Download 6D quad quiz review

Honors Math 2
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Quadrilaterals Quiz Review
This quiz will cover Investigation 6D in our textbook. This includes section 6.15-6.18.
Here is a summary of some key theorems for proving special types of quadrilaterals.
However, this is not a comprehensive list of all theorems proved in this unit.
Ways to prove a parallelogram
• If a quadrilateral has both pairs of opposite sides congruent, then the quadrilateral
is a parallelogram.
• If a quadrilateral has each pair of consecutive angles supplementary, then the
quadrilateral is a parallelogram.
• If a quadrilateral has both pairs of opposite angles congruent, then the
quadrilateral is a parallelogram.
• If each diagonal of a quadrilateral divides it into two congruent triangles, then the
quadrilateral is a parallelogram.
• If a quadrilateral has diagonals that bisect each other, then the quadrilateral is a
parallelogram.
• If a quadrilateral has one pair of opposite sides congruent and parallel, then the
quadrilateral is a parallelogram.
Ways to prove a rectangle
• If a parallelogram has at least one right angle, then the parallelogram is a
rectangle.
• If a parallelogram has congruent diagonals, then the parallelogram is a rectangle.
Ways to prove a rhombus
• If a parallelogram has two adjacent sides congruent, then the parallelogram is a
rhombus.
• If a quadrilateral has four congruent sides, then the quadrilateral is a rhombus.
• If a parallelogram has perpendicular diagonals, then the parallelogram is a
rhombus.
• If one diagonal of a parallelogram bisects two opposite angles, then the
parallelogram is a rhombus.
Part A. Visual Definitions
Draw and label a picture representing our definition for each of the special quadrilateral
types we have studied. Make sure you include only what’s in the definition, and not other
properties that we may have proved later.
For example, our definition of rectangle is: a parallelogram
with four right angles. So, the picture at the right represents
this definition.
Now you do the same for the other types of quadrilaterals.
rectangle
Part B. Quadrilateral types from diagonal properties
Here is a fact proved in homework (page 518 exercise 4):
If the diagonals of a quadrilateral bisect each other, then the quadrilateral must be a
parallelogram.
Let’s write similar statements where the conclusions involve other quadrilateral types,
then write proofs of the statements. Fill in the blanks with whatever properties are
needed, then write a proof. Note: we already have similar theorems but they require
starting with a parallelogram (not a quadrilateral).
1. If the diagonals of a quadrilateral _________________________________________,
then the quadrilateral must be a rectangle.
Proof:
2. If the diagonals of a quadrilateral _________________________________________,
then the quadrilateral must be a rhombus.
Proof:
3. If the diagonals of a quadrilateral _________________________________________,
then the quadrilateral must be a kite.
Proof:
4. If the diagonals of a quadrilateral _________________________________________,
then the quadrilateral must be an isosceles trapezoid.
To prove this one requires some knowledge of similar triangles, so OK to leave this
statement as a conjecture for now.
Homework: reviewing for Monday’s quiz on 6D: Quadrilaterals
The best way to prepare for the quiz on quadrilaterals is to thoroughly review the work
you did since vacation from sections 6.14-6.18.
If there are any problems that you missed the first time around, try to complete them now.
Here are some problems that would be particularly valuable to review and possible re-do:
p. 514 exercises 19-20
p. 518 exercises 1-5
NEW: p. 524 all
p. 519 exercise 30
p. 523 exercises 18-24
p. 526 Ch. review and test (focus on 6D)