Grade 5 · 5.G.B.4

Classifying shapes in a hierarchy

A square is also a rectangle, a rhombus and a parallelogram. How can one shape belong to so many groups, and how do the groups fit inside one another?

You need

Parallelograms, rectangles, rhombuses and squares: what each one guaranteesGrade 5 Classifying trianglesGrade 4

Where you are 15 in Two-dimensional shapes

Geometry11

Two-dimensional shapes15

▴ 10 above Triangles, quadrilaterals, pentagons and hexagons from their sides and angles Regular polygons and their angles

Classifying shapes in a hierarchy

Regular polygons and their circles The apothem of a regular polygon

Why this exists

The museum of shapes has a sorting game. Each visitor gets a stack of cards, and each card shows a shape along with the facts about it: how many sides, which sides are equal, which sides are parallel, and how big the angles are. Around the room stand bins with labels such as “rectangle”, “rhombus” and “triangle”. Fatoumata runs the game, and Kahiau and Murat play it.

Kahiau picks up a card showing a square and stops. It fits the “rectangle” bin, because it has four right angles. It fits the “rhombus” bin, because its four sides are equal. It fits “parallelogram” and “quadrilateral” too. A shape can belong to more than one group at once, as in Shapes in more than one category, and quadrilaterals in none. So which bin is right?

All of them are. The groups are not separate piles. They fit inside one another, like boxes inside boxes. A map of which group sits inside which is called a hierarchy. With that map, one look tells you every group a shape belongs to, and so every fact it is promised.

Definition

A property of a shape is a fact about its sides or angles, such as “has four right angles” or “has two pairs of parallel sides”. A category is the group of all shapes that have a certain list of properties.

A category B is a subcategory of a category A when every shape in B is also in A. This happens when B asks for every property that A asks for, and more. The rectangles are a subcategory of the parallelograms, because a rectangle is a parallelogram that also has four right angles.

A hierarchy is a diagram of categories in which each category is drawn below the categories it sits inside, joined to them by lines. Reading upward from a category gives every bigger category that holds it.

Polygons. A polygon is a closed flat shape made only of straight sides, joined end to end, with no two sides crossing (Polygons). Polygons are named by their number of sides (Naming and classifying polygons): a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8. A circle is not a polygon, because it has no straight sides.

Triangles are sorted in two ways, as in Classifying triangles. By sides:

A scalene triangle has no two sides of the same length.
An isosceles triangle has at least two sides of the same length.
An equilateral triangle has all three sides of the same length.

By angles:

An acute triangle has three acute angles, each less than 90^\circ.
A right triangle has one right angle, exactly 90^\circ.
An obtuse triangle has one obtuse angle, more than 90^\circ.

Quadrilaterals use the definitions of Parallelograms, rectangles, rhombuses and squares: what each one guarantees. Two sides are parallel when the lines they lie on never meet (Parallel lines), and perpendicular when they meet in a right angle (Perpendicular lines).

A parallelogram is a quadrilateral with two pairs of parallel sides.
A rectangle is a parallelogram with four right angles.
A rhombus is a parallelogram with four sides of equal length.
A square is a parallelogram with four right angles and four equal sides.
A trapezoid is a quadrilateral with at least one pair of parallel sides.
A kite is a quadrilateral with two pairs of equal sides, where the two sides in each pair are next to each other.

As in that lesson, a trapezoid needs at least one pair of parallel sides, so every parallelogram is a trapezoid. A book that says “exactly one pair” builds a different hierarchy, in which trapezoids and parallelograms share no shape.

Regular polygons. A polygon is regular when all of its sides have the same length and all of its angles have the same size.

Intuition

Part 1: categories and subcategories

A category is decided by properties and nothing else. Size, color, and which way the card is turned do not count. A tiny square printed on its corner is still a square.

Moving down to a subcategory means asking for one more property. Start with the parallelograms. Ask for four right angles as well, and the group shrinks to the rectangles. Ask for four equal sides on top of that, and it shrinks again, to the squares. Each new property lets fewer shapes in, and each shape that gets in has more facts in its favor.

That is why these two sentences say different things.

