created by chris markstrum © Proving Quadrilaterals are Parallelograms California State Standards for Geometry 4: Prove basic theorems involving congruence & similarity. 7: Prove & use theorems involving parallel lines and properties of quadrilaterals. 12: Find & use side and angle measures in triangles and polygons 17: Prove theorems using coordinate geometry

created by chris markstrum 2005 If a quadrilateral has both pairs of opposite sides are congruent, theorem AB CD p a r a l l e l o g r a m then it is a parallelogram

created by chris markstrum 2005 theorem If both pairs of opposite angles of a quadrilateral are congruent, AB CD parallelogram then it is a parallelogram

created by chris markstrum 2005 theorem If a quadrilateral has one angle that is supplementary to both of its consecutive angles, AB CD p a r a l l e l o g r a m then it is a parallelogram.

created by chris markstrum 2005 theorem if a quadrilateral’s diagonals bisect each other, AB CD M p a r a l l e l o g r a m then it is a parallelogram

created by chris markstrum 2005 AB CD theorem If a quadrilateral has one pair of opposite sides parallel and congruent, p a r a l l e l o g r a m then it is a parallelogram

created by chris markstrum 2005 Example Decide whether you are given enough information to determine that the quadrilateral is a parallelogram. Explain your reasoning. If a quadrilateral has one angle that is supplementary to both of its consecutive angles, then it is a parallelogram. YES

created by chris markstrum 2005 Example Decide whether you are given enough information to determine that the quadrilateral is a parallelogram. Explain your reasoning. If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. YES

created by chris markstrum 2005 Example Decide whether you are given enough information to determine that the quadrilateral is a parallelogram. Explain your reasoning. Definition of a parallelogram YES

created by chris markstrum 2005 Example Describe how you would prove that ABCD is a parallelogram. If a quadrilateral has one pair of opposite sides parallel and congruent, then it is a parallelogram. Congruent alternate interior angles make parallel lines.

created by chris markstrum 2005 Example Definition of parallelogram Congruent alternate interior angles make parallel lines. Describe how you would prove that ABCD is a parallelogram.

created by chris markstrum 2005 Example Describe how you would prove that ABCD is a parallelogram. Definition of parallelogram Congruent alternate interior angles make parallel lines. Congruent corresponding angles make parallel lines.

created by chris markstrum 2005 Given: Prove: PQRS is a parallelogram StatementReason

created by chris markstrum 2005 Given: Prove: ABCD is a parallelogram A B C D StatementReason

created by chris markstrum 2005 P Q R S T StatementReason

created by chris markstrum 2005 To get a parallelogram Show opposite sides || Show opposite sides Show opposite angles Show consecutive angles are supplementary Show diagonals bisect each other Show one pair of opposite sides are both || and ( – 1, 2) A C (1, – 2) x y ( 3, 2) B D ( – 3, – 2) Show that quadrilateral ABCD is a parallelogram Which of the theorems can be used? 1 1 –3 –3 –4 –4 Not on the coordinate plane.

created by chris markstrum 2005 x y ( – 1, 2) A C (1, – 2) ( 3, 2) B D ( – 3, – 2) Show that quadrilateral ABCD is a parallelogram 1 1 –3 –3 –4 –4 Show opposite sides ||Show opposite sides

created by chris markstrum 2005 Summary Six ways to show a quadrilateral is a parallelogram 1) Show two pairs of opposite parallel sides. 2) Show two pairs of opposite congruent sides. 3) Show two pairs of opposite congruent angles. 4) Show one angle is supplementary to both consecutive angles. 5) Show the diagonals bisect each other. 6) Show that one pair of opposite sides are both congruent and parallel.