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Conditions for parallelograms

Conditions for parallelograms. Geometry CP1 (Holt 6-3) K.Santos. Theorem 6-3-1. If one pair of opposite sides of a quadrilateral is both congruent and parallel then the quadrilateral is a parallelogram. Q R P S Given: and

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Conditions for parallelograms

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  1. Conditions for parallelograms Geometry CP1 (Holt 6-3) K.Santos

  2. Theorem 6-3-1 If one pair of opposite sides of a quadrilateral is both congruent and parallel then the quadrilateral is a parallelogram. Q R P S Given: and Then: quadrilateral PQRS is a parallelogram

  3. Theorem 6-3-2 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Q R P S Given: and Then: quadrilateral PQRS is a parallelogram

  4. Theorem 6-3-3 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Q R P S Given: < P < R and < Q < S Then: quadrilateral PQRS is a parallelogram

  5. Theorem 6-3-4 If an angle of a quadrilateral is supplementary to both of is consecutive angles then the quadrilateral is a parallelogram. A B D C Given: <A is supplementary to <B <A is supplementary to <D Then: parallelogram ABCD

  6. Theorem 6-3-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Q R P S Given: & bisect each other Then: quadrilateral PQRS is a parallelogram

  7. Ways to Prove a quadrilateral is a Parallelogram Show one of the following is true with the quadrilateral: --Both pairs of opposite sides are parallel (definition) --One pair of opposite sides is both congruent and parallel --Both pairs of opposite sides are congruent --Both pairs of opposite angles are congruent --One angle is supplementary to both of its consecutive angles --Both diagonals bisect each other

  8. Example—Is it a parallelogram??? Can you prove the quadrilateral is a parallelogram from what is given? Explain. 1. 75 105 2. 7 7 105 Yes—both pairs of opposite angles congruent yes—one pair of opposite sides parallel & congruent

  9. Example—Is it a parallelogram Determine if each quadrilateral must be a parallelogram. Justify your answer. Parallelogram--The diagonals bisect each other. No—different pairs of opposite sides congruent and parallel

  10. Example—show quad. Is Parallelogram Show PQRS is a parallelogram for a = 2.4 and b = 9. Q 10b-16 9b+25 R 2a+12 7a P S Substitute the values in for a and b 7a= 7(2.4) = 16.8 2a + 12= 2(2.4) + 12 = 16.8 opposite sides congruent 10b – 16 = 10(9) -16 = 74 9b + 25 = 9(9) + 25 = 106 consecutive angles supplementary (parallel sides) Parallelogram---one pair of opposite sides is both parallel and congruent

  11. Example--Numerical For what value of the variables must the quadrilateral be a parallelogram? Q a + 40a R 3c – 3 c + 1 P S Opposite sides in parallelogram are congruent (QP = RS) 3c – 3 = c + 1 2c – 3 = 1 2c = 4 c = 2 Consecutive angles in a parallelogram are supplementary (<Q & <R) a + 40 + a = 180 2a + 40 = 180 2a = 140 a = 70

  12. Proof Given: parallelogram ABCD A B parallelogram EFGD Prove: < B < F E F D G C Statements Reasons 1.Parallelogram ABCD 1. Given 2. < B < D 2. Opposite angles of a parallelogram are congruent 3. parallelogram EFGD 3. Given 4. <F < D 4. opposite angles of a parallelogram are congruent 5. <B < F 5. Transitive Property of Congruence (2,4)

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