Every square is a rectangle. A square has every property a rectangle needs: two pairs of parallel sides and four right angles. It also has one more, four equal sides.
Some rectangles are squares. A rectangle has to have four right angles, but its sides do not have to be equal. A door is shaped like a rectangle, and it is much taller than it is wide.

So “every” does not turn around. When B is a subcategory of A, the sentence “every B is an A” is true, and turning it around gives a sentence that is usually false. What is true the other way is “some A are B”.

Part 2: two ways to sort triangles

Sorted by sides, the triangles nest. Every equilateral triangle has three equal sides, so it certainly has at least two: it is isosceles. The equilateral triangles are a subcategory of the isosceles triangles, which are a subcategory of the triangles. The scalene triangles sit beside the isosceles ones, sharing no shape with them, since “no two sides equal” and “at least two sides equal” cannot both hold.

Sorted by angles, the triangles do not nest at all. The angles of a triangle add to 180^\circ (Triangle angle sum), so a triangle has at most one angle of 90^\circ or more, and its largest angle decides its name. Every triangle is exactly one of acute, right and obtuse.

The two sorts cut across each other, so every triangle has an angle name and a side name. Not every pair can happen. An equilateral triangle has three equal angles, and three equal angles that add to 180^\circ are 60^\circ each. So every equilateral triangle is acute, and a right equilateral triangle cannot exist. The full list of which pairs happen is worked out in Classifying triangles.

Part 3: the family tree of quadrilaterals

Figure 1 draws the whole hierarchy. Follow any line down, and every shape in the lower box is also in the upper box. The quadrilateral part is the busiest. Under the quadrilateral sit the trapezoid and the kite. Under the trapezoid sits the parallelogram, and under the parallelogram the rectangle and the rhombus. The square is at the bottom.

Figure 1The hierarchy of flat shapes. A line going down means every shape in the lower box is also in the box above it.

Two boxes have two lines going up. The square is below the rectangle and below the rhombus, because it is both. The rhombus is below the parallelogram and below the kite: its four equal sides make each pair of sides that meet at a corner equal, which is what a kite asks for.

Figure 2 shows the same categories as regions, one drawn inside another. The tree is good for climbing, from a shape up to every name above it. The regions are good for seeing overlaps. The squares are exactly where the rectangle region and the rhombus region overlap. The kite region reaches into the trapezoid region only as far as the rhombuses, because a kite that has a pair of parallel sides is always a rhombus (see “Why it is true”).

Figure 2The same quadrilateral categories drawn as regions. A region inside another is a subcategory, and an overlap holds the shapes that are in both.

The hierarchy answers questions of the form “is an X a Y?” in three ways.

Always, when the X box sits below the Y box in Figure 1. Every X is a Y.
Never, when the X and Y regions do not overlap at all. No shape is both.
Sometimes, when the regions overlap but the X region is not wholly inside the Y region. Then one shape shows “yes” and another shows “no”.

Part 4: polygons at the top, and regular polygons

At the very top of Figure 1 sits the polygon. Every triangle, every quadrilateral and every pentagon is a polygon. Directly below it, the categories are sorted by number of sides, and those never overlap: a shape has one number of sides, so no shape is both a triangle and a quadrilateral.

Being regular is a property that cuts across the branches. It asks for equal sides and equal angles together. In Figure 3, shapes 1, 4 and 5 have both. The rhombus, shape 2, has equal sides but two sizes of angle, and the rectangle, shape 3, has equal angles but two lengths of side.

Figure 3Matching ticks mean equal sides, matching arcs mean equal angles, and a small square marks a right angle. A regular polygon needs both kinds of match.

So one shape can carry names from several branches at once. An equilateral triangle is isosceles (a side name), acute (an angle name) and regular. A square sits below two boxes in the tree, and it is regular as well. The deeper a shape sits and the more lines run above it, the more names it carries.

Results

Subcategories pass properties down. If B is a subcategory of A, then every shape in B has every property of A. “Every B is an A” is true, and “every A is a B” is false unless the two categories are the same. What is true is “some A are B”.

Triangles by sides. Every equilateral triangle is isosceles. No triangle is both scalene and isosceles.

