unit 7 - quad (quadrilaterals)
quad 1-define quadrilateral and parallelogram
A shape is a quadrilateral iff it is a polygon with four sides.
A quadrilateral is a parallelogram iff it has two sets of parallel lines.
A quadrilateral is a parallelogram iff it has two sets of parallel lines.
quad 2-know and apply attributes for parallelograms to solve problems
quad 3/9/15/16-correctly establish "always, sometimes, never" problems with parallelograms and quadrilaterals by establishing examples and counterexamples
For the problem to the right, we are given the statement, "If parallelogram, then diagonals are bisected." We classify this as always because we have the example as parallelograms and no counter examples. If there are no examples and only counter examples, then the statement is never true, and if there are both examples and counter examples, then the statement is sometimes true.
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quad 4-fill in blanks in scaffolded proofs of the attributes in parallelograms. "iff parallelogram, then..."
An example of this kind of proof is the example at right which is proving that if parallelogram, then opposite angles are congruent. To do this, we first must prove that AIA are congruent, and use reflexive property in order to get ASA to prove that the two triangles are congruent. After this, we can use CPCTC to prove that opposite angles are congruent.
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quad 5-fill in blanks in scaffolded proofs proving a shape is a parallelogram. "If... then parallelogram"
One of the proofs you can do if you have a if.....then parallelogram statement is "If one pairs of opposite sides is both congruent and parallel, then parallelogram." After using reflexive property and using AIA to prove the first two AIA angles are congruent, and the given, we can prove the two triangles are congruent. Then we know that the other two angles are congruent AIA, so we can use that to prove the lines are parallel. If both sets of lines are parallel, it is a parallelogram.
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quad 6/13-understand quadrilateral family hierarchies and that attributes are inherited down the family
In quadrilaterals, there are different subgroups to each group. For example, quadrilateral leads to parallelograms, trapezoids, and other quadrilaterals. Parallelograms lead to rectangles and rhombi and so on. Each smaller group has the same attributes as all the groups its connected to before it, along with some new attributes. Here is a diagram to show what the hierarchy looks like.
quad 7-define rectangle, rhombus, and square
Though each shape has its own individual attributes, the broad definitions for rhombus, rectangle, and square are:
- Rectangle: A quadrilateral with 4 right angles.
- Rhombus: A quadrilateral with 4 congruent sides.
- Square: A quadrilateral with 4 right angles and 4 congruent sides.
quad 8/11-know and apply attributes for rectangle, rhombus, and square to solve problems
Rectangles, Rhombi, and Squares have all the attributes of a parallelogram plus....
- Rectangles -all interior angles are right and diagonals are congruent
- Rhombi-all 4 sides are congruent, opposite angles are bisected by diagonals, and diagonals are perpendicular
- Square-rectangle and rhombus attributes
quad 10-prove various theorems beginning with if a)rectangle, b)rhombus, C) square, then...
quad 12-define each type of quadrilateral. include all parallelograms, trapezoid, isosceles trapezoid, and kite
Definitions
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quad 14-know all the attributes for each quadrilateral
*For attributes for parallelogram, see QUAD 2, and for attributes for rectangle, rhombus, or square, see QUAD 8/11
Attributes of...
Attributes of...
- Quadrilateral
- 4 coplanar sides
- 4 angles whose measures add up to 360 degrees
- Trapezoid
- Exactly one pair of parallel sides
- Two pairs of adjacent, supplementary angles
- Isosceles Trapezoid (All the attributes of trapezoid and..)
- Non-parallel sides are congruent
- 2 pairs of base angles are congruent
- Diagonals are congruent
- Kite
- 2 pairs of congruent adjacent sides
- Exactly one pair of opposite angles is congruent
- Diagonals are perpendicular and bisect exactly one pair of opposite angles
Quad 17-given a quadrilateral's name, prove a specific property of the quadrilateralUsing knowledge of the attributes, you can then prove a certain attribute, given the name.
For example, -If rectangle, then diagonals are congruent -If rhombus, then diagonals are perpendicular *See QUAD 10 for full proof -If parallelogram, then opposite sides congruent |
quad 18-given a set of properties, prove the quadrilateral's specific name (based upon definition)You can then do the reverse of QUAD 17 by using attributes to prove a certain quadrilateral.
For example, -If two pairs of opposite sides are congruent, then parallelogram. -If one opposite side is congruent and parallel, then parallelogram. |
quad 19-know and use triangle and trapezoid midsegment theorems to solve problems algebraically
In a triangle, the midsegment is a line segment that whose endpoints are the midpoints of the two sides and is parallel to the base of the triangle.
For a trapezoid, the same rule applies, except it is parallel to both bases.
Here a few formulas that will be helpful in finding the midsegment.
For a trapezoid, the same rule applies, except it is parallel to both bases.
Here a few formulas that will be helpful in finding the midsegment.
- Midpoint Formula
- Distance Formula
quad 50-construct figures, by definition, Best described as parallelogram, rectangle, rhombus, and square
In constructing a parallelogram by definition,
- First, start off by making a set of parallel lines. This can be done several ways, such as by using AIA or corresponding angles, as we have learned in a previous unit.
- Next, you are going to want to making a second set of parallel lines intersecting with the previous lines you made. Use the same steps from step 1.
- Make the proper constructions marks to show parallel lines.
- First make a straight line using a straight edge.
- Mark 2 points near each end point of the line
- Draw two perpendicular bisectors between each point by extending your compass over halfway and making an arc on either side. Now you will have 3 of the 4 sides of your rectangle, and 2 right angles made.
- To create your last side, make sure that you are not making a square by measuring out the length of the first line and transferring that to one of the side lines.
- Create a perpendicular bisector anywhere that would not make a square, and draw the line.
- You have now created your rectangle and just need to mark the proper marks to show that it has 4 right angles.
- Start off by drawing a point on your paper.
- Using your compass, make an arc however big you want, and draw with the point on your previous drawn point.
- On that arc, draw any two points. Since any point on a circle is equidistant from its vertex, it doesn't matter where you place these points.
- From each point you created, replicate the same exact arc you used for the first point.
- Those arcs should intersect at a point, which is where your fourth vertex is.
- Just connect the 4 points, and make a mark to show congruence in all 4 sides, and you have made a rhombus.
- Start off by making a perpendicular bisector.
- With your compass on the intersection point, make a mark somewhere on the line.
- Copy this same mark all the way around on all 4 lines.
- Connect all 4 points
- Mark the right angles and congruent sides.
quad 51-construct each shape by constructing the associated attributes
a.) If diagonals are perpendicular AND congruent, then square.
b.) If, in a quadrilateral, one angle is supplementary to both of its adjacent angles, then parallelogram.
*See constructions below*
b.) If, in a quadrilateral, one angle is supplementary to both of its adjacent angles, then parallelogram.
*See constructions below*
quad 52-Verify each attribute by constructing the shape by definition first, then confirming the attribute's validity in the construction
a.) If rectangle, then diagonals are congruent.
b.) If square, then diagonals are congruent and perpendicular.
Also, construct figures that are BEST described as trapezoid, isosceles trapezoid, kite, and quadrilateral
*Below are the final products of some of these constructions.*
b.) If square, then diagonals are congruent and perpendicular.
Also, construct figures that are BEST described as trapezoid, isosceles trapezoid, kite, and quadrilateral
*Below are the final products of some of these constructions.*