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# Geom 6point3 97

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### Geom 6point3 97

1. 1. 6.3 Proving Quadrilaterals are Parallelograms <ul><li>Objective: </li></ul><ul><li>Prove that a quadrilateral is a parallelogram </li></ul><ul><li>Use coordinate geometry with parallelograms. </li></ul>
2. 2. <ul><li>If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. </li></ul>Theorem
3. 3. Theorem <ul><li>If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. </li></ul>
4. 4. Theorem <ul><li>If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. </li></ul>x˚ 180 - x˚ x˚
5. 5. Theorem <ul><li>If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. </li></ul>
6. 6. Proof <ul><li>Prove Theorem 6.6 - If both pairs of opposite sides are  , then the quadrilateral is a parallelogram. </li></ul><ul><li>Given: AB  CD and AD  CB </li></ul><ul><li>Prove: ABCD is a parallelogram </li></ul>A B C D
7. 7. Proof <ul><li>AB  CD and AD  CB </li></ul><ul><li>AC  AC </li></ul><ul><li>∆ ABC  ∆CDA </li></ul><ul><li><BAC  <DCA, <DAC  <BCA </li></ul><ul><li>AB || CD, AD || CB </li></ul><ul><li>ABCD is a parallelogram. </li></ul><ul><li>Given </li></ul><ul><li>Reflexive </li></ul><ul><li>SSS </li></ul><ul><li>CPCTC </li></ul><ul><li>Alternate Interior Angles Converse </li></ul><ul><li>Definition of a parallelogram </li></ul>A B C D
8. 8. Theorem <ul><li>If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram </li></ul>
9. 9. Summary - How can we prove quadrilaterals are parallelograms? <ul><li>Show that both pairs of opposite sides are || </li></ul><ul><li>Show that both pairs of opposite sides are  </li></ul><ul><li>Show that both pairs of opposite angles are  </li></ul><ul><li>Show that one angle is supplementary to both consecutive angles </li></ul><ul><li>Show that the diagonals bisect each other </li></ul><ul><li>Show that one pair of opposite sides are  and || </li></ul>
10. 10. Using Coordinate Geometry <ul><li>When a figure is in the coordinate plane, you can use the Distance Formula to prove that the sides are congruent and you can use the slope formula to prove that sides are parallel. </li></ul><ul><li>Go through Example 4 on p. 341 </li></ul><ul><li>Do p. 342 1-8 </li></ul>
11. 11. Homework: p. 342 10-24 evens