Additional Exercises:
- TRAP is a trapezoid. The median has length 4 cm, and one of the bases has length 7 cm. What is the length of
the other base?
Answer: 1 cm
Seven is three more than four, so the other base must be three less than four.
OR solve the equation 4= ( 7 +x)÷ 2
2.W XY Zis a trapezoid. The length of one base is twice the length of the other base, and the median is 9 cm. How
long is each base?
Answer:
(x+ 2 x)÷ 2 = 9 The bases are 6 cm and 12 cm.
x= 6
Kites
Only One Congruent Set –It is important to note that in a kite, only one set of interior angles are congruent, and
only one of the diagonals is bisected. Sometimes students struggle with identifying where these properties hold. It
is the nonvertex angles that are congruent, and the diagonal connecting the nonvertex angles that is bisected. The
single line of symmetry of a kite shows both these relationships.
Break it Up –When working with a kite, it is sometimes easier to think of it as two isosceles triangles, or four right
triangles, instead of one quadrilateral.
At this point in the class, students have had extensive experience working with isosceles triangles, and can easily
apply the Base Angle theorem to see that the nonvertex angles of the kite are congruent. They have also seen that
the segment from the vertex angle creates many symmetries in the triangle, and it will make since to them that the
diagonal connecting the nonvertex angles is bisected.
They can also think of a kite as four right triangles. This will help them remember that the diagonals are perpendic-
ular, and remind them that the Pythagorean theorem can be used to find missing segment measures. Noticing that
the right triangles are in two congruent sets will help them identify congruent segments and angles.
Additional Exercises:
- Refer to the kite used to prove the diagonal properties on page 396.
Prove that 4 AY Ris congruent to 4 T Y R.
Answer:
TABLE2.9:
Statement Reason
AR∼=T R Given
AT⊥PR Kite Diagonal Theorem
(^6) AY Ris right Definition of Perpendicular
(^6) T Y Ris right Definition of Perpendicular
(^6) T Y R∼= (^6) AY R Right Angle Theorem
4 AY R∼= 4 T Y R HL
2.6. Quadrilaterals