Discovering Kite Properties
Name ________________________________ hr_____
A kite is a quadrilateral with exactly two pairs of distinct congruent consecutive sides.
If you construct two different isosceles triangles on opposite sides of a common base and then remove the base, you have constructed
a kite. In an isosceles triangle, the angle between the congruent sides is called the vertex angle. Therefore let’s call the two angles
between each pair of congruent sides of a kite the vertex angles of the kite and let’s call the other pair the nonvertex angles. What
properties can you discover about kites? Let’s investigate.
Investigation
In this investigation you will look at the diagonals of a kite. Perform the following steps (the first two give you a kite), then compare
your results with your group.
Step 1: On a patty paper, draw two segments of different lengths as shown.
Step 2: Fold through the endpoints and trace the two segments on the back of the patty paper.
Step 3: Unfold the patty paper and draw both diagonals.
How do the diagonals intersect? Share your observations with others near you. State your observations as your next theorem.
Kite DiagonalsTheorem: The diagonals of a kite are _____________________________________________.
Step 4: How do the diagonals divide each other? Does either one bisect the other? Test by comparing the lengths of the segments on
both diagonals. Share your observations with the others near you. State your next theorem.
Kite Diagonal Bisector Theorem:
The diagonal connecting the vertex angles of a kite is the __________________________________ of the other diagonal
What else seems to be true about kites?
Step 5: Compare the size of each pair of opposite angles in your kite by folding an angle onto the opposite angle. Do this for both
pair of opposite angles.
Are both pairs of opposite angles congruent? One pair? Which pair? Are any pairs complementary? Supplementary? Share your
observations with others near you. State your observations as your next conjecture.
Kite Angles Theorem:
The _______________________________________ angles of a kite are ______________________________________.
Step 6: How do the diagonals divide the opposite angles? Does either diagonal bisect one of the angles? Test by folding on both
diagonals. Share your observations with others near you. State your next theorem.
Kite Angle Bisector Theorem:
The __________________________ angles of a kite are ___________________________ by a(n) ____________________________.
Discovering Properties of Midsegments and Medians
The segment connecting the midpoints of two sides of a triangle is called a midsegment of a triangle.
Investigation:
Step 1: Draw a triangle onto a patty paper, pinch to locate midpoints of the sides, and draw in the midsegments.
Step 2. Label the angles as shown. Place a second patty paper over the first and copy one of the four triangles.
Step 3: Compare all four triangles by sliding the copy of one of them over the other three.
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Are the four triangles congruent? Compare your results with the results of others near you. State your observations as your next
conjecture.
Conjecture: The three midsegments of a triangles divide the triangle into _____________________________________.
A midsegment connects the midpoints of two sides of a triangle. Can you see any relationships between a midsegment and the
triangle’s third side (the side not containing either endpoint)? Discuss this with others and state your conjecture.
Conjecture:
A midsegment of a triangle is _________________ to the third side and __________________the length of ____________________.
The line segment connecting the midpoints of the two nonparallel sides of a trapezoid is called the median.
Step 1: Draw a small trapezoid on the left side of a patty paper. Locate the midpoints of the nonparallel sides. Draw in the median.
Step 2: Label the angles as indicated. Place a second patty paper over the first and make a copy of the trapezoid and the median.
Step 3: Compare a pair of base angles with the corresponding angles at the median by sliding the copy up over the original.
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Is 13? If so, the median is parallel to the bases. Compare your results with others and state your conjecture below.
Conjecture: The median of a trapezoid is ________________________________________________________.
What else can you discover about the median of a trapezoid? In a triangle, the midsegment is half the length of the third side. How
does the length of the median of a trapezoid compare to the two bases?
Step 4: On the original trapezoid, extend the longer base to the right by at least the length of the shorter base.
Step 5: Slide the second patty paper under the first. Show the sum of the lengths of the two bases by marking a point on the extension
of the longer base.
Step. 6: Check how many times the median fits on the segment representing the sum of the lengths of the two bases.
What do you notice about the length of the median and the sum of the lengths of the two bases? Compare your results.
Conjecture: The median of a trapezoid is equal in length to ___________________________________________________.
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