ParallelogramRhombusRectangleSquareTrapezoidIsosceles Trapezoid Exactly 1 pair of parallel sides 2 pairs of parallel sides Exactly 1 pair of congruent

Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid

Exactly 1 pair of parallel sides

2 pairs of parallel sides

Exactly 1 pair of congruent sides

2 pairs of congruent sides

All sides congruent

Perpendicular diagonals

Congruent diagonals

Diagonals bisect angles

Diagonals bisect each other

Opposite angles congruent

Four right angles

Base angles congruent

Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid

Exactly 1 pair of parallel sides

2 pairs of parallel sides

Exactly 1 pair of congruent sides

2 pairs of congruent sides

All sides congruent

Perpendicular diagonals

Congruent diagonals

Diagonals bisect angles

Diagonals bisect each other

Opposite angles congruent

Four right angles

Base angles congruent

What does it take to prove that a quadrilateral is a rhombus?We know a rhombus has perpendicular diagonals.

Does this mean that if we have a quadrilateral with perpendicular diagonals, it must be a rhombus?

Ways to prove that a quadrilateral is a parallelogram


Show both pairs of opposite sides parallel


Show both pairs of opposite sides parallelShow one pair of opposite sides parallel and congruent

A B

CD


Show both pairs of opposite sides parallelShow one pair of opposite sides parallel and congruentShow both pairs of opposite sides congruentShow both pairs of opposite angles congruentShow that the diagonals bisect each other


Show both pairs of opposite sides parallelShow one pair of opposite sides parallel and congruentShow both pairs of opposite sides congruentShow both pairs of opposite angles congruentShow that the diagonals bisect each other Ways to prove that a quadrilateral is a rhombus

parallelogram and two adjacent sides congruent



parallelogram and two adjacent sides congruentparallelogram and perpendicular diagonals

A B

CD



parallelogram and two adjacent sides congruentparallelogram and perpendicular diagonalsparallelogram and diagonals bisect its angles



parallelogram and two adjacent sides congruentparallelogram and perpendicular diagonalsparallelogram and diagonals bisect its angles Ways to prove that a quadrilateral is a rectangle

parallelogram and one right angleparallelogram and congruent diagonals Ways to prove that a quadrilateral is an isosceles trapezoid

trapezoid and non-parallel sides congruenttrapezoid and one pair of congruent base anglestrapezoid and congruent diagonals

The median of a triangle is a segment from a vertex to the midpoint of the opposite side.

A

B

CM

AM is a median of triangle ABC.

We know that an altitude of a triangle is a segment from a vertex that is perpendicular to the opposite side(AD in the diagram).

A

B

C

D

MN is the median or midsegment of trapezoid ABCD

MN is a midsegment of triangle ABC

A

B C

M N

A B

CD

M N

The segment joining the midpoints of two sides of a triangle (the midsegment) is parallel to the third side and half its length.

The median of a trapezoid is parallel to the bases and is equal in length to the average of the lengths of the bases.

x

½x

b2

b1

½ (b1 + b2)

A

B C

M N

A B

CD

M N

A

B C

M N


P

A

B C

M N


P

AM PC, A PCA (CPCTC) ( alternate interior angles)

MP BC

A

B C

M N


P


MP BC

A

B C

M N


P


MP BC

A

B C

M N


P


MP BC

A

B C

M N


P


MP BC

A

B C

M N


P


MP BC

A

B C

M N


P


MP BC

A B

CD

M N


A B

CD

M N

Proof:


P

Because MP is the midsegment of ABD, MP is parallel to AB by the previous theorem.

A B

CD

M N

Proof:


P

Because MP is the midsegment of ABD, MP is parallel to AB by the previous theorem. Since MP is part of MN, MN is parallel to AB.

A B

CD

M N

Proof:


P


Similarly, PN is the midsegment of ADC, so that PN is parallel to DC. Since PN is part of MN, MN is parallel to DC.

Therefore, median MN is parallel to bases AB and CD.

