1. By Ethan Arteaga and Alex Goldschmidt
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. H4
Given
∠1 ≅ ∠2
∠3 ≅ ∠4
3. ΔEHG ≅ ΔGFE
4. EH ≅ GF
HG ≅ EF
1. EFGH
2. AIA
3. ASA
4. CPCTC 2 1 E 3 F

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

6. Example 1 10 x y 8 a 62 b 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 80 30 y 50 8 b a a x 70 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
3x + 5
7(y-1)
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 8 12 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.