The quadrilateral family tree
A square has four right angles, so is it a rectangle? It has four equal sides, so is it a rhombus too? How do all the named four-sided shapes fit together, and what does a shape inherit from each name it carries?
Where you are 7 in Quadrilaterals
Why this exists
Four-sided shapes come with a crowd of names: trapezoid, parallelogram, rectangle, rhombus, square, kite. Learned one at a time, they look like separate boxes, and a question such as “is a square a rectangle?” turns into an argument about what the shape looks like.
The definitions settle it. Each name is a short list of demands, and many of the lists contain one another. A square meets every demand a rectangle makes, so it is a rectangle, in the same way that every right triangle is a triangle. Put every name in one picture, with each shape placed under all the names whose demands it meets, and the arguments stop.
The picture also saves work. A fact proved once for every parallelogram is true for every rectangle, rhombus and square without being proved again, because each of them is a parallelogram. Knowing where a shape sits tells you, at a glance, every fact it is owed.
Definition
The names, each one built from an earlier one by adding a demand. They are the definitions used in Properties of a parallelogram, Rectangles, rhombuses and squares and Trapezoids and kites, collected in one place. A quadrilateral is a polygon with four sides; for quadrilateral ABCD the sides are \overline{AB}, \overline{BC}, \overline{CD} and \overline{DA}, and two sides that share a vertex are neighbors.
- A trapezoid is a quadrilateral with at least one pair of parallel sides.
- An isosceles trapezoid is a trapezoid whose legs (the two sides that are not the chosen parallel pair) are congruent and not parallel.
- A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
- A rectangle is a parallelogram with four right angles.
- A rhombus is a parallelogram with four congruent sides.
- A square is a parallelogram that is both a rectangle and a rhombus.
- A kite is a quadrilateral with two pairs of congruent sides, where the two sides of each pair are neighbors, for example AB = AD and CB = CD.
These definitions are inclusive: a name never says “and nothing more”. A trapezoid needs at least one pair of parallel sides, so a shape with two pairs still qualifies. A kite needs AB = AD and CB = CD, and a rhombus, with all four sides equal, meets that.
Saying “a square is a rectangle” means every square meets the definition of a rectangle. When that holds, draw the rectangle box above the square box and join them with a line. Do it for every pair of names and you get Figure 1.
Read a line going down as “every shape down here is also one of those up there”. Two boxes with no path between them going steadily downwards are side by side: neither name contains the other.
The most specific name of a shape is the lowest box it belongs to. It is the most useful name, because every other name the shape carries can be read off by climbing up from it.
Intuition
Moving down the tree adds demands. Each step down lets fewer shapes in, and each shape that gets in is owed more. A quadrilateral is owed nothing beyond its four sides. A parallelogram is owed equal opposite sides, equal opposite angles and diagonals that cut each other in half. A square, at the bottom, is owed all of that and more.
Look at what each shape in Figure 2 is marked with. The trapezoid has one pair of arrows. The parallelogram has two. The rectangle adds square corners, the rhombus adds equal ticks, and the square carries both. The kite stands apart: its marks are ticks on neighboring sides and nothing about parallels.
Most boxes in Figure 1 have one parent, but the rhombus has two. It is a parallelogram (two pairs of parallel sides) and a kite (neighboring sides equal), and neither of those families contains the other. So the rhombus inherits from both lines of the family: from the parallelogram, diagonals that bisect each other; from the kite, diagonals that cross at right angles and an area of half the product of the diagonals. The square sits below the rhombus and the rectangle, so it inherits everything above it, from every direction.
Results
The “is a” statements. Every line in Figure 1 is one statement, and a path of several lines is a statement too, because “is a” carries through: every square is a rectangle and every rectangle is a parallelogram, so every square is a parallelogram.
- Every parallelogram and every isosceles trapezoid is a trapezoid.
- Every rectangle and every rhombus is a parallelogram.
- Every rhombus is a kite.
- Every square is a rectangle, a rhombus, a parallelogram, a kite, a trapezoid and a quadrilateral. It is not an isosceles trapezoid.
