Classifying shapes by their properties

A triangle has one angle of 90°. Which angle classification applies, even if the drawing is rotated?

Read properties rather than relying on appearance

A polygon has a closed boundary made of straight line segments. In this topic, its sides meet only at neighbouring endpoints and do not cross

Each segment is a side and each corner is a vertex. Three sides make a triangle and four make a quadrilateral

Adjacent sides share a vertex. Opposite sides of a quadrilateral do not share a vertex

Parallel sides lie on lines that remain the same distance apart and do not meet, even when extended

Equal tick marks identify equal lengths. A small square marks a right angle

Use the stated facts and markings. A picture that looks almost square does not prove equal sides or right angles

Classify the side lengths

An equilateral triangle has three equal sides

An isosceles triangle has at least two equal sides. An equilateral triangle therefore also satisfies this inclusive definition

A scalene triangle has no equal pair of sides

When asked for the most specific side-length name, use equilateral for three equal sides, isosceles for exactly two equal sides, and scalene for three different sides

Isosceles and scalene triangles can each be acute, right-angled or obtuse. An equilateral triangle is always acute

Give a separate angle classification

An acute triangle has all three interior angles smaller than ninety degrees

A right-angled triangle has one interior angle equal to ninety degrees

An obtuse triangle has one interior angle greater than ninety degrees

Check the largest angle. One small angle does not make the whole triangle acute

Combine a side name and an angle name when both are requested. For example, a triangle can be isosceles and right-angled

Use both side and angle evidence

Triangle ABC has these side relationships: CA = CB, and AB has a different length

Its interior angles are A = 70°, B = 70° and C = 40°

Give the most specific side-length classification

Read the side relationships rather than the orientation

CA = CB, and AB has a different length

Exactly two sides are equal, so the most specific name is isosceles

Classify the triangle by its angles

Compare all three angles with 90°

The largest angle is 70°

Every angle is less than 90°, so the triangle is acute

Classify unequal sides and three acute angles

Recognise that two classifications can apply

A pair of equal sides does not force acute angles

Triangle ABC has these side relationships: CA = CB, and AB has a different length

Its interior angles are A = 30°, B = 30° and C = 120°

Give the most specific side-length classification

Read the side relationships rather than the orientation

CA = CB, and AB has a different length

Exactly two sides are equal, so the most specific name is isosceles

Classify the triangle by its angles

Compare all three angles with 90°

The largest angle is 120°

One angle is greater than 90°, so the triangle is obtuse

Keep side equality separate from a right angle

Classify using the largest angle

Choose the most specific equal-side class

Triangle ABC has these side relationships: AB = BC = CA

Its interior angles are A = 60°, B = 60° and C = 60°

Give the most specific side-length classification

Read the side relationships rather than the orientation

AB = BC = CA

All three sides are equal, so equilateral is the most specific name

Classify the triangle by its angles

Compare all three angles with 90°

The largest angle is 60°

Every angle is less than 90°, so the triangle is acute

Not every three positive lengths form a triangle

Imagine joining three rigid strips at their endpoints without stretching them

Put the lengths in increasing order and add the two shorter lengths

Their sum must be strictly greater than the longest length so the strips can bend around and enclose some area

Equality makes a straight, zero-area arrangement. A smaller sum leaves a gap

Use the same unit for all three lengths. Test the shortest pair, not just whichever two lengths are written first

Calculate a shortest-pair sum that exceeds the longest side

Three rigid strips have lengths 3, 4, 5 cm

They must join end to end without stretching to make a triangle with positive area

Calculate the sum of the two shorter lengths in centimetres

Order the lengths: 3, 4, 5 cm

The two shorter lengths are 3 cm and 4 cm

3 + 4 = 7

Their sum is 7 cm

Decide whether these lengths form a triangle

The two shorter lengths add to 7 cm

The longest length is 5 cm

7 > 5

A triangle requires the sum of the two shorter lengths to be strictly greater than the longest

A triangle can be formed

Order mixed lengths before testing

Explain the equality boundary

Three rigid strips have lengths 3, 4, 7 cm

They must join end to end without stretching to make a triangle with positive area

Calculate the sum of the two shorter lengths in centimetres

Order the lengths: 3, 4, 7 cm

The two shorter lengths are 3 cm and 4 cm

3 + 4 = 7

Their sum is 7 cm

Decide whether these lengths form a triangle

The two shorter lengths add to 7 cm

The longest length is 7 cm

7 = 7

A triangle requires the sum of the two shorter lengths to be strictly greater than the longest

Only a straight, zero-area shape is possible

Do not accept equality as a triangle

Explain why the shorter strips cannot meet

Three rigid strips have lengths 3, 4, 8 cm

They must join end to end without stretching to make a triangle with positive area

Calculate the sum of the two shorter lengths in centimetres

Order the lengths: 3, 4, 8 cm

The two shorter lengths are 3 cm and 4 cm

3 + 4 = 7

Their sum is 7 cm

Decide whether these lengths form a triangle

The two shorter lengths add to 7 cm

The longest length is 8 cm

7 < 8

A triangle requires the sum of the two shorter lengths to be strictly greater than the longest

The two shorter sides cannot meet across the longest side

Distinguish an actual gap from a straight arrangement

Define the main quadrilateral families

A parallelogram has both pairs of opposite sides parallel

A rectangle is a quadrilateral with four right angles. Its opposite sides are parallel, so every rectangle is also a parallelogram

A rhombus has four equal sides. Its opposite sides are parallel, so every rhombus is also a parallelogram

A square has four equal sides and four right angles. It meets the definitions of both a rectangle and a rhombus

