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Classify Triangles and Quadrilaterals by Sides and Angles

Name triangles by sides and angles, sort squares, rectangles, rhombuses, parallelograms and trapezoids, and read tick marks and right-angle marks.

Grades 4–5 · 30 practice questions · Original material
Learning objectiveClassify triangles as equilateral, isosceles or scalene by their sides and as acute, right or obtuse by their angles; use the 180° angle sum of a triangle; name squares, rectangles, rhombuses, parallelograms and trapezoids from their properties; explain why a square is also a rectangle and a rhombus; and read tick marks, square corner marks and parallel-side arrows.

The idea

A yield sign, a ramp seen from the side and a sandwich cut corner to corner are all triangles, but they are not the same kind. To name a shape exactly, check two kinds of evidence: its sides and its angles. Diagrams show this evidence with marks. Sides with the same number of tick marks are equal. A small square in a corner means a right angle (90°). Matching arrowheads on two sides mean those sides are parallel. Trust the marks and measurements, not how a drawing looks or which way it is turned.

By sides, a triangle is equilateral if all three sides are equal, isosceles if two sides are equal, and scalene if no sides are equal. A triangle with two equal sides is not equilateral: equilateral needs all three.

By angles, a triangle is right if one angle is 90°, obtuse if one angle is more than 90°, and acute if all three angles are less than 90°. The three angles of every triangle add to 180°. That is why a triangle can never have two right angles or two obtuse angles: 90° + 90° already uses all 180°. A triangle gets one name from each list, such as "isosceles right triangle".

A quadrilateral has four sides, and its angles add to 360°. Parallel sides run in the same direction and never meet, like the rails of a ladder. In this lesson a trapezoid has at least one pair of parallel sides. (Some books say exactly one pair; check which definition your class uses. Either way, a shape with exactly one pair is a trapezoid.) A parallelogram has two pairs of parallel sides. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square has four equal sides and four right angles.

These names include one another. A square has everything a rectangle needs (four right angles) and everything a rhombus needs (four equal sides), so a square is a rectangle, a rhombus and a parallelogram too. It does not work backwards: a rectangle with long and short sides is not a square, and a leaning rhombus is not a square. Give the most specific name, but the other names are still true.

Worked examples

Naming triangles by sides and angles, and how quadrilateral names include one anotherTop left: three triangles named by sides. Equilateral has one tick mark on each of its three sides, so all three sides are equal. Isosceles has one tick mark on two sides, so those two sides are equal. Scalene has one, two and three tick marks, so no sides are equal. Top right: three triangles named by angles. A right triangle has one square corner mark, which means 90 degrees. An acute triangle has three angles less than 90 degrees. An obtuse triangle has one angle more than 90 degrees, marked with an arc. The three angles of every triangle add to 180 degrees. Bottom: a chain of quadrilaterals. A trapezoid has at least one pair of parallel sides, shown with matching arrow marks. A parallelogram has two pairs of parallel sides. Adding four right angles gives a rectangle; adding four equal sides gives a rhombus. A square has four equal sides and four right angles, so it is also a rectangle, a rhombus and a parallelogram.Name a shape from its marks, not its looksTick marks: equal sides. Square corner: 90°. Matching arrows: parallel sides.Triangles by sidesTriangles by angles (sum 180°)Equilateral3 equal sidesIsosceles2 equal sidesScaleneno equal sidesRightone 90° angleAcuteall angles under 90°Obtuseone angle over 90°Quadrilaterals: each arrow adds a rule; the new shape keeps every earlier nameTrapezoidat least 1 pair parallelParallelogram2 pairs parallelRectangle+ 4 right anglesRhombus+ 4 equal sidesSquareboth: 4 equal sidesand 4 right anglesA square is a rectangle and a rhombus. A rectangle or a rhombus is not always a square.
Original schematic. Matching tick marks show equal sides, a small square shows a 90° angle and matching arrowheads show parallel sides. This lesson uses the inclusive definition of a trapezoid (at least one pair of parallel sides); some books use exactly one pair.