Properties pass down. A fact proved for every member of one box is true for every member of every box below it. The table lists what each family is owed, and every row is one box together with everything beneath it, or two boxes with everything beneath both.
| Property | Always true for |
|---|---|
| at least one pair of parallel sides | trapezoid, isosceles trapezoid, parallelogram, rectangle, rhombus, square |
| both pairs of opposite sides parallel and congruent | parallelogram, rectangle, rhombus, square |
| opposite angles congruent, diagonals bisect each other | parallelogram, rectangle, rhombus, square |
| four right angles | rectangle, square |
| four congruent sides | rhombus, square |
| two pairs of congruent neighboring sides | kite, rhombus, square |
| diagonals on perpendicular lines | kite, rhombus, square |
| area is half the product of the diagonals | kite, rhombus, square |
| congruent diagonals | isosceles trapezoid, rectangle, square |
| a line of symmetry through the middles of two opposite sides | isosceles trapezoid, rectangle, square |
Where side-by-side boxes overlap. When two boxes sit side by side, the shapes in both form a box of their own, or there are none:
- rectangle and rhombus overlap in the squares;
- parallelogram and kite overlap in the rhombuses;
- trapezoid and kite overlap in the rhombuses;
- rectangle and kite overlap in the squares;
- the isosceles trapezoid overlaps neither the parallelogram nor the kite.
So no isosceles trapezoid is a parallelogram or a kite.
One most specific name. Every quadrilateral has exactly one lowest box in Figure 1, and the boxes it belongs to are that one and every box on the way up from it.
Why it is true
The lines. Each line in Figure 1 joins a definition to one that adds a demand to it, so meeting the lower definition means meeting the upper one. A parallelogram has two pairs of parallel sides, so it has at least one, which makes it a trapezoid. The isosceles trapezoid, the rectangle, the rhombus and the square are each defined as members of the boxes just above them, so those lines hold by definition. A rhombus has AB = BC = CD = DA, so in particular AB = AD and CB = CD, which makes it a kite.
Properties pass down. A proof that every parallelogram has some property uses only the definition of a parallelogram. A rectangle meets that definition, so the same proof, word for word, works for the rectangle. That is the whole reason the table’s rows look the way they do.
Parallelogram and kite give a rhombus. In a parallelogram ABCD opposite sides are congruent: AB = CD and BC = DA (properties of a parallelogram). A kite has two pairs of congruent neighbors, so either AB = AD and CB = CD, or AB = BC and CD = DA. In the first case AB = AD = BC = CD; in the second AB = BC = CD = DA. Either way all four sides are congruent, so the shape is a rhombus. Every rhombus is both, so the overlap is exactly the rhombuses.
Trapezoid and kite give a rhombus. A kite with one pair of parallel sides is a rhombus; this is proved in trapezoids and kites. Every rhombus is both.
Rectangle and rhombus give a square by the definition of a square. Rectangle and kite give a square: a rectangle is a parallelogram, so a rectangle that is a kite is a rhombus by the argument above, and a rectangle that is a rhombus is a square.
No isosceles trapezoid is a parallelogram. An isosceles trapezoid’s legs are not parallel. In a parallelogram both pairs of opposite sides are parallel, so whichever pair is called the bases, the legs are parallel. No isosceles trapezoid is a kite either: such a shape would be a kite with a pair of parallel sides, so a rhombus, so a parallelogram, which was just ruled out.
One most specific name. The pairs of boxes in Figure 1 that sit side by side are the trapezoid and the kite, the kite and the parallelogram, the kite and the rectangle, the rectangle and the rhombus, and the isosceles trapezoid with each of the kite, parallelogram, rectangle, rhombus and square. The overlaps above cover every one of them: the overlap of two side-by-side boxes is always another box below both, or empty. Now take any shape and the boxes it belongs to. Any two of them are either one above the other, or side by side with an overlap that also holds the shape and sits below both. So the boxes holding the shape never branch: there is one at the bottom, and the rest are the boxes on the way up from it.
Worked examples
Every name for a marked shape
Quadrilateral ABCD in Figure 3 has four congruent sides and a right angle at A. List every name from the family tree that it carries, and give its most specific name.
Only the marks count, so start from them and earn each name.
Opposite sides are congruent (AB = CD and BC = DA, since all four are equal), and a quadrilateral whose opposite sides are congruent is a parallelogram, by the tests for a parallelogram. A parallelogram with four congruent sides is a rhombus.
In a parallelogram, neighboring angles are supplementary and opposite angles are congruent. So m\angle B = 180^\circ - 90^\circ = 90^\circ, then m\angle C = m\angle A = 90^\circ and m\angle D = m\angle B = 90^\circ. Four right angles make it a rectangle.
