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Transcript
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: ππ β ππ and ππ || ππ Then: quadrilateral PQRS is a parallelogram 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: ππ β ππ and ππ β ππ Then: quadrilateral PQRS is a parallelogram 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 β < R and < Q β < S Then: quadrilateral PQRS is a parallelogram 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 Given: <A is supplementary to <B <A is supplementary to <D Then: parallelogram ABCD C THEOREM 6-3-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Q R P S Given: ππ & ππ bisect each other Then: quadrilateral PQRS is a parallelogram 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 EXAMPLEβIS IT A PARALLELOGRAM??? Can you prove the quadrilateral is a parallelogram from what is given? Explain. 1. 75° 105° 2. 7 7 105° Yesβboth pairs of opposite angles congruent yesβone pair of opposite sides parallel & congruent EXAMPLEβIS IT A PARALLELOGRAM Determine if each quadrilateral must be a parallelogram. Justify your answer. Parallelogram--The diagonals bisect each other. Noβdifferent pairs of opposite sides congruent and parallel EXAMPLEβSHOW QUAD. IS PARALLELOGRAM Show PQRS is a parallelogram for a = 2.4 and b = 9. Q 10b-16 2a+12 P 9b+25 R 7a 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 β 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 EXAMPLE--NUMERICAL For what value of the variables must the quadrilateral be a parallelogram? Q a + 40° a° R 3c β 3 c+1 P S Opposite sides in parallelogram are congruent (QP = RS) 3c β 3 = c + 1 2c β 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° PROOF Given: parallelogram ABCD parallelogram EFGD Prove: < B β < F Statements 1. Parallelogram ABCD 2. < B β < D 3. parallelogram EFGD 4. <F β < D 5. <B β < F A B E F D G Reasons 1. Given 2. Opposite angles of a parallelogram are congruent 3. Given 4. opposite angles of a parallelogram are congruent 5. Transitive Property of Congruence (2,4) C