sides of a parallelogram are congruent. Theorem 5-2 Opposite angles of a parallelogram are congruent. Theorem 5-3 Diagonals of a parallelogram bisect each other. 1 Section 1: Properties of Parallelograms Theorem 5-1 Opposite sides of a parallelogram are congruent. H 4 1 G E 2 3 F 2

2 Section 1: Properties of Parallelograms Theorem 5-2 Opposite angles of a parallelogram are congruent. H 4 1 G E 2 3 F 3 Section 1: Properties of Parallelograms Theorem 5-3 Diagonals of a parallelogram bisect each other. H 1 3 G E 4 2 F 4

3 Section 1: Properties of Parallelograms 1-3: Refer to ABCD. 1. Name all pairs of parallel lines. 2. Name all pairs of congruent angles. 3. Name all pairs of congruent segments. 4-5: Given RSUT, find the values of x, y, a, and b. x 4. R 6 S 5. R y x 9 9 b U a T U B A b T y a S C D 6. Find the perimeter of PINE if PI=12 and IN=8. 5 Section 1: Properties of Parallelograms 7. ABCD is a ; CEFG is a A F A F G D B E C 8. Three coordinates of XWYZ are as follows: X(4,3), W(1,-1), Y(-5,-1), Z(?,?) 9. 3y x 13 4y x+8 22 (4y+5) 80 (11x) 45 6

4 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Theorem 5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. E HE FG, HG FE Quad. EFGH is a. Theorem 5-5 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. H F G Theorem 5-6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 5-7 If a diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 7 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Theorem 5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. E HE FG, HG FE Quad. EFGH is a. H F G 8

5 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Five Ways to Prove that a Quadrilateral is a Parallelogram 1. Show that both pairs of opposite are. 2. Show that both pairs of opposite are. 3. Show that one pair of opposite sides are and. 4. Show that both pairs of opposite are. 5. Show that the diagonals each other. 9 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Exercise Always, sometimes, or never. 1. The diagonals of a quadrilaterals bisect each other. 2. If the measure of two angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. 3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. 4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 5. To prove that a quadrilateral is a parallelogram, it is enough to show that one pair of opposite sides is parallel. 10

6 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Exercise Is the Quadrilateral ABCD parallelogram? If yes, state why. (1) (2) (3) (4) (5) (6) (7) (8) (9) 11 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms Exercise What values must x and y have to make the quadrilateral a parallelogram? (10) (11) BO=x, CO=3x-40, AO=y 2, DO=y+30 m 1=7y-2, m 2=4x+1, m 3=9y, m 4=3x (12) ABCD; W, X, Y, Z are midpoints of segments AO, BO, CO, and DO. WXYZ is a 12

7 Section 2: Ways to Prove that Quadrilaterals Are Parallelograms (12) ABCD; W, X, Y, Z are midpoints of segments AO, BO, CO, and DO. WXYZ is a 13 Section 3: Theorems Involving Parallel Lines Theorem 5-8 If two lines are parallel, then all points on one line are equidistant from the other line. Theorem 5-9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 14

8 Section 3: Theorems Involving Parallel Lines Theorem 5-8 If two lines are parallel, then all points on one line are equidistant from the other line. 15 Section 3: Theorems Involving Parallel Lines Theorem 5-9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 16

9 Section 3: Theorems Involving Parallel Lines Theorem 5-10 A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. Theorem 5-11 The segment that joins the midpoints of two sides of a triangle (1) is parallel to the third side (2) is half as long as the third side. 17 Section 3: Theorems Involving Parallel Lines Theorem 5-10 A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. 18

10 Section 3: Theorems Involving Parallel Lines Theorem 5-11 The segment that joins the midpoints of two sides of a triangle (1) is parallel to the third side (2) is half as long as the third side. 19 Section 3: Theorems Involving Parallel Lines Exercise M, N, and T are midpoints of the sides of XYZ. 1. If XZ=10, then MN= 2. If TN=7, then XY= 3. If ZN=8, then TM= 4. If XY=k, then TN= 5. Suppose XY=16, YZ=12, and XZ=10. What are the lengths of the three sides of (a) TNZ (b) MYN (c) XMT (d) NTM 6. AR BS CT; RS ST a) If RS=12, the ST= b) If AB=8, the BC= c) If AC=20, the AB= d) If AC=10x, the BC= 20