A rectangle that is also a rhombus is a square, which is the lowest box it can reach. Climbing up from the square gives every other name. In all, the shape is a square, a rectangle, a rhombus, a parallelogram, a kite, a trapezoid and a quadrilateral.
It is not an isosceles trapezoid, because its two pairs of opposite sides are both parallel. Its most specific name is square.
Always, sometimes or never
Decide whether each statement is always, sometimes or never true.
(a) A parallelogram is a trapezoid. (b) A rectangle is a rhombus. (c) A square is a kite. (d) A kite is a parallelogram. (e) An isosceles trapezoid is a parallelogram. (f) A trapezoid is a kite.
SolutionThe tree answers each one. “X is a Y” is always true when the box for X lies below the box for Y. It is never true when the two boxes do not overlap. Otherwise it is sometimes true, and the honest answer shows one shape each way.
(a) Always. The parallelogram box is below the trapezoid box: two pairs of parallel sides include at least one pair.
(b) Sometimes. The rectangle and rhombus boxes sit side by side. A 4 by 9 rectangle has neighboring sides of different lengths, so it is not a rhombus; a square is both.
(c) Always. The square box is below the rhombus box, which is below the kite box.
(d) Sometimes. The kite and parallelogram boxes overlap in the rhombuses. A rhombus is a kite that is a parallelogram; a kite with sides 5, 5, 8, 8 in order around it is not a rhombus, so it is not a parallelogram.
(e) Never. The two boxes do not overlap: an isosceles trapezoid’s legs are not parallel, and a parallelogram’s are.
(f) Sometimes. The trapezoid and kite boxes overlap in the rhombuses only. A rhombus is a trapezoid that is a kite; an isosceles trapezoid is a trapezoid that is not.
Borrowing from every ancestor
Square ABCD has diagonals \overline{AC} and \overline{BD} crossing at E, with AE = 2x + 3 and BE = 5x - 9. Find x, the length AC, m\angle AEB and the area of the square.
SolutionNo fact about squares alone is needed. Each step borrows from a box above the square, and each box is named.
From the parallelogram: the diagonals bisect each other, so AE = \tfrac{1}{2}AC and BE = \tfrac{1}{2}BD.
From the rectangle: the diagonals are congruent, AC = BD. Half of equal lengths are equal, so AE = BE:
So AE = 2(4) + 3 = 11, and BE = 5(4) - 9 = 11 agrees. Then AC = 2 \cdot 11 = 22, and BD = 22 as well.
From the kite (or the rhombus): the diagonals lie on perpendicular lines, so m\angle AEB = 90^\circ. The kite also supplies the area formula, half the product of the diagonals:
Check. \triangle AEB has a right angle at E and legs 11 and 11, so by the Pythagorean theorem AB^2 = 11^2 + 11^2 = 242. The area of a square is its side squared, AB^2, which is 242 again.
The most specific name from the diagonals
In quadrilateral ABCD (Figure 4) the diagonals cross at E, with AE = EC, BE = ED and \overline{AC} \perp \overline{BD}. What is the most specific name the shape is guaranteed?
Diagonals that bisect each other make a quadrilateral a parallelogram, by the tests for a parallelogram. That places the shape in the parallelogram box. Now try to move it lower.
Compare \triangle AEB and \triangle AED. They share the side \overline{AE}, the angles at E are both right angles, and BE = ED. So the triangles are congruent by SAS (the congruence criteria), and their third sides match: AB = AD. A parallelogram has AB = CD and AD = BC, so all four sides are congruent. The shape is a rhombus.
Can it be pushed down to the square? Nothing given forces it. A rhombus with diagonals 6 and 8 meets all three conditions, and its diagonals are not congruent, so it is not a rectangle (every rectangle’s diagonals are congruent), and so not a square. A square also meets all three conditions, so the shape might be one; it is just not guaranteed.
The most specific name guaranteed is rhombus.
A rectangle that is a kite
Show that a quadrilateral that is both a rectangle and a kite is a square.
SolutionName it ABCD. As a rectangle it is a parallelogram, so its opposite sides are congruent:
As a kite, its sides come in two pairs of congruent neighbors. Neighbors meet at a vertex, and the two pairs meet at opposite vertices, so the pairs are either at A and C (AB = AD and CB = CD) or at B and D (BA = BC and DA = DC).
