Ninth Grade Mathematics: Trapezoids, Isosceles Trapezoids & Kites
Beyond the family of parallelograms lie quadrilaterals with asymmetrical or partial parallel properties. In this chapter, you will master the geometric properties of trapezoids (where only one pair of opposite sides is parallel), examine the symmetry of isosceles trapezoids, and explore kites (which feature adjacent side equality rather than opposite side parallelism).
The Anatomy of a Trapezoid
A trapezoid (known as a trapezium in British English) is a quadrilateral having at least one pair of parallel opposite sides.
Anatomy of a Trapezoid:
Base 1 (b1)
*-------------*
/ \
Leg (L1) / \ Leg (L2)
/ \
*---------------------*
Base 2 (b2)
(Base 1 || Base 2)
The two parallel sides are called the bases of the trapezoid. The two non-parallel opposite sides are called the legs. The consecutive angles between the two parallel bases along each leg are supplementary: Angle A + Angle D = 180 degrees, and Angle B + Angle C = 180 degrees.
The Mid-Segment (Median) of a Trapezoid
The line segment connecting the midpoints of the two non-parallel legs is called the mid-segment or median of the trapezoid:
The Trapezoid Mid-Segment Theorem:
The mid-segment MN of a trapezoid is parallel to both bases,
and its length is equal to the arithmetic mean (average) of the two base lengths:
Base 1 + Base 2
MN = -------------------
2
The Isosceles Trapezoid and Axis of Symmetry
An isosceles trapezoid is a special trapezoid in which the two non-parallel legs are equal in length (AD = BC).
Properties of an Isosceles Trapezoid:
1. Non-parallel legs are equal: AD = BC.
2. Base angles are equal: Angle A = Angle B, and Angle D = Angle C.
3. Diagonals are strictly equal in length: AC = BD.
4. It possesses a single vertical axis of reflectional symmetry.
5. It is always a cyclic quadrilateral (its vertices lie on a circle).
D C
*---------------------------*
/| |\
/ | | \
AD / | h h | \ BC (AD = BC)
/ | | \
*----+---------------------------+----*
A P Q B
Base Angle A == Base Angle B
Proof of base angle equality: Drop perpendicular heights DP and CQ to the lower base AB. In right triangles APD and BQC: DP = CQ = h (perpendicular distance between parallel lines is constant). Hypotenuse AD = BC (given). By the RHS Congruence Criterion, Triangle APD =~ Triangle BQC. By CPCTC, Angle A = Angle B.
The Kite: Adjacent Equality and Perpendicularity
A kite is a quadrilateral that has two distinct pairs of equal adjacent sides, but opposite sides are not equal.
Properties of a Kite:
1. Two pairs of adjacent sides are equal: AB = AD, and CB = CD.
2. Diagonals are perpendicular to each other: AC is perpendicular to BD.
3. The main diagonal (the line of symmetry AC) bisects the other diagonal BD:
BM = MD.
4. The main diagonal bisects the corner angles at A and C.
5. Exactly one pair of opposite angles is equal: Angle B = Angle D.
A
*
/ | \
AB / | \ AD (AB = AD)
/ | \
*-------+-------*
B M D
\ | /
BC \ | / CD (BC = CD)
\ | /
\ | /
*
C
Notice the critical distinction: in a kite, the main diagonal AC bisects the secondary diagonal BD, but BD does not bisect AC.
Chapter Exercises
Exercise 1: In a trapezoid ABCD with AB parallel to CD, Angle A = 55 degrees and Angle C = 125 degrees. Find the measures of Angle D and Angle B.
Exercise 2: The parallel bases of a trapezoid have lengths 14 cm and 22 cm. Calculate the length of the mid-segment. If the non-parallel legs each have length 5 cm, classify the trapezoid and determine whether its diagonals are equal.
Exercise 3: In a kite ABCD with AB = AD and CB = CD, the diagonals intersect at point M. If Angle B = 110 degrees, Angle A = 80 degrees, and diagonal AC has length 18 cm, find the measure of Angle C and state which diagonal bisects the other.
Solutions and Explanations
Solution 1: In trapezoid ABCD with AB || CD, the consecutive interior angles along leg AD are supplementary: Angle A + Angle D = 180 degrees. Substitute Angle A = 55 degrees: 55 + Angle D = 180, so Angle D = 180 - 55 = 125 degrees. Similarly, the consecutive interior angles along leg BC are supplementary: Angle B + Angle C = 180 degrees. Substitute Angle C = 125 degrees: Angle B + 125 = 180, so Angle B = 180 - 125 = 55 degrees. The missing angles are Angle D = 125 degrees and Angle B = 55 degrees.
Solution 2: Apply the Trapezoid Mid-Segment Theorem: the length of the mid-segment is the average of the two bases: (14 + 22) / 2 = 36 / 2 = 18 cm. Because the two non-parallel legs have equal lengths (5 cm and 5 cm), the figure is an isosceles trapezoid. In any isosceles trapezoid, the diagonals are strictly equal in length; therefore, its diagonals are equal.
Solution 3: In kite ABCD with AB = AD and CB = CD, the non-vertex opposite angles are equal: Angle D = Angle B = 110 degrees. By the Angle Sum Property of a quadrilateral, all four interior angles sum to 360 degrees: Angle A + Angle B + Angle C + Angle D = 360. Substitute the known angles: 80 + 110 + Angle C + 110 = 360. Combine terms: 300 + Angle C = 360, so Angle C = 360 - 300 = 60 degrees. In a kite, the main diagonal connecting the vertices between the pairs of equal sides is the axis of symmetry: diagonal AC bisects the secondary diagonal BD (BM = MD).