Task 5 Worksheet – Special Quadrilaterals
In this task, you will be investigating how 2 congruent triangles can be combined to form
special quadrilaterals such as parallelograms, rectangles, rhombi (plural of rhombus),
squares, trapezoids and kites. The special properties of these quadrilaterals stem from
the fact that they can be constructed from congruent triangles.
Part A.
One member of your group will cut out 2 scalene congruent triangles. Arrange these
‘back-to-back’ to form a quadrilateral, so that opposite sides appear parallel, then
answer the following questions:
1. To the best of your ability to determine, do both pairs of opposite sides appear
parallel?
Mark a pair of alternate interior angles that are congruent. Recall that when these
angles are congruent, then the Converse of the Alternate Interior Angles Theorem
states that the non-common sides must be parallel. This proves that the figure you just
created is a parallelogram.
Now, we will use this to explore some other properties of parallelograms. Either
measure lengths of both pairs of opposite sides or recall how the triangle sides match.
2. Opposite sides of a parallelogram are __________________ (fill in the blank)
Now measure opposite angles of the parallelogram or add the angles of the triangle that
form them. Draw a conclusion about opposite angles.
3. Opposite angles of a parallelogram are _________________ (fill in the blank)
From the value you measured, look at consecutive angles of the parallelogram.
4. Consecutive angles of a parallelogram are _______________ (fill in the blank).
Connect opposite vertices of the parallelogram with 2 diagonal lines. Measure the
lengths of each part of the diagonals (a total of 4 measurements). Are any of these 4
lengths congruent?
5. Diagonal lines of a parallelogram ________________ each other. (fill in the
blank)
Finally, move one of the triangles by one of the transformation methods so that it ends
on top of the other triangle.
6. The primary transformation used here is a
_______ Translation
__________ Rotation _________ reflection
Part B
The second member of your group will cut out a pair of congruent right triangles, or use
a pair of congruent right triangles previously constructed. Place the triangles back-toback so that the opposite sides are parallel. Measure the vertex angles produced.
7. The most precise name for this figure is a ______________ (fill in the blank)
Since this figure has opposite, parallel sides it is a parallelogram and therefore has all of
the properties of parallelograms discovered above. In addition, it has special properties
such as the right angles. Now draw the diagonals and measure them.
8. The diagonals of a rectangle are __________________ (fill in the blank)
Note that the diagonals of the rectangle divide it into 4 triangles. Measure the sides and
angles of these triangles.
9. Opposite triangles formed from the diagonals of a rectangle are ____________
and _____________ (fill in the blanks)
Part C
The third member of your group will cut or use two isosceles triangles. Place the
triangles back-to-back on the non-congruent sides. Thus all 4 of the congruent sides
will be on the outside of the figure.
10. The most precise name for this figure is a _________________ (fill in the blank)
This figure is also a special parallelogram and will have all of the parallelogram
properties. In addition, it has special properties such as all 4 sides are congruent. Now
draw the 2 diagonals. Measure the 4 angles formed from the intersection of the
diagonals.
11. The diagonals of a rhombus intersect at _______________ angles.
Move one of the triangles by one (or more) of the transformation methods so that it ends
on top of the other triangle.
12. Which method(s) can be used to accomplish this?
_______ Translation
__________ Rotation _________ reflection
Part D
Cut or use a pair of congruent isosceles right triangles (45-45-90 triangles). Place
them back-to-back along their hypotenuses. Go through the questions for Part B and
Part C.
13. Does this figure have all of the properties of rectangles? ________ (Y/N)
14. Does this figure have all of the properties of rhombi?
________ (Y/N)
15. The most precise name for this figure is a ________________.
Part E
Take any pair of congruent, acute triangles and place them back-to-back so that one is
the reflection of the other.
16. The most precise name for this figure is a ________________.
Draw the diagonals of the figure.
17. In a kite, the diagonals cross at ________________angles.
Bonus
See if you can produce an isosceles trapezoid from a pair of congruent triangles. Show
your result to the instructor.