Question

unit 2 day 12 proofs about parallelograms practice name: sean breyer per: 3
part i: the diagonals of a parallelogram
conjecture: the diagonals of a parallelogram bisect each other.

1. construct a parallelogram and its diagonals using a straightedge and protractor. make a case that this conjecture is true using your tools.
2. what information is needed to prove that the diagonals of a parallelogram bisect each other?
3. label all information as you determine it and finish the proof however you see fit.

given: abcd is a parallelogram
prove: \\( \overline { a o } \cong \overline { c o } \\) and \\( \overline { b o } \cong \overline { d o } \\)
(hint: try to prove opposite triangles congruent with asa)
4.
given: abed is an isosceles trapezoid
prove: \\( \overline { a d } \cong \overline { b e } \\)
(hint: try to prove diagonals are congruent with sas)
part ii: other diagonals

Explanation:

Step1: Use properties of parallelograms

In parallelogram \(ABCD\), \(AB\parallel CD\) (by definition of a parallelogram). So, \(\angle BAO=\angle DCO\) (alternate - interior angles). Also, \(AB = CD\) (by definition of a parallelogram). And \(\angle BOA=\angle DOC\) (vertical angles).

Step2: Prove triangles congruent

By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle ABO\cong\triangle CDO\).

Step3: Conclude on segment congruence

Since \(\triangle ABO\cong\triangle CDO\), then \(AO = CO\) and \(BO = DO\) (corresponding parts of congruent triangles are congruent).

For the isosceles trapezoid \(ABED\):

Step1: Use properties of isosceles trapezoids

In isosceles trapezoid \(ABED\), \(AE = BD\) (legs of an isosceles trapezoid are congruent), \(\angle AED=\angle BDE\) (base angles of an isosceles trapezoid are congruent), and \(ED = DE\) (common side).

Step2: Prove triangles congruent

By the SAS (Side - Angle - Side) congruence criterion, \(\triangle AED\cong\triangle BDE\).

Step3: Conclude on segment congruence

Since \(\triangle AED\cong\triangle BDE\), then \(AD = BE\) (corresponding parts of congruent triangles are congruent).

Answer:

For the parallelogram \(ABCD\), we have proved \(AO\cong CO\) and \(BO\cong DO\). For the isosceles trapezoid \(ABED\), we have proved \(AD\cong BE\).