11 Section 3: Theorems Involving Parallel Lines (7) Given that A is the midpoint of segment OX; AB XY; BC YZ AC XZ 21 Section 4: Special Parallelograms Definitions A rectangle is a quadrilateral with four right angles. A rhombus is a quadrilateral with four congruent sides A square is a quadrilateral with four right angles and four congruent sides. 22

12 Section 4: Special Parallelograms Theorem 5-13 The diagonals of a rectangle are congruent. Theorem 5-14 The diagonals of a rhombus are perpendicular. Theorem 5-15 Each diagonal of a rhombus bisects two angles of the rhombus. 23 Section 4: Special Parallelograms Theorem 5-15 The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Theorem 5-16 If an angle of a parallelogram is right angle, then the parallelogram is a rectangle. Theorem 5-17 If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. 24

13 Section 4: Special Parallelograms Exercise Quadrilateral ABCD is a rhombus. Find the measure of each angle. 1. ACD 2. DEC 3. EDC 4. ABC Quadrilateral MNOP is a rectangle. Complete. 5. m PON= 6. m PMO= 7. PL= 8. MO= ABC is right ; M is the midpoint of segment AB. 9. If AM=7, then MB=, AB=, and CM=. 10. If AM=7, then MB=, AB=, and CM=. 25 Section 4: Special Parallelograms Practice Always, sometimes, or never. 1. A square is a rhombus. 2. The diagonals of a parallelogram bisect the angles of the parallelogram. 3. A quadrilateral with one pair of sides congruent and one pair parallel is a parallelogram. 4. The diagonals of a rhombus are congruent. 5. A rectangle has consecutive sides congruent. 6. A rectangle has perpendicular diagonals. 7. The diagonals of a rhombus bisect each other. 8. The diagonals of a parallelograms are perpendicular bisectors of each other. 26

14 Section 4: Special Parallelograms Quadrilateral MORE is a square. 9. If ME=7x-3 and MO=6x, find the value of x. 10. If MT=4x and TE=3x+2, fine OE. Segment XW is a median of right ZYX. 11. If m 2=m 3, find m If YW=3x-2 and WZ=x+8, find YZ. 13. If m 1=40, find m 2, m 3, and m The coordinates of three vertices of a rectangle are A(-3,2), B(4,2), and C(4,-2). Plot the points and find the coordinates of the fourth vertex. Is the rectangle square? 27 Section 5: Trapezoids Definitions A quadrilateral with exactly one pair of parallel sides is called a trapezoid. A trapezoid with congruent legs is called an isosceles trapezoid. The median of a trapezoid is the segment that joining the midpoint of the legs. 28

15 Section 5: Trapezoids Theorem 5-18 Base angles of an isosceles trapezoid are congruent. Theorem 5-19 The median of a trapezoid (1) is parallel to the base; (2) has a length equal to the average of the base lengths. 29 Section 5: Trapezoids Exercise In trapezoid ABCD, segment EF is a median. Complete. 1. AB=25, DC=13, EF=. 2. AE=11, FB=8, AD=, BC=. 3. AB=29, EF=24, DC=. 4. AB=7y+6, EF=5y-3, DC=y-5, y=. In KLM, points H, J, and I are midpoints. 5. Name the three trapezoids in the figure. 6. Name an isosceles trapezoid and its perimeter. 30

16 Section 5: Trapezoids Practice Each diagram show a trapezoid and its median. Find the value of x Quadrilateral TUNE is an isosceles trapezoid with m T=62. Find m U, m N, and m E. (Draw the diagram) 4. Name four trapezoids in the figure. 5. If segment XZ is a median of trapezoid TRAP, then XZ= and XY=. 31