1. 9-5: Quadrilaterals
Proof Geometry

2. Definition Write your own definition for a quadrilateral.
Remember a good definition will include all elements that you want in the set while excluding those that you do not want.
Switch definitions with your partner
They will circle all the shapes on the sheet that qualify as quadrilaterals according to your definition.
Switch back, revise your definition, and repeat until you feel your definition is exact.

3. Definition from text book
Let A, B, C, and D be four coplanar points. If no three of these points are collinear, and the segments , , , and intersect only at their end points, then the union of these four segments is called a quadrilateral.
The four segments are called its sides.
The points A, B, C, and D are called its vertices.

4. Convex and Concave
A quadrilateral is called convex if no two of its vertices lie on opposite sides of a line containing a side of the quadrilateral.
Convex
Not Convex

5. More Definitions In a quadrilateral:
Sides are opposite if they do not intersect.
Angles are opposite of they do not have a side in common.
Sides are consecutive if they have a common endpoint.
Angles are consecutive if they have a side in common.
A diagonal is a segment joining two nonconsecutive vertices.

6. Definitions
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel

7. Definitions
A trapezoid is a quadrilateral in which only one pair of opposite sides are parallel. The parallel sides are called bases. The segment joining the midpoints of non-parallel sides is called the median.

8. Properties of Parallelogram
These 4 Theorems are assigned for Homework
Properties of Parallelogram
The opposite sides are parallel. (Definition)
The opposite sides are congruent. (Theorem)
The opposite angles are congruent. (Theorem)
The diagonals bisect each other. (Theorem)
Any two consecutive angles are supplementary. (Theorem)
m1 + m 2 = 180
m2 + m 3 = 180
m3 + m 4 = 180
m4 + m 1 = 180

9. How to prove a polygon is a parallelogram
Both pairs of opposite sides are parallel. (Definition)
Both pairs of opposite sides are congruent. (Theorem)
One pair of opposite sides are congruent and parallel. (Theorem)
Both pairs of opposite angles are congruent. (Theorem)
The diagonals bisect each other. (Theorem)

10. The Midline Theorem
The segment joining the midpoints of two sides of a triangle is:
Parallel to the third side.
Half as long as the third side.

11. Proof of The Midline Theorem
Let F be a point on such that EF = DE
1. EF = DE Point Plotting Theorem
2. EB = EC Def of midpoint
3. BED  FEC Vert. Angle Thm
4.  EDB   EFC SAS
5. DBE  ECF CPCTC
Alt. Int. Angle Theorem

12. Proof of The Midline Theorem (Continue…)
7. AD = DB Def of midpoint
8. DB = CF CPCTC
9. AD= CF Transitive prop of equality
10. ADFC is a One set of sides parallel parallelogram and congruent
Def. of parallelogram

13. Proof of The Midline Theorem (Continue…)
12. DF = DE + EF Def of between
13. DF = 2DE Substitution 1 into 12
14. DE =½ DF Transitive prop of eq
15. AC = DF Opp .sides of parallelogram congruent
16. DE =½ AC Substitution 15 into 14

14. Two lines parallel to the same line Theorem
In a plane, if two lines are each parallel to a third line, then they are parallel to each other.

15. Example Given: PQRS is a parallelogram PW = PS and RU = RQ
Prove: SWQU is a parallelogram
1. PQRS is a parallelogram Given
Def. of Parallelogram
3. m1 = m5 Alt. Angle || Converse
4. PW = PS Given
5. m1 = m2 Isosc. Triangle Thm.
6. m2 = m5 Substitute 5 into 3

16. Example 7. RU = RQ Given 8. m5 = m6 Isosc. Triangle Thm.
9. m2 = m6 Substitute 8 into 6
10. PS = RQ Opp. Sides of parallelogram congruent
11. SWP   QUR ASA
12. SW = QU CPCTC
10. m3 = m7 Third Angles Theorem
11. m4 = m8 Supplement Theorem
Alt. Int. Angle Theorem
13. SWQU is a parallelogram One set of sides parallel and congruent

17. Homework
p : # 2-3, 6, 12, 13, 16, 20