Example 1 · Read the marks

A triangle has one tick mark on each of two sides, no tick mark on the third side, and a small square in the corner between the two marked sides. Name it by sides and by angles.

  1. The two sides with one tick each are equal, and the third side is different: isosceles.
  2. The square corner mark means that angle is 90°: right.
  3. Put the two names together.

An isosceles right triangle. Its other two angles are 45° each.

Example 2 · Find the missing angle, then classify

A triangle has angles of 40° and 70°. Find the third angle and name the triangle by its angles.

  1. The angles of a triangle add to 180°.
  2. 40° + 70° = 110°, so the third angle is 180° − 110° = 70°.
  3. All three angles (40°, 70°, 70°) are less than 90°.

70°. It is an acute triangle.

Example 3 · Test a claim

Sam says he drew a triangle with two right angles. Is that possible?

  1. Two right angles add to 90° + 90° = 180°.
  2. The three angles of a triangle must add to exactly 180°.
  3. That leaves 0° for the third angle, so the two sides would never meet to close the shape.

No. A triangle can have at most one right angle.

Example 4 · Give every name

A quadrilateral has four equal sides and four right angles. Which names fit it?

  1. Four equal sides and four right angles: square.
  2. Four right angles: it is also a rectangle. Four equal sides: it is also a rhombus.
  3. It has two pairs of parallel sides, so it is a parallelogram (and a trapezoid, with the "at least one pair" definition).

Square is the most specific name; rectangle, rhombus and parallelogram are also true.

Example 5 · One pair or two?

A quadrilateral has matching arrowheads on its top and bottom sides only. Its left and right sides lean in different directions. What is it?

  1. The matching arrows show that the top and bottom sides are parallel.
  2. The left and right sides are not parallel, so there is only one pair.
  3. A parallelogram would need two pairs.

A trapezoid, not a parallelogram.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Saying a square is not a rectangle.

Why this fails: A rectangle only needs four right angles. A square has four right angles, so it is a rectangle (and a rhombus) as well as a square.

Thinking a triangle can have two right angles.

Why this fails: The angles of a triangle add to 180°. Two right angles already make 180°, leaving nothing for the third angle.

Calling a triangle with two equal sides equilateral.

Why this fails: Equilateral means all three sides are equal. Two equal sides and one different side make an isosceles triangle.

Naming a shape by the way it is turned, such as calling a tilted square a "diamond".

Why this fails: Turning a shape does not change its side lengths or angles. Check the marks and measurements, not the direction it faces.

Thinking every shape with four equal sides is a square.

Why this fails: Four equal sides make a rhombus. It is a square only if its angles are also all 90°.

Foundation

Foundation practice

Name each shape from its sides or angles. Check the marks before you choose.

Name by sides

1. A triangle has sides of 5 cm, 5 cm and 8 cm. What is its name by sides?

  1. equilateral
  2. scalene
  3. isosceles
  4. none; it needs three equal sides to have a name
Name by sides

2. A triangle has sides of 4 cm, 6 cm and 9 cm. What is its name by sides?

  1. isosceles
  2. scalene
  3. equilateral
  4. it cannot be named without its angles
Tick marks

3. In a triangle diagram, two sides each have one tick mark and the third side has two tick marks. What kind of triangle is it?

  1. isosceles: the two sides with one tick are equal
  2. scalene: the sides have different marks
  3. equilateral: every side is marked
  4. it has 4 cm, 4 cm and 8 cm sides, one centimetre per tick
Name by angles

4. A triangle has angles of 90°, 35° and 55°. What is its name by angles?

  1. acute
  2. obtuse
  3. straight
  4. right
Name by angles

5. A triangle has angles of 110°, 40° and 30°. What is its name by angles?

  1. acute
  2. obtuse
  3. right
  4. scalene
Most specific name

6. A quadrilateral has four right angles. Its long sides are 6 cm and its short sides are 3 cm. What is its most specific name?