At A and C: AB = AD, and AD = BC, and AB = CD, so all four sides equal AB. At B and D: AB = BC, and BC = DA, and AB = CD, so again all four sides equal AB.
Either way the rectangle has four congruent sides, so it is a rhombus as well, and a shape that is both a rectangle and a rhombus is a square. On the tree this is the overlap of two side-by-side boxes, the rectangle and the kite, landing exactly on the square.
A parallelogram named from its numbers
ABCD is a parallelogram with AB = 2x + 5, BC = 4x - 3, CD = 13 and m\angle B = (10x + 10)^\circ. Find x, then give the most specific name of ABCD.
SolutionOpposite sides of a parallelogram are congruent, so AB = CD:
Then AB = 13 and BC = 4(4) - 3 = 13, and opposite sides give DA = BC = 13. All four sides are 13, so ABCD is a rhombus.
Is it lower still? m\angle B = 10(4) + 10 = 50^\circ, which is not a right angle, so ABCD is not a rectangle and therefore not a square. The most specific name is rhombus, and it also carries the names parallelogram, kite, trapezoid and quadrilateral. Its angles are 50^\circ at B and D and 130^\circ at A and C.
One formula, three families
The trapezoid area formula, from area of parallelograms and trapezoids, says a trapezoid with parallel sides a and b a distance h apart has area
Use it to find the area of a parallelogram with a base of 9 and height 4, and of a square with side 6. Then find the square’s area again with the kite formula.
SolutionThe formula was proved for every trapezoid, and under the inclusive definition every parallelogram is a trapezoid, so it applies. Take the two sides of length 9 as the parallel pair: a = b = 9 and h = 4.
That is base times height, 9 \cdot 4, the parallelogram formula, which the trapezoid formula turns into whenever a = b.
A square is a trapezoid too, with a = b = 6 and the distance between them h = 6:
A square is also a kite, so its area is half the product of its diagonals. Each diagonal is the hypotenuse of a right triangle with legs 6 and 6, so it has length \displaystyle \sqrt{72} = 6\sqrt{2}:
Three formulas, from three different boxes, give one answer, as they must: the square sits below all three. Had trapezoids been defined with “exactly one pair” of parallel sides, the trapezoid formula would not cover parallelograms at all, and this check would need a separate proof.
Boundaries
The tree depends on the definitions. Some books define a trapezoid as having exactly one pair of parallel sides. Then parallelograms leave the trapezoid box, the parallelogram box joins straight onto the quadrilateral box, and “every square is a trapezoid” becomes false. Some books require a kite’s two pairs to have different lengths, which takes the rhombus and the square out of the kite box. Some define an isosceles trapezoid by its congruent base angles, which lets rectangles in. Each choice redraws part of Figure 1, so a question about which shape “is a” which only has an answer once the definitions are fixed. This lesson uses the inclusive ones throughout.
It is not really a tree. In a true tree each box has one parent. The rhombus has two, the parallelogram and the kite, and the square has two as well, the rectangle and the rhombus, so the picture is a family with shared descendants rather than a tree. The name stays because it is the usual one.
Properties pass down, never up. Every square has congruent diagonals, but that does not make it true of every rhombus, the box above. And a property shared by two side-by-side boxes need not belong to the box above both of them. Only a fact proved for a whole box travels, and it travels downwards.
Converses do not come free. “Every square is a rectangle” is true and “every rectangle is a square” is false. Climbing up the tree is automatic; climbing down needs evidence, which is what the tests for a rectangle or a rhombus supply.
Only what is stated counts. A shape that looks like a square is a quadrilateral until its marks or its description earn it more. A drawing is a sketch, and two sides that look parallel prove nothing.
Most quadrilaterals have no special name. A quadrilateral with sides 3, 4, 5 and 6 and no parallel sides belongs to the top box only. A dented kite, the dart described in Trapezoids and kites, is a kite and belongs to no box below it. The family tree names the special cases; the general four-sided shape is not on any lower branch, as shapes in more than one category also shows.
Where this goes next
The family tree is a map, and the lessons around it fill in the routes. What each box guarantees, and how those guarantees combine, is collected in what each special quadrilateral guarantees, and a shape that carries several names at once is the subject of shapes in more than one category.
