Proving Parallelogram for Grade 9 Students

Proving Parallelogram for Grade 9 Students

1. CHAPTER 2CHAPTER 2
Quadrilaterals

2. SECTION SECTION
3-13-1
Properties of
Parallelograms

3. •Quadrilateral - a
closed plane figure
that has four sides

4. •Opposite sides - two
sides that do not
share a common
endpoint

5. •Opposite angles -
two angles that do
not share a common
side

6. •Parallelogram - a
quadrilateral with
both pairs of opposite
sides parallel.

7. THEOREM 3 -1THEOREM 3 -1
•Opposite sides of a
parallelogram are
congruent

8. THEOREM 3 - 2THEOREM 3 - 2
•Opposite angles of a
parallelogram are
congruent

9. THEOREM 3 - 3THEOREM 3 - 3
•Diagonals of a
parallelogram bisect each
other

10. SECTION SECTION
3-23-2
Ways to Prove that
Quadrilaterals Are
Parallelograms

11. THEOREM 3 - 4THEOREM 3 - 4
•If both pairs of opposite
sides of a quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.

12. THEOREM 3 - 5THEOREM 3 - 5
•If one pair of opposite sides
of a quadrilateral are both
congruent and parallel,
then the quadrilateral is a
parallelogram.

13. THEOREM 3 - 6 THEOREM 3 - 6
•If both pairs of opposite
angles of a quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.

14. THEOREM 3 - 7THEOREM 3 - 7
•If the diagonals of a
quadrilateral bisect each
other, then the quadrilateral
is a parallelogram.

15. Ways to Prove that Quadrilaterals
Are Parallelograms
1.Show that both pairs of
opposite sides are parallel.
2.Show that both pairs of
opposite sides are congruent
3.Show that one pair of
opposite sides are both
congruent and parallel.

16. Ways to Prove that Quadrilaterals
Are Parallelograms
4. Show that both pairs of
opposite angles are
congruent.
5. Show that the diagonals
bisect each other

17. SECTION SECTION
3-33-3
Theorems Involving
Parallel Lines

18. THEOREM 3 - 8THEOREM 3 - 8
•If two lines are parallel,
then all points on one
line are equidistant
from the other line.

19. THEOREM 3 - 9THEOREM 3 - 9
•If three parallel lines cut
off congruent segments on
one transversal, then they
cut off congruent
segments on every
transversal.

20. THEOREM 3 - 10THEOREM 3 - 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.

21. THEOREM 3 - 11THEOREM 3 - 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

22. SECTION SECTION
3 - 43 - 4
Special Parallelograms

23. •RectangleRectangle -- is a
quadrilateral with
four right angles.

24. •Square -Square - is a
quadrilateral with four
right angles and four
sides of equal length.

25. •Rhombus -Rhombus - is a
quadrilateral with
four sides of equal
length.

26. THEOREM 3 - 12THEOREM 3 - 12
•The diagonals of a
rectangle are
congruent.

27. THEOREM 3 - 13THEOREM 3 - 13
•The diagonals of a
rhombus are
perpendicular.

28. THEOREM 3 - 14THEOREM 3 - 14
•Each diagonal of a
rhombus bisects two
angles of the rhombus

29. THEOREM 3 - 15THEOREM 3 - 15
•The midpoint of the
hypotenuse of a right
triangle is equidistant
from the three vertices

30. THEOREM 3 - 16THEOREM 3 - 16
•If an angle of a
parallelogram is a right
angle, then the
parallelogram is a
rectangle.

31. THEOREM 3 - 17THEOREM 3 - 17
•If two consecutive sides
of a parallelogram are
congruent, then the
parallelogram is a
rhombus.

32. SECTION SECTION
3 - 53 - 5
Trapezoids

33. •Trapezoid - a
quadrilateral with
exactly one pair of
parallel sides.

34. •Bases - the sides that
are parallel in a
trapezoid.

35. •Legs -- the
nonparallel sides of a
trapezoid.

36. •Base angles -- angles
that share a base.
Trapezoids have two
pairs of base angles.

37. •Isosceles Trapezoid -
a trapezoid with legs
of equal length.

38. THEOREM 3 - 18THEOREM 3 - 18
•Base angles of an
isosceles trapezoid are
congruent.

39. •Median -- the
segment that joins
the midpoints of the
legs.

40. THEOREM 3 - 19THEOREM 3 - 19
The median of a trapezoid
1. is parallel to the bases;
2. length is equal to the
½(sum of the two bases)

41. • END Chapter 2END Chapter 2