Triangles by angles. Every triangle is exactly one of acute, right and obtuse. Each angle of an equilateral triangle is 60^\circ, so every equilateral triangle is acute. No triangle is right and equilateral, and none is obtuse and equilateral.

The quadrilateral hierarchy.

Every square is a rectangle and a rhombus.
Every rectangle and every rhombus is a parallelogram.
Every parallelogram is a trapezoid.
Every rhombus is a kite.
Every trapezoid and every kite is a quadrilateral, and every quadrilateral is a polygon.
A kite is a trapezoid only when it is a rhombus.

Regular polygons. Every equilateral triangle is regular, and the regular quadrilaterals are exactly the squares.

Why it is true

Why “every square is a rectangle”. A rectangle is defined as a parallelogram with four right angles. A square is a parallelogram with four right angles and four equal sides. The square’s list contains the rectangle’s whole list, so every square passes the rectangle test. The same reading of the definitions gives every line of Figure 1.

Why “every” does not turn around. One shape that breaks an “every” sentence is enough to make it false. A door-shaped rectangle breaks “every rectangle is a square”, because its sides are not all equal.

Why every equilateral triangle is acute. In a triangle, two equal sides come with two equal angles, the angles across from them (Classifying triangles). In an equilateral triangle the first and second sides are equal, so the angles across from them are equal. The second and third sides are equal too, so their angles are equal. That makes all three angles the same size. They add to 180^\circ, so each is 180 \div 3 = 60 degrees. Every angle is less than 90^\circ, so the triangle is acute, and it has no right angle and no obtuse angle.

Why a triangle cannot have two right angles. Two right angles already add to 90^\circ + 90^\circ = 180^\circ, which leaves nothing for the third angle. So a right triangle has only one right angle, and the same reasoning rules out two obtuse angles.

Why a regular quadrilateral is a square. Cut a quadrilateral along a line from one corner to the corner across from it. That makes two triangles, and their angles together are the quadrilateral’s angles, so the four angles add to 180^\circ + 180^\circ = 360^\circ. In a regular quadrilateral the four angles are equal, so each is 360 \div 4 = 90 degrees. Four right angles make a rectangle, and four equal sides make a rhombus, so the shape is both: a square. A square, in turn, has equal sides and four equal angles, so it is regular.

Why a kite with a pair of parallel sides is a rhombus. Fold a kite along the line through the corner where its two short sides meet and the corner where its two long sides meet. Each side lands on its equal neighbor. If one side is parallel to the side across from it, the fold carries that pair onto the other pair, so the other pair is parallel too. The kite is then a parallelogram, so its opposite sides are equal. Its neighboring sides were already equal, so all four sides are equal, and it is a rhombus.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

The first tray of cards

Fatoumata deals Kahiau six cards.

Card A: four sides of 5 cm and four right angles.
Card B: sides of 8 cm, 3 cm, 8 cm and 3 cm in order, and four right angles.
Card C: four sides of 6 cm, and angles of 60^\circ, 120^\circ, 60^\circ and 120^\circ.
Card D: two pairs of parallel sides, sides of 7 cm, 4 cm, 7 cm and 4 cm in order, and angles of 70^\circ, 110^\circ, 70^\circ and 110^\circ.
Card E: four sides, with exactly one pair of parallel sides.
Card F: three sides.

Which cards go in the parallelogram bin, the rectangle bin, the rhombus bin and the square bin?

Solution

Test each card against the properties each bin asks for.

Card F has three sides, so it is not a quadrilateral, and it goes in none of the four bins. Card E has only one pair of parallel sides. Every one of the four bins asks for two pairs, so Card E goes in none of them either. It belongs in a trapezoid bin.

Card D has two pairs of parallel sides, so it is a parallelogram. It has no right angle, so it is not a rectangle. Its 7 cm and 4 cm sides meet at a corner, so its sides are not all equal, and it is not a rhombus. A square has to be both, so it is not a square.

Card C has four equal sides. Four equal sides always make a rhombus (Parallelograms, rectangles, rhombuses and squares: what each one guarantees), and every rhombus is a parallelogram. Its angles are not 90^\circ, so it is not a rectangle or a square.

