Math Work !!
Diagonals of Parallelograms
In this section, you will explore properties of diagonals of rhombuses and
rectangles.
1. Under Condition, select A diagonal bisects one angle.
a. Which angle is bisected? Since the diagonal is an angle
bisector, which angles are congruent?
Angle A was bisected. The angles which are congruent
are angles DAC and BAC.
b. Name another angle that is congruent to DAC. (Hint:
Keep in mind that ABCD is a parallelogram.) How do you know that these angles
are congruent? Name another angle that is congruent to BAC. To check
these congruences, click on Click to measure angles and use the interactive
protractors to measure the angles you think are congruent. (For help using
the protractor, click on Gizmo help, below the Gizmotm.)
Another angle that is congruent to DAC is angle ACB.
I know these are congruent because I checked the angle with the angle measuring
tool. Another angle that is congruent to angle BAC is angle ACD.
c. Drag the vertices of this parallelogram to see other
parallelograms with the same property. Is this parallelogram a rhombus, a
rectangle, or neither? Click on Click to measure lengths and use the interactive
rulers to see test your hypothesis. (For help using the ruler, click on Gizmo
help, below the Gizmo.) Check your hypothesis by clicking Show shape name.
The shape is a rhombus. You can’t change the shape
but you can change the size in length and the angles but still stays the
same parallelogram.
2. Turn off Show shape name, and under Condition, select
A diagonal bisects two angles.
a. Which angles are bisected? Name the two angle pairs
that are marked as congruent. Name two other angle pairs that must also be
congruent because ABCD is a parallelogram. Explain why this is true.
The angles that are bisected are angles A and C. Two
pairs of congruent angles are angles ACB, CAB, and CAD, ACD. Another two
pairs of congruent angles are angles BAD, BCD, and ABC, ADC. The angle measurement
tools to help you prove the angles because they tell you the angle measurement.
b. Drag the vertices of this parallelogram to see other
parallelograms with the same properties. Is this parallelogram a rhombus,
a rectangle, or neither? Use the rulers or protractors to verify your hypothesis.
Check your answer by clicking Show shape name.
The parallelogram is a rhombus. But if you change the
size and the angles measurement you will have a bigger have parallelogram
size wise.
3. Turn off Show shape name, and under Condition, select
Diagonals are perpendicular.
a. Name all the right angles in this figure. Name other
angle pairs that are congruent, based on the fact that ABCD is a parallelogram.
Which triangles in the figure are congruent? Explain.
The names of all the right angles in the rhombus are
AED, DEC, BEC, and AEB. The names of all the congruent angles are DAC, DCA,
and ACB, BAC. Also DCB is congruent to DAB. Also ADC is congruent to ABC.
The triangles that are congruent in this shape is ADC, and ABC.
b. Drag the vertices of this parallelogram to see other
parallelograms with the same property. What is this quadrilateral called?
Use the rulers or protractors to verify your hypothesis. Check your answer
by clicking Show shape name.
Again the size changes about the characteristics but
the form of the rhombus doesn’t change its shape. All the sides are congruent
but we don’t know about the angles.
4. Turn off Show shape name, and under Condition, select
Diagonals are congruent.
a. Which segments are congruent? (Remember, the entire
diagonals are congruent.) Which triangles are congruent? Briefly outline
the proof of this.
b. Resize this parallelogram by dragging the vertices.
Is this a rhombus or a rectangle? Explain your reasoning. Check your answer
by clicking Show shape name.
5. Use what you have learned to make a hypothesis about
what kind of parallelogram is formed if the diagonals are both congruent
and perpendicular. Justify your answer using the Gizmo and an informal proof.
More Special Parallelogram Explorations
In this section, you will explore parallelograms with
special conditions involving right angles and side lengths.
1. Turn off Show shape name and, under Condition, select
One angle is a right angle.
a. Which angle is marked as a right angle? Use the definition
of a parallelogram to explain why D must also be a right angle. Do
the same for B. If three angles in a quadrilateral are right angles,
what must be true about the fourth angle? Why?
The angle that is marked as a right angle is angle
A. Angle D must be a right angle because it is parallel to angle A. Yes the
reason is the same of angle B as well. The last all of the shape should be
a right angle because the all are congruent to make a perfect rectangle.
b. Drag the vertices of this parallelogram to see other
parallelograms with one right angle. Are the other angles always right angles?
Yes all the angles are always right because the shape
doesn’t change only the size changes.
c. What is this quadrilateral called? Check your answer
by clicking Show shape name.
The quadrilateral is a rectangle and not a square because
the shape doesn’t show the lengths of the sides.
2. Turn off Show shape name, and under Condition, select
All four angles are right angles.
a. Since all four angles of this parallelogram are right
angles, what is this figure called? Click Show shape name to check your answer.
This would be called a rectangle that has all sides
that are equal but the sides weren’t congruent.
b. In addition to all four angles being right angles, if
all four sides were congruent, what would this shape be called?
All the angles and all sides are congruent so that
would make this shape a square not a rectangle.
3. Turn off Show shape name, and under Condition, select
All four sides are congruent.
a. Since all four sides of this parallelogram are congruent,
what is this figure called? Click Show shape name to check your answer.
This shape is a rhombus.
b. Drag the vertices of this parallelogram so that all
four angles appear to be congruent. If you could get all four angles to be
congruent, what would be the measures of the angles? What would the name
for that figure be?
The size changes in different ways but the shape always
stays the same at the end.
c. When is a rhombus a square? When is a rectangle a square?
Explain your answers.
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