7.1 Properties of Parallelograms

Transcription

1 PropertiesofParallelograms nb Properties of Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Notice that the definition states that both pairs of opposite sides are parallel. Parallelograms have many other properties but those properties will be the result of proofs. These proofs will be based on what we have learned about parallel lines and congruent triangles. Theorem: Opposite sides of a parallelogram are congruent. Look at the proof below. Given: Parallelogram ABCD êêêêê êêêê Prove: DC;AD@ BC D C 1 2 A B Statements Reasons êêêêê 1. Draw AC 1. Through any 2 pts. there is exactly one line. 2. Parallelogram ABCD 2. Given. DC»» AB. Def. of 2, If 2»» lines are cut by a transversal, A.I.A. 5. AC 5. Reflexive 6. DCAD 6. ASA Postulate êêêêê êêêê 7. DC;AD@ BC 7. C.P.C.T.C.

2 PropertiesofParallelograms nb 2 So now we know that opposite sides of a parallelogram are parallel (from the definition) and that opposite sides of a parallelogram are congruent (from the first theorem). There are several more properties that can be proven as well. Look at the next theorem. Theorem: Opposite angles of a parallelogram are congruent. The proof of this theorem actually follows from the theorem above. In fact we can conclude that B as an 8th step (from C.P.C.T.C). And with the Angle Addition Postulate, Addition Property of Equality, and substitution we can prove the other pair of opposite angles are congruent as well. Look at the next theorem. Theorem: Diagonals of a parallelogram bisect each other. Given: Parallelogram with diagonals AC and DB. Prove: AC and DB bisect each other. D 1 C M A 2 B Plan for proof: You can prove that DCMD using ASA AB since opposite sides of a parallelogram are congruent) and 2 (since they are alternate interior angles). Then MBand MC by C.P.C.T.C.

3 PropertiesofParallelograms nb The following table summarizes the properties of parallelograms. Properties of Parallelograms Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Diagonals bisect each other. So now we know that if a quadrilateral is a parallelogram then it exhibits the four properties above. What if we wanted to prove that a quadrilateral that exhibits one of the properties above is a parallelogram? Certainly, if a quadrilateral has both pairs of opposite sides parallel we can conclude that the quadrilateral is a parallelogram. This follows directly from the definition since, by their very nature, definitions are biconditional. What about the converses of the above theorems? Are they also true? As it turns out the answer is yes. We will prove one of them and list the others without proof. We will also state one other theorem as well. Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. êêêêêê êêêê Given: XYand ZY Prove: Quad. XYZW is a parallelogram W 2 Z X 1 Y

4 PropertiesofParallelograms nb Statements êêêêêê êêêê 1. XYand ZY êêêê êêêê 2. XZ 1. Given Reasons 2. Reflexive property. DZWX. SSS Postulate. C.P.C.T.C. êêêêêê êêêê 5. WZ»» XYand WX»» ZY 5. If two lines are cut by a trans. so that A.I.A. the lines are»». The next three theorems are listed without proof. Theorem: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If the diagonals of a parallelogram bisect each other, then the quadrilateral is a parallelogram. The following table summarizes the different ways to prove that a quadrilateral is a parallelogram.

5 PropertiesofParallelograms nb 5 Five Ways to Prove that a Quadrilateral Is a Parallelogram 1. Show that both pairs of opposite sides are parallel. 2. Show that both pairs of opposite sides are congruent.. Show that one pair of opp. sides are both»» Show that both pairs of opposite angles are congruent. 5. Show that the diagonals bisect each other.