Card B has four right angles, so it is a rectangle, and so a parallelogram. Its 8 cm and 3 cm sides meet, so it is not a rhombus or a square.

Card A has four right angles and four equal sides, so it is a rectangle, a rhombus, a square and a parallelogram.

Bin	Cards
parallelogram	A, B, C, D
rectangle	A, B
rhombus	A, C
square	A

Every card in the square bin is also in the rectangle and rhombus bins, and every card in those two bins is also in the parallelogram bin. The bins nest.

Example 2

Every and some

Kahiau says, “Every square is a rectangle, so every rectangle is a square too.” Murat says, “Card B says no.” Who is right? What true sentence can Kahiau say instead?

Solution

Kahiau’s first sentence is true. A square is a parallelogram with four right angles and four equal sides, so it has everything a rectangle needs.

His second sentence claims something about every rectangle, and one rectangle that is not a square is enough to break it. Card B is a rectangle, but its sides are 8 cm and 3 cm, not all equal, so it is not a square. Murat is right: “every rectangle is a square” is false.

The true sentence the other way round is “some rectangles are squares”. Card A shows one. In the language of categories, the squares are a subcategory of the rectangles: the square box sits inside the rectangle box, and the rectangle box is bigger. The same holds for the rhombuses. Every square is a rhombus, Card C is a rhombus that is not a square, and so only some rhombuses are squares.

Example 3

Two names for every triangle card

Murat’s next six cards are triangles.

Card	Sides	Angles
M	all three marked equal	60^\circ, 60^\circ, 60^\circ
N	no two equal	25^\circ, 65^\circ, 90^\circ
P	two marked equal	70^\circ, 70^\circ, 40^\circ
Q	two marked equal	35^\circ, 35^\circ, 110^\circ
R	no two equal	40^\circ, 65^\circ, 75^\circ
S	no two equal	30^\circ, 45^\circ, 105^\circ

Sort each card by its sides and by its angles, and place it in a table with one row for each side name and one column for each angle name.

Solution

First, each card is a real triangle: every row of angles adds to 180^\circ. For example, Card S gives 30 + 45 + 105 = 180.

By sides: cards N, R and S have no two sides equal, so they are scalene. Cards P and Q have two equal sides, so they are isosceles. Card M has three equal sides, so it is equilateral, and it is isosceles too, because three equal sides include two.

By angles, the largest angle decides. Card N has a 90^\circ angle, so it is right. Cards Q and S have largest angles of 110^\circ and 105^\circ, more than 90^\circ, so they are obtuse. Cards M, P and R have largest angles of 60^\circ, 70^\circ and 75^\circ, all less than 90^\circ, so they are acute.

To give every card exactly one row, the middle row takes the isosceles cards that are not equilateral.

	acute	right	obtuse
scalene	R	N	S
isosceles, not equilateral	P		Q
equilateral	M

Card M is the only equilateral card, and it can only ever land in the acute column, because every angle of an equilateral triangle is 60^\circ.

Example 4

Cards that cannot be printed

Murat asks Fatoumata to print four more triangle cards: (a) right and isosceles, (b) right and equilateral, (c) obtuse and scalene, (d) obtuse and isosceles. Which can be printed? For each one that can, give its three angles.

Solution

(b) Cannot be printed. Every angle of an equilateral triangle is 60^\circ, so none of them is 90^\circ.

(a) Can. In an isosceles triangle, the two equal sides come with two equal angles (Classifying triangles). The right angle cannot be one of the equal pair, since two right angles would leave nothing for the third angle. So the right angle is the odd one, and the two equal angles share what is left, 180^\circ - 90^\circ = 90^\circ. Each is 90 \div 2 = 45 degrees. The card shows 45^\circ, 45^\circ, 90^\circ.

(c) Can. A scalene triangle has three different angles. Pick an obtuse angle, 115^\circ, and split the remaining 180^\circ - 115^\circ = 65^\circ into two different parts, 25^\circ and 40^\circ. The card shows 25^\circ, 40^\circ, 115^\circ.