A B

CD

M N

Proof:


P


Similarly, PN is the midsegment of ADC, so that PN is parallel to DC. Since PN is part of MN, MN is parallel to DC.

Therefore, median MN is parallel to bases AB and CD.

Where is the flaw in this proof?

A B

CD

M N

Proof:


First we prove the following lemma:

If two lines are parallel to the same line, they are parallel to each other.

l If m // l and n // l, prove m // n

A B

CD

M N

Proof:


l



If m // l and n // l, prove m // n

A B

CD

M N

Proof:


l



If m // l and n // l, prove m // n

A B

CD

M N

Proof:


P

MP is parallel to AB. DC is parallel to AB.

PN is parallel to DC.

It is a midsegment of ABD Definition of trapezoid

It is a midsegment of BDCBut Euclid’s Parallel Postulate says there can

only be one line through P parallel to DC.

Therefore, PN and MP must be the same line, i.e. MN. Therefore, MN is parallel to both AB and DC.

A B

CD

M N

Proof:


b1

b2

½ b1½ b2

P

MN = ½ b1 + ½ b2 = ½ (b1 + b2)



A

B C

M N

x

½x

b2

b1

½ (b1 + b2)

A B

CD

M N

42 inches

29 inches

The altitude of a parallelogram

DC

B A

The altitude of a trapezoid

A B

C D

The following construction problems should be worked on and solved in groups of three people. Each group will be assessed three times—once for each category. When your group is ready to be assessed, notify me and indicate on which category the group wants to be assessed. Your group will choose the problem in that category on which to be assessed. The group member who will be asked to solve the problem will be selected randomly. Whatever grade is assigned to that student will be the grade for the group. Be sure each group member can solve each problem correctly. The following scoring guidelines will be used to assess each problem. The maximum possible point total is 45 points.

Construction

0 Cannot do the construction correctly even with hints

3 Can do the construction correctly but only after hints are given

6 Can do the construction correctly without hints

The following construction problems should be worked on and solved in groups of three people. Each group will be assessed three times—once for each category. When your group is ready to be assessed, notify me and indicate on which category the group wants to be assessed. Your group will choose the problem in that category on which to be assessed. The group member who will be asked to solve the problem will be selected randomly. Whatever grade is assigned to that student will be the grade for the group. Be sure each group member can solve each problem correctly. The following scoring guidelines will be used to assess each problem. The maximum possible point total is 45 points.

Proof

0 Cannot do the proof correctly even with hints

3 Can do the proof correctly but only after hints are given

6 Can do the proof correctly without hints

Sample: Construct a rhombus given one side and an altitude

1. Construct a rhombus (not a square), given

a. one side and one angle (1 point) b. one angle and a diagonal (2 points)

c. the altitude and one diagonal (3 points)

2. Construct a parallelogram (not a rhombus), givena. one side, one angle, and one diagonal (1 point)b. two adjacent sides and an altitude (2 points)c. one angle, one side, and the altitude on that side (3 points)

3. Construct an isosceles trapezoid, given a. the diagonal, altitude, and one of the bases (1 point) b. one base, the diagonal, and the angle included by them (2 points) c. the bases and one angle (3 points)

Construct a rhombus given one side and an altitude.










Conclusion: ARTB is a rhombus with the required information.


Conclusion: ARTB is a rhombus with the required information.

Brief Proof:

Because AR, AB, and RT, are all radii of congruent circles, they have the same length. Because RT and AB are both perpendicular to PQ, RT is parallel to AB. Therefore, ARTB a parallelogram (one pair of opposite sides are both congruent and parallel). It is now also a rhombus because two adjacent sides are congruent (AR AB).The sides of the rhombus are all congruent to AB, which was the given side. Since QT is parallel to AB, the distance from QT to AB is always the same, and that distance is equal to PQ, the given altitude.

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ParallelogramRhombusRectangleSquareTrapezoidIsosceles Trapezoid Exactly 1 pair of parallel sides 2 pairs of parallel sides Exactly 1 pair of congruent