Geometry Theorems — Trapezoid and Kite
Geometry Theorems — Trapezoid and Kite. Practice questions to deepen understanding of the trapezoid and kite. Online math practice with full solutions and step-by-step explanations.
Trapezoid and kite practice — definitions, isosceles trapezoid, midsegment, kite and its diagonals. Properties and theorems.
Definition of a trapezoid (one pair of parallel sides), isosceles trapezoid, equal base angles (isosceles trapezoid), diagonals.
▱ Definition:
A trapezoid is a quadrilateral in which:
Trapezoid = a quadrilateral with one pair of opposite sides parallel
AB ∥ CD (bases)
BC is not parallel to AD (legs)
This is the basic definition! ✓
• The parallel sides: bases
• The non-parallel sides: legs
▱ Special trapezoid:
An isosceles trapezoid is a trapezoid in which:
Isosceles trapezoid = a trapezoid whose two legs are equal in length
BC = AD ✓
A special case of a trapezoid!
∠ Property:
In an isosceles trapezoid, the base angles are:
In an isosceles trapezoid:
• Angles on the same base are equal!
∠A = ∠B (on the upper base)
∠D = ∠C (on the lower base) ✓
Because of the congruence of the triangles formed when altitudes are dropped from both ends! ✓
✖️ Diagonals:
In an isosceles trapezoid, the diagonals are:
In an isosceles trapezoid, the diagonals are equal in length!
AC = BD ✓
A special property!
The diagonals are not necessarily:
• Bisecting each other ✗
• Perpendicular to each other ✗
Only equal! ✓
📏 Midsegment:
A segment connecting the midpoints of the legs of a trapezoid:
A segment connecting the midpoints of the legs of a trapezoid:
1. Is parallel to the bases ✓
2. Its length = average of the bases ✓
MN = (AB + CD) / 2
If AB = 10 cm and CD = 18 cm
then MN = (10 + 18) / 2 = 14 cm ✓
🔢 Computation:
In an isosceles trapezoid, if an angle on the upper base is 70°, the angle on the lower base (on the same leg) is:
Isosceles trapezoid
∠A = 70° (on the upper base)
Solution:
Angles on the same leg are supplementary to 180°:
∠A + ∠D = 180° (on leg AD)
70° + ∠D = 180°
∠D = 110° ✓
🔢 Computation:
In a trapezoid, the bases are 12 cm and 20 cm. The length of the midsegment of the legs is:
base 1 = 12 cm
base 2 = 20 cm
Solution:
Midsegment = average of bases
MN = (12 + 20) / 2
MN = 32 / 2
MN = 16 cm ✓
◆ Kite:
A kite is a quadrilateral in which:
Kite = a quadrilateral with two pairs of adjacent sides equal
AB = AD
CB = CD ✓
A special shape!
✖️ Kite diagonals:
The diagonals of a kite are:
In a kite, the diagonals are perpendicular to each other!
AC ⊥ BD ✓
A property unique to the kite!
One diagonal (the longer one) bisects the other!
But not necessarily both ✓
∠ Kite angles:
In a kite, the angles between the non-equal sides are:
In a kite, the angles between the non-equal sides are equal!
∠B = ∠D ✓
(The angles at the ends of the axis of symmetry)
▱ Right trapezoid:
A right trapezoid is a trapezoid in which:
Right trapezoid = a trapezoid in which one leg is perpendicular to the bases
AD ⊥ AB and AD ⊥ CD
∠A = ∠D = 90° ✓
🔍 Identification:
A trapezoid whose diagonals are equal is:
If in a trapezoid the diagonals are equal,
then the trapezoid is isosceles! ✓
AC = BD ⇒ BC = AD
A way to identify!
◆ Relationship:
A rhombus is:
Rhombus = a special case of a kite!
In a kite: two pairs of adjacent equal sides
In a rhombus: all sides equal
Rhombus ⊂ Kite ✓
In a rhombus:
AB = BC = CD = DA
This also means:
AB = AD (one pair)
CB = CD (second pair)
This satisfies the kite definition! ✓
🔢 Computation:
In a trapezoid, the midsegment is 15 cm and the small base is 12 cm. The large base is:
Midsegment MN = 15 cm
Small base AB = 12 cm
Large base CD = ?
Solution:
MN = (AB + CD) / 2
15 = (12 + CD) / 2
30 = 12 + CD
CD = 18 cm ✓
∠ Angles:
The sum of the angles in a trapezoid is:
The sum of the angles in any quadrilateral is 360°!
∠A + ∠B + ∠C + ∠D = 360° ✓
This is true for a trapezoid too!
Any quadrilateral can be divided into two triangles
2 × 180° = 360° ✓
🔍 Identification:
A quadrilateral whose diagonals are perpendicular and one bisects the other is:
If in a quadrilateral:
• The diagonals are perpendicular ✓
• One of them bisects the other ✓
then the quadrilateral is a kite!
A way to identify a kite!
🔢 Computation:
In kite ABCD, if ∠A=80° and ∠B=120°, then ∠D equals:
∠A = 80°
∠B = 120°
∠D = ?
Solution:
In a kite: ∠B = ∠D (equal angles)
∠D = 120° ✓
Simple!
🔢 Computation:
In an isosceles trapezoid, the bases are 10 and 16 cm and the leg is 5 cm. The perimeter is:
Isosceles trapezoid
Base 1: AB = 10 cm
Base 2: CD = 16 cm
Leg: BC = AD = 5 cm (equal!)
Solution:
Perimeter = 10 + 5 + 16 + 5
Perimeter = 36 cm ✓
📊 Relationship:
A parallelogram is:
Parallelogram = a special case of a trapezoid!
Trapezoid: one pair of parallel sides
Parallelogram: two pairs of parallel sides
Parallelogram ⊂ Trapezoid ✓
In a parallelogram there is also one pair of parallel sides (and another!)
Therefore every parallelogram is also a trapezoid ✓
📚 Summary:
Which property is not true?
"In a trapezoid, the legs are always equal"
This is false! ✗
Only in an isosceles trapezoid are the legs equal
In a regular trapezoid, the legs can be different! ⚠️
✓ In an isosceles trapezoid: diagonals are equal
✓ In an isosceles trapezoid: angles on the same base are equal
✓ Midsegment = average of the bases
✓ In a kite: diagonals are perpendicular
✓ In a kite: angles between non-equal sides are equal