(d) Can. As in (a), the obtuse angle is the odd one, and the two equal angles share what is left. With an obtuse angle of 140^\circ, the two equal angles share 40^\circ, so each is 20^\circ. The card shows 20^\circ, 20^\circ, 140^\circ.

Three of the four cards can be printed. The right equilateral card is not missing by bad luck; the definition of equilateral rules it out.

Example 5

One card in both pictures

Kahiau’s card shows a rectangle 9 cm long and 4 cm wide. Use Figure 1 and Figure 2 to list every category it belongs to, and name the quadrilateral categories it misses.

Solution

In Figure 1, find the rectangle box and climb. A line goes up to the parallelogram, from there to the trapezoid, then to the quadrilateral, then to the polygon. Each line up is an “every” sentence, so the card is a rectangle, parallelogram, trapezoid, quadrilateral and polygon.

Figure 2 shows what it misses. The card lies in the rectangle region. To be in the rhombus region its four sides would have to be equal, but its 9 cm and 4 cm sides meet at a corner. So it lies outside the rhombus region, and therefore outside the square region, which is part of the rhombus region. It is also outside the kite region: a kite needs two pairs of equal sides that meet, and at every corner of this card a 9 cm side meets a 4 cm side.

So the card misses rhombus, square and kite. The tree told us which names it gains by climbing, and the regions told us which neighboring names it misses.

Example 6

Murat’s statement cards

Murat draws five statement cards. Say whether each is always, sometimes or never true.

(a) A trapezoid is a quadrilateral. (b) A kite is a square. (c) A parallelogram is a rhombus. (d) A rectangle is a triangle. (e) A square is a trapezoid.

Solution

(a) Always. The trapezoid box sits directly below the quadrilateral box: a trapezoid has four sides.

(b) Sometimes. Every square is a rhombus and every rhombus is a kite, so a square is a kite that is a square. A kite with sides of 3 cm, 3 cm, 5 cm and 5 cm in order is a kite that is not a square, since its sides are not all equal.

(c) Sometimes. Card C from Example 1 is a parallelogram that is a rhombus. Card D is a parallelogram that is not, since its 7 cm and 4 cm sides meet.

(d) Never. A rectangle has four sides and a triangle has three. In Figure 1 they sit on different branches below the polygon, and no shape is in both.

(e) Always. Climb from the square: square, rectangle, parallelogram, trapezoid. Every square has two pairs of parallel sides, which is at least one pair.

A “sometimes” answer needs two shapes, one for “yes” and one for “no”. An “always” answer needs a path up the tree, and a “never” answer needs a property that the two categories cannot share.

Example 7

Which shapes are regular?

Use the marks in Figure 3 to decide which of the five shapes are regular.

Solution

A regular polygon needs two things: all sides equal and all angles equal. Check each shape for both.

Shape 1, the triangle, has one tick on every side and one arc in every corner. Both tests pass, so it is regular.

Shape 2, the rhombus, has one tick on every side, so its sides are equal. But two of its corners carry one arc and two carry two arcs, so its angles come in two sizes. It is not regular.

Shape 3, the rectangle, has a right angle in every corner, so its angles are equal. But its long sides carry one tick and its short sides two, so its sides come in two lengths. It is not regular.

Shape 4, the hexagon, has matching ticks on all six sides and matching arcs in all six corners. It is regular.

Shape 5, the square, has matching ticks on all four sides and a right angle in every corner. It is regular.

So shapes 1, 4 and 5 are regular. The rhombus has half of what “regular” asks for, and the rectangle has the other half; it takes both halves together.

Example 8

The card with the most names

Fatoumata offers a prize to the card with the most names. The finalists are an equilateral triangle, a square and a regular hexagon. Count every name each one has, from every way of sorting in this lesson.

Solution

The equilateral triangle. It is a polygon and a triangle. By sides it is isosceles and equilateral, but not scalene. By angles, every angle is 60^\circ, so it is acute. Its sides are equal and its angles are equal, so it is a regular polygon. That makes 6 names.

