GSP LAB: KITES AND ISOSCELES TRAPEZOIDS
PART ONE: KITES
Definition: A kite is a quadrilateral with two pairs of congruent adjacent sides.
Your GSP sketch should look as follows:
D
C
A
B
1.
Measure angles A, B, C and D in GSP.
Which statement(s) are (is) true?
a) Both sets of opposite angles in a kite are congruent.
b) The angles formed by non-congruent sides of a kite are congruent.
c) The angles formed by congruent sides of a kite are congruent.
2. Create the two diagonals of kite ABCD.
Which statement(s) are (is) true?
a) The diagonals of a kite form right angles.
b) Each diagonal bisects the opposite angles of the kite.
c) The diagonal that connects the vertices between the congruent sides bisects a
pair of opposite angles of the kite.
d) The diagonal that connects the vertices between the non-congruent sides
bisects a pair of opposite angles of the kite.
e) The diagonal formed by the vertices between congruent sides bisects the other
diagonal.
f) The diagonal formed by the vertices between non-congruent sides bisects the
other diagonal.
PART TWO: ISOSCELES TRAPEZOIDS
Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides (per
our text).
Definition: An isosceles trapezoid is a trapezoid in which the nonparallel sides are
congruent. Note: The nonparallel sides are called the legs of the trapezoid. The
parallel sides are called the bases of the trapezoid.
Also note: Each trapezoid has two pairs of base angles. Each pair of base angles are
on the same base and adjacent to each other. One set is called the “upper base
angles”. One set is called the “lower base angles”.
Let’s create a dynamic sketch of an isosceles trapezoid in GSP. (i.e. make sure it has
complete “dragability”.)
If you need help:
Construct circle O. Label the control point as C. Choose the line tool (not the segment or
ray tool!) and draw a line through point O that attaches to the circle at new point B (thus, do
not use the control point C – make sure to start the line at point O to guarantee that the line
goes through point O, and then make sure to lock the line onto circle O at some point we will
call B.) Plot point A at the other point where the line intersects the circle. Hide the line
and create segment AB.
At this point, your GSP sketch should look as follows:
B
O
A
C
Now, highlight segment AB and point C and choose Parallel Line from the construct menu. Plot
point D at the intersection of this new line and the circle. Now, hide the line and create
segments AD, DC and CB to create the isosceles trapezoid. At this point, your sketch should
look approximately as follows:
O
A
D
B
C
Now hide the circle and point O.
1.
Measure angles A, B, C and D in GSP.
Which statement(s) are (is) true?
a) Both pairs of base angles (upper and lower) are congruent in an isosceles
trapezoid.
b) Adjacent angles, one formed at a vertex of the lower base and one formed at a
vertex of the upper base, are supplementary.
c) The bases of an isosceles trapezoid are congruent.
d) The legs of an isosceles trapezoid are congruent.
2. Create the two diagonals of isosceles trapezoid ABCD.
Which statement(s) are (is) true?
a) The diagonals of an isosceles trapezoid are congruent.
b) Both diagonals bisect each other.
c) The diagonals bisect the opposite angles of the isosceles trapezoid.
d) The diagonals intersect to form right angles.
PART THREE: PROOF!
1. Construct a proof to show that the two angles formed by the non-congruent sides
of a kite are congruent. (Make use of the definition of kite.)
Include a diagram, given statement(s), a “to prove” statement, and your proof.
2. Suppose we have just proven that the upper base angles DAB and CBA are
congruent in isosceles trapezoid ABCD shown below. Prove that the diagonals are
congruent. (Make use of the definition of isosceles trapezoid.)
Include a diagram, given statement(s), a “to prove” statement, and your proof.
B
A
C
D
If you run out of time on GSP, use these diagrams to complete the lab.
D
mEDC = 63.83
mADE = 38.29
mDEC = 90.00
mDAE = 51.71
A
C
mECB = 26.17
E
mEAB = 51.71
mDCE = 26.17
mABE = 38.29 mEBC = 63.83
m ED = 5.01 cm
m EB = 5.01 cm
m AE = 3.96 cm
B
m EC = 10.20 cm
m DA = 7.57 cm
m CB = 7.57 cm
m AB = 18.28 cm
B
m DC = 12.01 cm
m AC = 16.64 cm
m DB = 16.64 cm
mABD = 24.46
mDBC = 41.07
A
mCAB = 24.46
mCAD = 41.07
mACB = 90.00
mACD = 24.46
mADB = 90.00
mBDC = 24.46
D
C