Climbing down the tree takes evidence, and proofs are where the evidence is assembled: proving a quadrilateral is a parallelogram gets a shape into the parallelogram box, and proving a parallelogram is a rectangle or a rhombus moves it lower.
On a coordinate grid the same question, “which box?”, is answered with numbers: slopes decide which sides are parallel or perpendicular, and distances decide which are congruent. That is classifying a quadrilateral from coordinates.
The tree is also a tree of symmetries. A rectangle folds along the lines through the middles of its opposite sides and a rhombus along its diagonals, and the square, below both, has all four folds; Line symmetry and Rotational and point symmetry make that precise.
Exercises
Reading the family tree
- List every name from the family tree that applies to a rectangle that is not a square.
- List every name from the family tree that applies to a rhombus that is not a square.
- A kite has sides of 5, 5, 8 and 8 in that order around it. List every name from the family tree that applies to it.
- Which boxes in Figure 1 have the square box somewhere below them? Which box does not, and why?
Always, sometimes or never
Decide whether each statement is always, sometimes or never true, and give a reason.
- A square is a rhombus.
- A rhombus is a square.
- A trapezoid is a parallelogram.
- A rectangle is a kite.
- An isosceles trapezoid is a rectangle.
- A kite has a pair of parallel sides.
The most specific name
Give the most specific name the quadrilateral is guaranteed, with a reason.
- It has four right angles, and two of its neighboring sides each measure 7.
- It is a parallelogram with one right angle.
- Its four sides each measure 6, and one of its angles measures 70^\circ.
- It has exactly one pair of parallel sides, and its other two sides are congruent.
- It is a parallelogram whose diagonals are congruent.
Using the numbers
- ABCD is a parallelogram with AB = 3x - 4, CD = 2x + 6, BC = 26 and m\angle A = 90^\circ. Find x and give the most specific name of ABCD.
- Kite ABCD has AB = AD = 2y + 1 and CB = CD = 5y - 8, and \overline{AB} \parallel \overline{DC}. Find y and the length of each side, and name the shape.
- In square ABCD the diagonals cross at E, with BD = 3x + 1 and AE = x + 3. Find x and AC.
- Rectangle ABCD has diagonals crossing at E, with m\angle AEB = (4x)^\circ and m\angle BEC = (2x + 30)^\circ. Find x. Is ABCD a square?
- A rhombus has diagonals of 10 and 24. Find its area with the kite formula, then its side, then its height to that side using the fact that it is also a trapezoid.
Reasoning
- Show that a rhombus whose diagonals are congruent is a square.
- Explain why Figure 1 is not a tree in the strict sense, and name two families that a rhombus inherits from even though neither family contains the other. Give a shape in each family that is not in the other.
- Under the exclusive definition of a trapezoid (“exactly one pair of parallel sides”), is a square a trapezoid? What is lost for parallelograms when that definition is used?
- A property holds for every rectangle and for every rhombus. Must it hold for every parallelogram? Give a reason or an example.
- Show that no isosceles trapezoid is a kite.
Answers
- Rectangle, parallelogram, trapezoid, quadrilateral. Climb up from the rectangle box. It is not a rhombus (neighboring sides differ), not a kite (a rectangle that is a kite is a square) and not an isosceles trapezoid (both pairs of sides are parallel).
- Rhombus, parallelogram, kite, trapezoid, quadrilateral. The rhombus box has two parents, and the names above both of them all apply. It is not a rectangle, since a rhombus that is a rectangle is a square.
- Kite and quadrilateral only. A kite with a pair of parallel sides is a rhombus, and this one’s sides are not all equal, so it has no parallel sides. That rules out trapezoid and every box below it.
- Quadrilateral, trapezoid, parallelogram, rectangle, rhombus and kite. The isosceles trapezoid box does not: its legs must not be parallel, but a square has both pairs of opposite sides parallel.
- Always. A square is a parallelogram with four congruent sides, which is the definition of a rhombus.
- Sometimes. A rhombus with angles of 60^\circ and 120^\circ is not a square; a rhombus with a right angle is.
- Sometimes. Every parallelogram is a trapezoid that is a parallelogram; a trapezoid whose legs are not parallel is not.
- Sometimes. A rectangle is a kite exactly when it is a square (Example 5). A 3 by 5 rectangle is not a kite.
- Never. Every rectangle has both pairs of opposite sides parallel, and an isosceles trapezoid’s legs are not parallel.