The square. Climb Figure 1 from the square along every line. The rectangle line gives rectangle, parallelogram, trapezoid, quadrilateral and polygon. The rhombus line adds rhombus and kite. With square itself and regular polygon, that makes 9 names.

The regular hexagon. It is a polygon, a hexagon and a regular polygon. This lesson has no subcategories of hexagons, so that makes 3 names.

The square wins. It sits at the bottom of the quadrilateral tree with two lines running up from it, and the rhombus above it has two more, so it collects the names of every box those lines reach.

Boundaries

Properties decide, not looks. A card belongs in a category because its properties say so. A corner that looks square is not a right angle unless it is marked or measured as one.
Polygons have straight sides and are closed. A circle, a half circle, or a shape with a gap in its outline is not a polygon and has no place in Figure 1.
The definitions decide the hierarchy. With “exactly one pair of parallel sides” for a trapezoid, no parallelogram is a trapezoid, and the line from parallelogram up to trapezoid disappears. Some books also define isosceles as “exactly two equal sides”, and then no equilateral triangle is isosceles. Always check which definitions are in use.
“Every” does not turn around. “Every square is a rectangle” is true and “every rectangle is a square” is false. One example can prove a “some” sentence or break an “every” sentence, but no number of examples proves an “every”.
The two triangle sorts are separate. A side name says nothing directly about the angles, and an angle name says nothing directly about the sides, so a full description of a triangle uses one of each.
Regular needs both. Equal sides alone (a rhombus) or equal angles alone (a rectangle) are not enough.
Most polygons have no special name. A quadrilateral with four different sides and no parallel pair is a quadrilateral and a polygon, and nothing more.

Where this goes next

Because properties pass down a hierarchy, a fact proved once for a big category holds for every subcategory inside it. Later geometry uses this all the time. The quadrilateral family tree of high school geometry adds facts about the lines joining opposite corners, and proves each one once, high in the tree.

Regular polygons have their own lessons. Sharing the angle total equally among equal angles gives the angle of a regular polygon, and their centers lead to the central angle. The triangles return in Triangles, with the side and angle names used throughout.

Exercises

Part 1: categories and subcategories

A card shows four sides of 12 cm and four right angles. Which of these bins does it go in: parallelogram, rectangle, rhombus, square?
A card shows sides of 10 cm, 6 cm, 10 cm and 6 cm in order, and four right angles. Which of the four bins does it go in, and which does it miss?
True or false: (a) Every square is a rhombus. (b) Every rhombus is a square. (c) Some rhombuses are squares.
A card shows four sides of 9 cm and angles of 80^\circ, 100^\circ, 80^\circ and 100^\circ. Which of the four bins does it go in?
Fatoumata labels a new bin “four right angles”. Which of these go in it: a square, a rectangle that is not a square, a rhombus with angles of 75^\circ and 105^\circ, a parallelogram with angles of 50^\circ and 130^\circ?

Part 2: triangles sorted two ways

Name a triangle with sides of 4 cm, 7 cm and 4 cm by its sides.
Name a triangle with three sides of 6 cm by its sides. Give every side name that fits.
A triangle has angles of 35^\circ, 55^\circ and 90^\circ. Name it by its angles.
A triangle has two sides marked equal and angles of 55^\circ, 55^\circ and 70^\circ. Give its side name and its angle name.
A triangle has no two sides equal and angles of 20^\circ, 30^\circ and 130^\circ. Give its side name and its angle name.
Which of these triangles can exist? For each one that can, give three angles that work. (a) obtuse and equilateral (b) right and scalene (c) acute and isosceles (d) a triangle with two obtuse angles

Part 3: the quadrilateral family tree

Use Figure 1. List every category above the parallelogram.
Use Figure 1. Which category sits directly below the kite? What other box is it directly below?

In Exercises 14 to 17, say whether the statement is always, sometimes or never true, and give a reason.

A rhombus is a kite.
A trapezoid is a parallelogram.
A kite is a triangle.
A parallelogram is a kite.
Use Figure 2. Describe the region that holds the squares, and use it to explain why every square is a kite.

