1. § 7.1
1. Prove that in a parallelogram, the diagonals bisect each other. D C F A B

2. A C B D
2. Prove that a quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.

3. A C B D
3. Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. F

4. D C
4. Prove that in a rhombus the diagonals are perpendicular to one another. E A B

5. 5 Midpoint Connector Given: ΔABC, L and M midpoints.
Prove: LM ∥ BC and LM = ½ BC
(1) Extend LM to P so that LM = MP
Construction
(2)  ALM ≅  CPM
SAS
(3) AB ∥ PC
Alt inter s, A & ACP
(4) Construct LC
Construction
(5) BLC ≅ PCL
Alt inter s
(6)  LCP ≅  CLB
SAS
(7) LM = ½ LP = ½ BC
CPCTE
(8) LM ∥ BC
Alt inter s, BCL & PLC A L M P 5 B C

6. 6. Prove that the perpendicular bisectors of the two diagonals and the four sides of a cyclic quadrilateral are concurrent.
This one is easy. The four sides and the diagonals are all chords and the perpendicular bisectors of all chords go through the center of the circle.

7. 7. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
This takes some knowledge of high school geometry and inscribed angles. An inscribed angle is equal in measure to ½ the intercepted arc. The two opposite angles intercept arcs that total the measure of the entire circumference of the circle. Thus the measure of the two opposite angles is ½ of 360 or 180 and thus they are cupplementary.