DWG NO. 06.5 — Lesson 5 of 5

Trapezoids and Kites

Unit 6: Quadrilaterals and Polygons · ~15–30 min

Objective

Apply the properties of isosceles trapezoids, the trapezoid midsegment formula, and kite side/diagonal relationships to find missing measures.

Quadrilaterals with only one pair of parallel sides

A trapezoid has exactly one pair of parallel sides, called the bases; the other two sides are the legs. When the legs happen to be congruent, the trapezoid is isosceles and picks up two extra properties.

Isosceles Trapezoid Base Angles Theorem: Each pair of base angles in an isosceles trapezoid is congruent.
Isosceles Trapezoid Diagonals Theorem: The diagonals of an isosceles trapezoid are congruent.
Trapezoid Midsegment Theorem: The midsegment connecting the midpoints of the legs is parallel to both bases, and its length is the average of the two base lengths: midsegment = (base₁ + base₂) ÷ 2.

A kite is a different special case: a quadrilateral with two distinct pairs of consecutive congruent sides (not opposite sides, like a rhombus).

Kite Diagonals Theorem: The diagonals of a kite are perpendicular.
Kite Angles Theorem: Exactly one pair of opposite angles in a kite is congruent — the pair between the two unequal side lengths.

Isosceles trapezoid ABCD, bases AB and DC, with midsegment connecting the midpoints of legs AD and BC.

Worked Example 1 · Trapezoid midsegment

ProblemA trapezoid has bases of 12 and 20. Find the length of its midsegment.

1.Midsegment = (base₁ + base₂) ÷ 2 = (12 + 20) ÷ 2.

2.(12 + 20) ÷ 2 = 32 ÷ 2 = 16.

Midsegment = 16

Worked Example 2 · Kite angle

ProblemKite WXYZ has WX = WZ (short sides) and XY = ZY (long sides). If m∠W = 100°, m∠X = 75°, and m∠Z = 75°, find m∠Y.

1.The quadrilateral's angles sum to 360°, and ∠X ≅ ∠Z as expected for a kite (the angles between unequal sides).

2.m∠Y = 360° − 100° − 75° − 75° = 110°.

m∠Y = 110°

Guided practice

1.A trapezoid has bases of 9 and 15. Find the midsegment length.
Show answer
(9 + 15) ÷ 2 = 24 ÷ 2 = 12.
2.An isosceles trapezoid has diagonals of length 4x − 3 and 2x + 9. Find x.
Show answer
The diagonals are congruent: 4x − 3 = 2x + 9, so 2x = 12, x = 6.
3.A trapezoid's midsegment is 18, and one base is 14. Find the other base.
Show answer
18 = (14 + b) ÷ 2, so 36 = 14 + b, giving b = 22.
4.In kite PQRS, the diagonals intersect at T. What is m∠PTQ?
Show answer
A kite's diagonals are perpendicular, so m∠PTQ = 90°.
5.Explain the key structural difference between a kite and a rhombus, even though both have two pairs of congruent sides.
Show answer
A rhombus has congruent opposite sides (all four sides equal), making it a parallelogram. A kite has congruent consecutive sides — two short sides next to each other and two long sides next to each other — so it's generally not a parallelogram at all.

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