Download Activity 6-6

Geometry
Activity 6-6: Discovering Properties of Kites and Trapezoids
Name ____________________________________
Trapezoids and Kites: Properties
Write each definition below backwards.
By definition, every trapezoid is a quadrilateral that has exactly one pair of parallel sides, and so
every ____________________ that _________________________________________________
__________________________________________is a trapezoid.
By definition, every kite is a quadrilateral that has two pairs of consecutive congruent sides, and
so every _______________________________ that ______________________________________
_____________________________________________________ is a _______________________.
By definition, every isosceles trapezoid is a trapezoid that has two congruent (but nonparallel)
legs, and so every ________________________that ______________________________________
__________________________________________ is an __________________ _______________.
By definition, every midsegment of a trapezoid is a line segment that has its endpoints at the
midpoints of the two (nonparallel) legs of a trapezoid, and so every _______________________
that _____________________________________________________________________________
__________________________________________________________is a ___________________.
Answer the questions about the figures below:
Label the vertices of the trapezoid A, B, C, and D. Measure all four interior angles. Are any of the
four angles congruent? If so, which sides? ______________________________________________
Measure the lengths of the two non-parallel sides. Are the two non-parallel sides congruent? _____
Measure the lengths of the two parallel sides. Are the two parallel sides congruent? _____
Draw the two diagonals. Measure their lengths, and the angle they form with one another.
Are the diagonals congruent? ______ Are the diagonals perpendicular? _________
Label the vertices of the isosceles trapezoid A, B, C, and D. Draw the two diagonals. Measure their
lengths, and the angle they form with one another.
Are the diagonals congruent? _______ Are the diagonals perpendicular? _______
Therefore, the diagonals of an isosceles trapezoid are ________________________ to one
another.
Measure the two legs (the non-parallel sides). Are they congruent? _________
Therefore, the two legs of an isosceles trapezoid are ______________.
Measure the length of the two parallel sides. Are the two parallel sides congruent? ________
Measure the two acute base angles. Compare their measures. Therefore, the two acute base
angles of an isosceles trapezoid are __________________ to each other.
Measure the two obtuse base angles. Compare their measures:
Therefore, the two obtuse base angles of an isosceles trapezoid are __________________ to
each other.
Add the measures of one acute and one obtuse angle. What do you notice about the sum?
Therefore, in an isosceles trapezoid, an obtuse base angle and an acute base angle are always
_____________________________________.
Label the vertices of the kite A, B, C, and D. Draw the two diagonals. Measure their lengths, and the
angle they form with one another.
Are the diagonals congruent? _________
Are the diagonals perpendicular? _________
Therefore, the diagonals of a kite are _____________________ to each other, and they form
four ___________________________ triangles.
Measure all four interior angles. Label the two congruent angles.
Are these two congruent angles formed by the pairs of congruent sides, or are these two angles
formed by the non-congruent sides? _____________________________________
Therefore, in a kite, the two angles formed by the __________________________ sides are
congruent to each other.
Examine the angles in the figure above. Highlight the diagonal that bisects the opposite angles.
Is the diagonal joining the two pairs of congruent sides, or does it join the two pairs of noncongruent sides? _________________________________________________________________
Therefore, in a kite, the diagonal joining the two pairs of __________________________ sides
bisects those two opposite angles.
Measure the top base. The top base measures ___________ cm. Measure the bottom base. The
bottom base measures __________ cm.
Now, average the two bases. (Add them up and divide by 2) The average of the bases measure
______________ cm.
Now, measure the midsegment. The midsegment is ________________ cm.
What do you notice about the length of the midsegment compared to the average of the two bases?
_____________________________
Therefore, the length of the midsegment of a trapezoid is equal to the _________________ of
the two bases.
In other words, if the top base measures b1, and the bottom base measures b2, then the
formula for the length of the midsegment is ___________________________________.
Measure all four angles along the left side of the figure, and label the angle measures on the diagram.
What do you notice about all of the acute angles? ________________________________________
Therefore, all of the acute angles on the left side of the figure are ______________________ to
each other.
What do you notice about all of the obtuse angles? _______________________________________
Therefore, all of the obtuse angles on the left side of the figure are ______________________
to each other.
Since the top and bottom bases of the trapezoid are parallel, and the left side of the trapezoid is a
transversal, and the corresponding angles are congruent, what can we conclude about the orientation
of the midsegment? In other words, the midsegment is _______________________ to the two
bases.
Repeat the steps above for the right side of the figure. What do you notice about all pairs of
corresponding angles on the right side? ________________________________________________