- Sometimes. A kite with a pair of parallel sides is a rhombus, so a rhombus kite has them; a kite with sides 5, 5, 8, 8 does not.
- Square. Along each side the two right angles at its ends add to 180^\circ, so they are co-interior angles that make the two sides they stand on parallel; both pairs of opposite sides are therefore parallel. A parallelogram with four right angles is a rectangle, and opposite sides of a parallelogram are congruent. The two neighboring sides of 7 then force the other two to be 7, so it is a rhombus too, and a rectangle that is a rhombus is a square.
- Rectangle. Neighboring angles of a parallelogram are supplementary, so the neighbors of the right angle are 90^\circ, and the fourth angle is opposite the first, so it is 90^\circ. Nothing makes the sides equal, so a square is not guaranteed.
- Rhombus. Four congruent sides make opposite sides congruent, so it is a parallelogram, and then a rhombus. An angle of 70^\circ is not a right angle, so it is not a rectangle and not a square.
- Isosceles trapezoid. It is a trapezoid, and its legs are the other two sides. They are congruent, and they are not parallel because only one pair of sides is.
- Rectangle. In parallelogram ABCD with AC = BD, triangles \triangle ABC and \triangle BAD share \overline{AB}, have BC = AD and have AC = BD, so they are congruent by SSS and \angle ABC \cong \angle BAD. These two angles are neighbors, so supplementary, so each is 90^\circ, and then all four angles are right angles.
- x = 10; ABCD is a square with side 26. Opposite sides are congruent: 3x - 4 = 2x + 6, so x = 10 and AB = CD = 26. With BC = DA = 26 all four sides are equal, so it is a rhombus, and the right angle at A makes it a rectangle too.
- y = 3; every side is 7, and the shape is a rhombus. A kite with a pair of parallel sides is a rhombus, so all four sides are equal: 2y + 1 = 5y - 8, so 3y = 9. Check: 2(3) + 1 = 7 and 5(3) - 8 = 7.
- x = 5 and AC = 16. The diagonals bisect each other (parallelogram) and are congruent (rectangle), so AE = \tfrac{1}{2}BD: x + 3 = \tfrac{1}{2}(3x + 1), so 2x + 6 = 3x + 1. Then BD = 16, AC = BD = 16, and AE = 8 is half of it.
- x = 25, and ABCD is not a square. \angle AEB and \angle BEC lie side by side along the diagonal \overline{AC}, so they add to 180^\circ: 6x + 30 = 180. They measure 100^\circ and 80^\circ, so the diagonals are not perpendicular, but a square’s diagonals are, because a square is a kite.
- Area 120, side 13, height \tfrac{120}{13}. The kite formula gives \tfrac{1}{2} \cdot 10 \cdot 24 = 120. The diagonals bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 5 and 12: \displaystyle \sqrt{25 + 144} = 13. As a trapezoid with a = b = 13, \tfrac{1}{2}(13 + 13)h = 120, so h = \tfrac{120}{13}.
- A rhombus is a parallelogram, and a parallelogram with congruent diagonals is a rectangle (Exercise 15). A shape that is both a rhombus and a rectangle is a square.
- In a tree each box has one parent, but the rhombus has two: the parallelogram and the kite (and the square has two, the rectangle and the rhombus). A parallelogram with sides 4 and 6 is not a kite, since no two neighboring sides are equal; a kite with sides 5, 5, 8, 8 is not a parallelogram, since it has no parallel sides.
- No: a square has two pairs of parallel sides, not exactly one. Parallelograms no longer sit below trapezoids, so a fact proved for every trapezoid, such as the area formula \tfrac{1}{2}(a + b)h or the supplementary angles along a leg, no longer applies to parallelograms automatically and would need its own proof.
- No. Every rectangle has a line of symmetry through the middles of two opposite sides, and every rhombus has a diagonal as a line of symmetry, so “has a line of symmetry” holds for both. A parallelogram with sides 4 and 6 and an angle of 60^\circ is neither a rectangle nor a rhombus and has no line of symmetry at all (Rectangles, rhombuses and squares).
- An isosceles trapezoid is a trapezoid, so a kite that was one would be a kite with a pair of parallel sides, which is a rhombus and therefore a parallelogram. But an isosceles trapezoid’s legs are not parallel, so it is never a parallelogram. So it is never a kite.