A rectangle need not have four equal sides, and a rhombus need not have right angles

A special name does not cancel a broader name. Select all applicable names when a question asks for every matching class

Use two parallel pairs without inventing extra properties

Quadrilateral ABCD has the following stated properties

AB is parallel to CD and BC is parallel to DA. AB = CD = 6 cm and BC = DA = 4 cm. Angle A is 60°

Give every applicable name from square, rectangle, rhombus and parallelogram

Read the stated side and angle properties

AB is parallel to CD and BC is parallel to DA. AB = CD = 6 cm and BC = DA = 4 cm. Angle A is 60°

Both pairs of opposite sides are parallel, so parallelogram also applies

Select every matching name: parallelogram

Include the broader parallel-side family

Equal sides do not guarantee right angles

Quadrilateral ABCD has the following stated properties

All four sides are equal. Both pairs of opposite sides are parallel. Angle A is 60°

Give every applicable name from square, rectangle, rhombus and parallelogram

Read the stated side and angle properties

All four sides are equal. Both pairs of opposite sides are parallel. Angle A is 60°

Four equal sides make a rhombus

Both pairs of opposite sides are parallel, so parallelogram also applies

Select every matching name: rhombus, parallelogram

Find all four applicable family names

Which statement follows from the definitions of square and rectangle?

Distinguish adjacent equal sides from parallel opposite sides

A kite has two pairs of adjacent equal sides. In the kite examples here, the two pairs have different lengths

The examples called trapeziums here have exactly one pair of opposite sides parallel

Some naming conventions use broader definitions. These examples explicitly state the side relationships, so the conclusion does not depend on an unstated convention

Parallel means a direction relationship between sides, not equality of their lengths

Adjacent equal sides meet at a vertex. They are not the same arrangement as two opposite equal sides

Read adjacent equal-side pairs

Quadrilateral ABCD has the following stated properties

AB = AD and BC = CD, with the first pair longer than the second pair. No opposite sides are parallel

Give every applicable name from square, rectangle, rhombus, parallelogram, kite and trapezium

Read the stated side and angle properties

AB = AD and BC = CD, with the first pair longer than the second pair. No opposite sides are parallel

Two pairs of adjacent equal sides give a kite

Select every matching name: kite

Use the stated number of parallel pairs

Compare fixed triangle sides with a hinged quadrilateral

When three valid side lengths are fixed, the triangle shape is fixed apart from moving, turning or reflecting it

With one side fixed as a base, the third vertex must be at the required distance from each endpoint. The two possible positions are mirror images

A four-sided hinged frame can change its angles without changing its four side lengths

Therefore four side lengths alone do not generally determine one quadrilateral shape. Additional information such as an angle or a diagonal can constrain it further

Start with the horizontal shift at zero, then move it towards two. Compare the corner angles while watching the unchanged side-length readouts. The frame remains a parallelogram, but a shifted frame is no longer a rectangle

What does this particular hinged frame demonstrate?

Three rigid strips of lengths 3 cm, 4 cm and 5 cm form a triangle. Can its angles change while all three lengths and all three joins stay fixed?

Count sides around a closed boundary

Count each straight side once as you go around the boundary

The number of sides equals the number of vertices

Five sides make a pentagon, six a hexagon, seven a heptagon and eight an octagon

A polygon does not have to be regular, upright or convex to keep its side-count name

Regular requires two conditions together

A regular polygon has all side lengths equal and all interior angles equal

An irregular polygon fails at least one of these conditions

Equal angles alone are not enough: a non-square rectangle is irregular

Equal sides alone are not enough: a non-square rhombus is irregular

A square and an equilateral triangle are regular because they satisfy both conditions

Use side count and both regularity conditions

This closed polygon has 5 straight sides

All its side lengths are equal and all its interior angles are equal

Name the polygon from its number of sides

Count the 5 sides once around the closed boundary

There are also 5 vertices

A 5-sided polygon is a pentagon

Decide whether the polygon is regular

Check both conditions for regularity

Every side length is equal and every interior angle is equal

Both conditions hold, so the polygon is regular

Classify a six-sided regular polygon

Keep the side count distinct from side length

Four equal angles do not guarantee regularity

All four angles are right angles. AB = CD = 6 cm and BC = DA = 4 cm

Name the polygon by its number of sides

Trace the closed boundary

There are four straight sides

A four-sided polygon is a quadrilateral

Decide whether this particular quadrilateral is regular

A regular polygon must have both equal sides and equal angles

The side lengths include both 6 cm and 4 cm

One of the two necessary conditions fails, so this polygon is irregular

Four equal sides do not guarantee regularity

An inward corner still belongs to the polygon

In a convex polygon, every interior angle is less than one hundred and eighty degrees

A concave polygon has an inward corner with a reflex interior angle greater than one hundred and eighty degrees

Trace the actual boundary, including the inward corner. Its two meeting sides still count

A simple concave polygon is still a polygon: its boundary is closed and straight-sided without crossing itself

The concave example here has unequal sides, so it is irregular. Concave and irregular describe different properties

Count through the inward corner and justify the classification

Name the shape and give the property that supports the name

Classify triangle sides and triangle angles separately, using the most specific side name when requested

Test triangle lengths by ordering them and comparing the sum of the two shorter lengths with the longest

Use the complete quadrilateral definitions and keep every applicable broader class

Count the whole polygon boundary and test both equal-side and equal-angle conditions for regularity

A change in orientation does not change a classification, and a visual resemblance does not replace given properties

Fixed triangle side lengths fix its shape, while a hinged quadrilateral can keep its side lengths and change its angles