Proving that a quadrilateral is a parallelogram PROVING THE PROPERTIES OF A PARALLELOGRAM
WE ARE PROVING THE “PROVEN PROPERTIES OF PARALLELOGRAMS,” SO YOU SHOULD NOT USE THOSE IN YOUR
PROOFS IN THIS SECTION.
AFTER YOU PROVE A PROPERTY, YOU MAY USE THAT IN FUTURE PROOFS
ALL OF THESE PROOFS WILL START WITH ONLY THE INFORMATION IN THE DEFINITION OF A PARALLELOGRAM .
1.
If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
(You only need to prove this for one pair and then state:
“The argument would be the same for the other consecutive angles”)
2.
If a quadrilateral is a parallelogram, then the opposite angles are congruent.
[HINTS ARE AVAILABLE ON THE LAST
PAGE OF THIS PACKET]
3.
If a quadrilateral is a parallelogram, then the opposite sides are congruent.
4.
If a quadrilateral is a parallelogram, then the diagonals bisect each other.
(You will need to use the proven property from #3)
NOW THAT YOU HAVE PROVEN THESE, YOU CAN USE THESE PROPERTIES IN ANY QUADRILATERAL
THAT IS ALSO A PARALLELOGRAM (RHOMBUS, RECTANGLE, OR SQUARE.)
PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM
TO PROVE SOMETHING IS A PARALLELOGRAM , WE NEED TO MATCH THE DEFINITION: A PARALLELOGRAM HAS 2
PAIRS OF OPPOSITE SIDES PARALLEL .
WE ARE PROVING THE “SUFFICIENT CONDITIONS OF PARALLELOGRAMS,” SO YOU SHOULD NOT USE THOSE IN
HOW DO WE PROVE LINES ARE PARALLEL ? TRY TO FIND AIA, SSIA, OR CORRESPONDING ANGLES.
1.
If the opposite sides are congruent, then the quadrilateral is a parallelogram.
2.
If the opposite angles are congruent, then the quadrilateral is a parallelogram.
3.
If one pair of opposite sides is both parallel and congruent, then the quadrilateral is a parallelogram.
4.
If the diagonals bisect each other, then the quadrilateral is a parallelogram.
NONE OF THESE PROOFS ARE 1 STEP. IF THE PROOF IS A THEOREM , THEN YOU ARE PROVING THAT THEOREM;
DO NOT USE IT IN YOUR PROOF.
1.
Prove the diagonals of an isosceles trapezoid are congruent.
2.
Prove that the line through the ends of a kite is a line of symmetry.
(This is a proof of the Kite Symmetry Theorem, so don’t use that.)
3.
Prove that the perpendicular bisector of the bases of an isosceles trapezoid is a line of symmetry.
(This is a proof of the Isosceles Trapezoid Symmetry Theorem, so don’t use that.)
4.
Prove that the diagonal through the ends of a kite is a perpendicular bisector of the other diagonal.
(This is a proof of the Kite Diagonal Theorem, so don’t use that.)
5.
Prove that a trapezoid with one right angle must have another right angle.
6.
Prove that a paralleogram with one right angle is a rectangle.
7.
Prove the diagonals of a rectangle are congruent.
8.
Prove that the diagonals of a rhombus bisect the vertex angle.
(This is a proof of the Rhombus Diagonal Theorem, so don’t use that.)
CHALLENGE: THESE PROOFS ARE MORE CHALLENGING AND WILL BE WORTH SOME EXTRA
CREDIT. THERE ARE NO HINTS FOR THESE PROOFS .
9.
Prove the diagonals of a rhombus are perpendicular bisectors of each other.
10. Prove the diagonals of a square are both congruent and also perpendicular bisectors of each other.
HINTS
PROVING THE PROPERTIES OF A PARALLELOGRAM
1.
2.
3.
4.
SSIA
Draw a diagonal and prove 2 congruent triangles
Draw a diagonal and prove 2 congruent triangles
Find a congruent triangle pair
PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM
1.
2.
3.
4.
Draw a diagonal and prove 2 congruent triangles
Make the opposite angles equal with variables. What do all 4 angles add up to?
Draw a diagonal and prove 2 congruent triangles
Draw both diagonals and prove 2 congruent triangles
1.
2.
3.
4.
5.
6.
7.
8.
Overlapping congruent triangles
Use triangle congruence to show reflection -or- Use perpendicular bisector theorem to show a perpendicular bisector.
See symmetry proofs
Perpendicular Bisector Theorem
SSIA + 90
SSIA + 90
Overlapping congruent triangles
AIA + ITBAT