1. The Distance Formula
Used to find the distance between two points: A( x1, y1) and B(x2, y2)
You also could just plot the points and use the Pythagorean Theorem!!

2. Find the distance between the two points
Find the distance between the two points. Round your answer to the nearest tenth.
T(5, 2) and R(-4, -1)
Take a look at example 2, p. 44
2. A( -2, -3) and B(1, 3)

3. Midpoint Formula Find the midpoint coordinates between 2 points
Find by averaging the x-coordinates and the y-coordinates of the endpoints
(x2, y2)
(x1, y1)

4. Find the coordinates of the midpoint of
Q(3, 5) and S(7, -9)
Q( -4, 4) and S(5, -1)

5. Special Quadrilaterals
Parallelogram – A quadrilateral with both pairs of opposite sides parallel.
Rhombus – A parallelogram with four congruent sides.
Rectangle – A parallelogram with four right angles.
Square – A parallelogram with four congruent sides and four right angles.
Kite – A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent.
Trapezoid – A quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid – A trapezoid whose nonparallel opposite sides are congruent.

6. Warm-up
Draw each figure on graph paper if possible. If not possible explain why.
A parallelogram that is neither a rhombus nor a rectangle
An isosceles trapezoid with vertical congruent sides
A trapezoid with only one right angle
A trapezoid with two right angles
A rhombus that is not a square
A kite with two right angles

7. Properties of Parallelograms
Theorem 6-1
Opposite sides of a parallelogram are congruent.
Theorem 6-2
Opposite angles of a parallelogram are congruent.
(Consecutive angles of a parallelogram are supplementary, they are same-side interior angles!)
Theorem 6-3
The diagonals of a parallelogram bisect each other.

8. Using Algebra Find the value of x in PQRS. Then find QR and PS.

9. Using Algebra
Find the value of y in EFGH. Then find m<E, m<F, m<G, m<H. E F
3y + 37
6y + 4 H G

10. TR=12 find VR
QS=10 find VS

11. Find x and the length of the side Find all angle measures
110̇
120 30

12. Theorem 6-4
Theorem 6-4:
If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

13. Measurement
In the figure, DH ll CG ll BF ll AE, and AB = BC = CD = 2, and EF = Find EH. H D 2 G C 2 F B 2
2.5 A E

14. Proving that a quadrilateral is a parallelogram.
(both pairs of opposite sides are parallel)
Theorem 6-5
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem 6-6
If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.
Theorem 6-7
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

15. Proving that a quadrilateral is a parallelogram.
(Both pairs of opposite sides are parallel)
Theorem 6-5
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem 6-6
If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.
Theorem 6-7
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

16. 5 ways to prove that a quadrilateral is a parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are  .
3. Show that one pair of opposite sides are both  and || .
4. Show that both pairs of opposite angles are  .
5. Show that the diagonals bisect each other .

17. Examples ……
Example 1:
Find the value of x and y that ensures the quadrilateral is a parallelogram.
y+2
6x = 4x+8
2x = 8
x = 4 units
2y = y+2
y = 2 unit 6x
4x+8 2y
Find the value of x and y that ensure the quadrilateral is a parallelogram.
Example 2:
2x + 8 = 120
2x = 112
x = 56 units
5y = 180
5y = 60
y = 12 units
(2x + 8)°
120°
5y°

18. Is the Quadrilateral a Parallelogram?
95°
95° 95 x x

19. Finding Values
Find the values of a and c for which PQRS must be a parallelogram. Q R
(a + 40) a
3c – 3
c + 1 S P

20. Objective 2: Using Coordinate Geometry
When a figure is in the coordinate plane, you can use the Distance Formula (see—it never goes away) to prove that sides are congruent and you can use the slope formula (see how you use this again?) to prove sides are parallel.

21. Ex. 4: Using properties of parallelograms
Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.

22. Ex. 4: Using properties of parallelograms
Method 1—Show that opposite sides have the same slope, so they are parallel.
Slope of AB.
3-(-1) = - 4
1 - 2
Slope of CD.
1 – 5 = - 4
7 – 6
Slope of BC.
5 – 3 = 2
Slope of DA.
- 1 – 1 = 2
AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.
Because opposite sides are parallel, ABCD is a parallelogram.

23. Ex. 4: Using properties of parallelograms
Method 2—Show that opposite sides have the same length.
AB=√(1 – 2)2 + [3 – (- 1)2] = √17
CD=√(7 – 6)2 + (1 - 5)2 = √17
BC=√(6 – 1)2 + (5 - 3)2 = √29
DA= √(2 – 7)2 + (-1 - 1)2 = √29
AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram.

24. Ex. 4: Using properties of parallelograms
Method 3—Show that one pair of opposite sides is congruent and parallel.
Slope of AB = Slope of CD = -4
AB=CD = √17
AB and CD are congruent and parallel, so ABCD is a parallelogram.

25. Theorem 6-9
Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 6-10
The diagonals of a rhombus are perpendicular.
Theorem 6-11
The diagonals of a rectangle are congruent.
Theorem 6-12
If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.
Theorem 6-13
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
Theorem 6-14
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

26. Use the properties of rectangles to find all missing angle measures, list the properties you used.32

27. Objective: To verify and use properties of trapezoids and kites

28. Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to the bases.
The length of the midsegment of a trapezoid is half the sum of the lengths of the bases.
If you are given the midsegment length and one base: -double the length of the midsegment
-subtract the other base.

29. Midsegment (Median) of a Trapezoid
Joins the midpoints of the nonparallel sides
Is parallel to the bases
Its length is ½ the sum of the bases
MN || BC
MN || AD
MN = ½(BC+AD)

30. Find the following:
EF:
86° 3 4
108°

31. Find x: x 8 12

32. Theorem 6-15
The base angles of an isosceles trapezoid are congruent.
Theorem 6-16
The diagonals of an isosceles trapezoid are congruent.
Theorem 6-17
The diagonals of a kite are perpendicular.

33. Theorem: The diagonals of a kite are perpendicular.
A kite has exactly one pair of opposite, congruent angles.

34. Find the measure of the missing angles.1
44°
112° 2
What is the sum of the angles in a quadrilateral?

35. TRAPEZOID LEG LEG BASE ANGLES Name the following: Bases: Legs:
The 2 parallel sides are the bases
The 2 non-parallel sides are the legs
BASE ANGLES B D
LEG
LEG A C
BASE ANGLES
Name the following:
Bases:
Legs:
2 Pairs of Base Angles:

36. Theorem:
The base angles of an isosceles trapezoid are congruent B D C A
Find x.

37. Theorem: The diagonals of an isosceles trapezoid are congruent.
EXAMPLE:
If BD= 2x+10 and AC=x+15, find x and the length of the diagonals.

38. 2 angles that share a leg are supplementary because they are same-side interior angles.

39. The measure of angle A= 110. find the measures of the other 3 angles.

40. One side of a kite is 4 cm less than two times the
length of another side. The perimeter of the kite is
58 cm. Find the lengths of the sides of the kite.

41. Midsegments of trapezoids
The midsegment of an isosceles trapezoid measures 14 cm. One of the bases measures 24 cm. Find the length of the other base.