created by chris markstrum © Proving Quadrilaterals are Parallelograms Objective: To learn how to prove quadrilaterals are parallelograms

created by chris markstrum If a quadrilateral has both pairs of opposite sides 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 2. If a quadrilateral has both pairs of opposite angles congruent, then it is parallelogram. AB CD parallelogram

created by chris markstrum 2005 theorem 3. 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 4. 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 5.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. One angle is supplementary to both of its consecutive angles 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. Opposite angles of a quadrilateral are congruent. 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. One pair of opposite sides are parallel and congruent. 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: ABCD is a parallelogram A B C D 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 Show that quadrilateral ABCD is a parallelogram

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

created by chris markstrum 2005 P Q R S T StatementReason

created by chris markstrum 2005 P Q R S T StatementReason

created by chris markstrum H L J K I M 2 StatementReason

created by chris markstrum 2005 Summary start w/ parallelogram Get opposite sides || Get opposite sides Get opposite angles Get supplementary consecutive angles Get diagonals that bisect each other 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

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