  1. square
  2. rhombus
  3. rectangle
  4. parallelogram
True or false

7. True or false: a small square drawn inside one corner of a triangle means that angle is exactly 90°.

  1. True
  2. False

Practice

Practice practice

Use the angle sums: 180° in a triangle, 360° in a quadrilateral. Then name the shape.

Missing angle

8. Two angles of a triangle are 50° and 60°. What is the third angle?

Missing angle

9. Two angles of a triangle are each 45°. What is the third angle, and what is the triangle's name by angles?

Name the quadrilateral

10. A quadrilateral has exactly one pair of parallel sides. Which name fits it?

  1. parallelogram
  2. rhombus
  3. rectangle
  4. trapezoid
Most specific name

11. A quadrilateral has four equal sides but no right angles. What is its most specific name?

  1. square
  2. rhombus
  3. rectangle
  4. parallelogram
Equilateral angles

12. All three angles of an equilateral triangle are equal. What does each angle measure?

Angle sum

13. Which set of angles could belong to a triangle?

  1. 100°, 45°, 35°
  2. 90°, 90°, 10°
  3. 70°, 70°, 70°
  4. 120°, 30°, 20°
Quadrilateral angle sum

14. A parallelogram has angles of 100°, 80° and 100°. The angles of any quadrilateral add to 360°. What is the fourth angle?

  1. 100°
  2. 160°
  3. 80°
  4. 260°
Name it twice

15. A triangle has sides of 9 cm, 12 cm and 15 cm, and a square corner mark between the 9 cm and 12 cm sides. Name it by sides and by angles.

Think

Think practice

These questions target common mistakes. Explain why the tempting answer fails.

Square or rectangle?

16. Jonah says, "A square is not a rectangle, because a rectangle has two long sides and two short sides." Which answer is correct?

  1. Jonah is right: a rectangle must have unequal sides.
  2. A square is a rectangle: it has four right angles, which is all a rectangle needs.
  3. A square is a rectangle only when it is turned on its side.
  4. A square is a rhombus, so it cannot also be a rectangle.
Two right angles?

17. Can a triangle have two right angles?

  1. Yes, if the third side is very short.
  2. Yes: a right isosceles triangle has two right angles.
  3. No, because right angles only fit in quadrilaterals.
  4. No: 90° + 90° = 180°, which leaves 0° for the third angle.
Isosceles or equilateral?

18. A triangle has sides of 6 cm, 6 cm and 10 cm. Priya calls it equilateral because it has equal sides. What is the correct name by sides?

  1. isosceles: only two sides are equal
  2. equilateral: two sides are equal
  3. scalene: one side is different
  4. equilateral, as long as its angles are equal
True or false

19. True or false: a rhombus is always a square.

  1. True
  2. False
Explain

20. A square is turned so that it stands on one corner. Some people call it a "diamond". Is it still a square? Explain.

Always true?

21. Which statement is true for every triangle?

  1. It has one right angle.
  2. An obtuse triangle has two obtuse angles.
  3. It has at least two acute angles.
  4. An isosceles triangle is always acute.
Trapezoid definition

22. This lesson uses the definition "a trapezoid has at least one pair of parallel sides." Using that definition, is a rectangle a trapezoid?

  1. Yes: a rectangle has two pairs of parallel sides, so it has at least one pair.
  2. No: a trapezoid must have slanted sides.
  3. No: every book says a trapezoid has exactly one pair of parallel sides.
  4. Only if the rectangle is a square.

Challenge

Challenge practice

Combine angle sums with side and angle names. Show your reasoning.