Part 4: polygons and regular polygons

Is a rectangle 13 cm long and 5 cm wide a regular polygon? Explain.
A pentagon has five sides of 14 cm and five angles of 108^\circ. Is it regular? Find the distance around it.
A regular octagon has a perimeter of 96 cm. How long is each side?
A card shows a triangle with two equal sides and one angle of 100^\circ. List every name it has from this lesson, say whether it is regular, and find its other two angles.

Reasoning

Kahiau says, “A square is not a rectangle, because squares have their own name.” Explain what is wrong.
Explain why every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.
Explain why every regular quadrilateral is a square, and why a rhombus can fail to be regular.

Answers

All four. Four right angles make it a rectangle, and so a parallelogram. Four equal sides make it a rhombus. Both together make it a square.
Parallelogram and rectangle; it misses rhombus and square. Four right angles make a rectangle, but its 10 cm and 6 cm sides meet at a corner, so its sides are not all equal.
(a) True: a square is a parallelogram with four equal sides, which is what a rhombus is. (b) False: a rhombus that leans has no right angle, so it is not a square. (c) True: every square is a rhombus that is a square.
Parallelogram and rhombus. Four equal sides make a rhombus, and every rhombus is a parallelogram. With no 90^\circ angle it is not a rectangle and not a square.
The square and the rectangle. The rhombus and the parallelogram have no 90^\circ angle.
Isosceles. The two 4 cm sides are equal, and the 7 cm side is different, so it is not equilateral.
Equilateral and isosceles. All three sides are equal, and three equal sides include two.
Right. The angles add to 35 + 55 + 90 = 180, and the largest angle is exactly 90^\circ.
Isosceles and acute. The angles add to 55 + 55 + 70 = 180, and the largest, 70^\circ, is less than 90^\circ.
Scalene and obtuse. The angles add to 20 + 30 + 130 = 180, and 130^\circ is more than 90^\circ.
(a) No: every angle of an equilateral triangle is 60^\circ. (b) Yes, for example 20^\circ, 70^\circ, 90^\circ. (c) Yes, for example 75^\circ, 75^\circ, 30^\circ. (d) No: two obtuse angles already add to more than 180^\circ.
Trapezoid, quadrilateral and polygon.
The rhombus. It is also directly below the parallelogram.
Always. All four sides of a rhombus are equal, so the two sides at every corner are equal, which is what a kite asks for.
Sometimes. A rectangle is a trapezoid that is a parallelogram. A trapezoid with exactly one pair of parallel sides is not.
Never. A kite has four sides and a triangle has three.
Sometimes. A rhombus is a parallelogram that is a kite. A parallelogram with sides of 11 cm and 6 cm is not, because at every corner an 11 cm side meets a 6 cm side.
The squares are where the rectangle region and the rhombus region overlap. That overlap is part of the rhombus region, and the rhombus region is inside the kite region, so every square is inside the kite region.
No. Its four angles are all 90^\circ, but its sides are 13 cm and 5 cm, so they are not all equal.
Yes: all five sides are equal and all five angles are equal. The distance around is 5 \times 14 = 70 cm.
96 \div 8 = 12 cm, since all eight sides are equal.
Polygon, triangle, isosceles and obtuse. The 100^\circ angle cannot be one of the equal pair, because two 100^\circ angles already add to more than 180^\circ. So the other two angles are the equal pair and share 180^\circ - 100^\circ = 80^\circ, so each is 40^\circ. It is not regular, because 40^\circ and 100^\circ are different.
A square has every property a rectangle needs: two pairs of parallel sides and four right angles. “Square” is a more exact name for a special rectangle, not a different kind of shape; the square category sits inside the rectangle category.
An equilateral triangle has all three sides equal, so it certainly has at least two equal sides, which makes it isosceles. A triangle with sides of 5 cm, 5 cm and 8 cm is isosceles but not equilateral, because the 8 cm side is different.
A regular quadrilateral has four equal sides, so it is a rhombus. Its four angles are equal and add to 360^\circ, so each is 360 \div 4 = 90 degrees, which makes it a rectangle. A rhombus that is also a rectangle is a square. A rhombus has equal sides, but its angles can be, for example, 70^\circ and 110^\circ, so they need not all be equal.