9-5: Quadrilaterals Proof Geometry

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.

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.

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

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.

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

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.

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

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)

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.

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 6. Alt. Int. Angle Theorem

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 11. Def. of parallelogram

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

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.

Example Given: PQRS is a parallelogram PW = PS and RU = RQ Prove: SWQU is a parallelogram 1. PQRS is a parallelogram Given 2. 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

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 12. Alt. Int. Angle Theorem 13. SWQU is a parallelogram One set of sides parallel and congruent

Homework p. 285-287: # 2-3, 6, 12, 13, 16, 20