Understanding Properties of Parallelograms and Their Theorems

1. Chapter 5 By Ethan Arteaga and Alex Goldschmidt

2. 5-1 :-: Properties of a Parallelogram :-: • A parallelogram is a quadrilateral with both pairs of opposite sides parallel. • The symbol for a parallelogram is

3. Theorem 5-1 • Opposite sides of parallelograms are congruent. • Given: EFGH • Prove: EF ≅ HG ; FG ≅ EH E H F G

4. Theorem 5-1 • Proving Opposite sides of parallelograms are congruent. H G 4 2 1 3 E F

5. Theorem 5–2 & 5-3 • Opposite angles of a parallelogram are congruent. • Diagonals of a parallelogram bisect each other. E H 3 2 ∠1 ≅ ∠2 ∠3 ≅ ∠4 4 1 F G

6. Example 1 • Solve for all the variables • X = 62 because of of theorem 5-2 • Y = 118 by subtracting 180 by 62 because you can derive a triangle by cutting the parallelogram in half. 10 x y 8 a 62 b

7. Practice • Solve for all the variables, assume the quadrilaterals are parallelograms. 15 30 80 y 50 8 b a a 70 x 9 11 33 b

8. Theorem 5-4 • This theorem proves that a quadrilateral is a a parallelogram. • It states if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

9. Theorem 5-5 • If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

10. Theorem 5-6 • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

11. Theorem 5-7 • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

12. Example • Is the following quadrilateral a parallelogram? • Yes because opposite angles are congruent.

13. Theorem 5-8 • If two lines are parallel, then all points on one line are equidistant from the other line.

14. Theorem 5-9 • If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

15. Theorem 5-10 & 11 • A line that contains the midpoint of one side o f a triangle and is parallel to another side passes through the midpoint of the third side. Any segment of a triangle that joins the midpoints of two sides of a triangle is not only parallel to the third side but is half as long as the third side as well.

16. Example • Solve for a • a= 6 because since the segment joins the midpoints of the sides it is half of the third side (12). a 12

17. Practice • Find the values of x and y. The red segment is the midpoint of the triangle. 4y + 2 7(y-1) 3x + 5 12x-8

18. Theorem 5-12 & 5-16 • The diagonals of a rectangle are congruent. • If a parallelogram has a right then the parallelogram is a rectangle.

19. Theorem 5-13 & 5-14 • The diagonals of a rhombus are perpendicular. • Each diagonal of a rhombus bisects two angles of the rhombus.

20. Theorem 5-17 • If two consecutive sides of a parallelogram are congruent then parallelogram is a rhombus.

21. Theorem 5-15 • The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices

22. Trapezoids • A quadrilateral with exactly one pair of parallel sides is called a trapezoid. The parallel sides are called the bases. The other sides are legs. Base Leg Leg Base

23. Theorem 5-18 • In another type of trapezoid, an isosceles trapezoid, the base angles and legs are congruent.

24. Theorem 5-19 • A median of a trapezoid is the segment that joins the midpoints of the legs. This median is parallel to the bases and has a length equal to the average of the base lengths. ½(b1+b2) will give you the median length.

25. Example • Find the length of the median. • The median is 10 because ½(8+12) = ½(20) = 10 8 12

26. Practice • Solve for x, assume the median. 18 8 x + 4

27. End - Credits Power Point Directed by Ethan Arteaga & Alex Goldschmidt.