Isosceles angles

23. An isosceles triangle has a top angle of 40°. Its two base angles are equal. What does each base angle measure?

  1. 140°
  2. 40°
  3. 50°
  4. 70°
Right isosceles

24. A right triangle is also isosceles, and its two equal angles are not the right angle. What do those two angles measure?

Both names

25. Which shape is always both a rectangle and a rhombus?

  1. a square
  2. a parallelogram
  3. a trapezoid
  4. any quadrilateral with four equal sides
All the names

26. A quadrilateral has four right angles and four sides of 5 cm each. List every name from this lesson that fits it.

Isosceles angles

27. An isosceles triangle has one angle of 100°. What are its other two angles?

  1. 80° and 80°
  2. 40° and 40°
  3. 50° and 50°
  4. 100° and 100°
Quadrilateral angle sum

28. A quadrilateral has angles of 90°, 90° and 120°. What is the fourth angle?

Name it twice

29. A triangle has angles of 30°, 60° and 90°. Its sides are 5 cm, about 8.7 cm and 10 cm. Name it by sides and by angles.

  1. isosceles right triangle
  2. scalene acute triangle
  3. scalene right triangle
  4. scalene obtuse triangle
Must be a parallelogram

30. Which description must be a parallelogram?

  1. a quadrilateral with exactly one pair of parallel sides
  2. a quadrilateral with two right angles
  3. a quadrilateral with two equal sides
  4. a quadrilateral with four equal sides

Answer key

  1. 1. isosceles
    Two sides are equal (5 cm and 5 cm), so it is isosceles. It is not equilateral, because that needs all three sides equal, and it is not scalene, because scalene means no equal sides. A triangle with only two equal sides still has a name.
  2. 2. scalene
    All three lengths are different, so no sides are equal: scalene. Isosceles needs two equal sides. You can always name a triangle by its sides alone; you do not need the angles.
  3. 3. isosceles: the two sides with one tick are equal
    Sides with the same number of ticks are equal. The two one-tick sides match, so the triangle is isosceles. The two-tick side is a different length. Tick marks never count the length in centimetres, and different numbers of ticks do not make all three sides different.
  4. 4. right
    One angle is exactly 90°, so it is a right triangle. It does have two acute angles, but every triangle does; the name comes from its largest angle. 90° is not more than 90°, so it is not obtuse.
  5. 5. obtuse
    One angle, 110°, is more than 90°, so the triangle is obtuse. Two of its angles are acute, but an acute triangle needs all three angles under 90°. No angle is 90°, so it is not right.
  6. 6. rectangle
    Four right angles make it a rectangle. It is also a parallelogram, but "rectangle" tells more. It is not a square or a rhombus, because its sides are not all equal.
  7. 7. True
    The square corner mark is the standard symbol for a right angle. A triangle with that mark is a right triangle, even if the drawing is tilted.
  8. 8. 70°
    The three angles of a triangle add to 180°. 50° + 60° = 110°, and 180° − 110° = 70°.
  9. 9. 90°, so it is a right triangle
    45° + 45° = 90°, and 180° − 90° = 90°. One angle is 90°, so the triangle is a right triangle.
  10. 10. trapezoid
    One pair of parallel sides is what a trapezoid needs. A parallelogram, rectangle or rhombus needs two pairs of parallel sides, so none of those fit.
  11. 11. rhombus
    Four equal sides make a rhombus. It is also a parallelogram, but "rhombus" is more specific. It is not a square, because a square also needs four right angles, and it is not a rectangle for the same reason.
  12. 12. 60°
    The angles add to 180°, and they are all the same size, so each one is 180° ÷ 3 = 60°. That also means every equilateral triangle is acute.
  13. 13. 100°, 45°, 35°
    Only 100° + 45° + 35° adds to 180°. 90° + 90° + 10° = 190°, 70° + 70° + 70° = 210° and 120° + 30° + 20° = 170°, so those cannot be the angles of one triangle.
  14. 14. 80°
    100° + 80° + 100° = 280°, and 360° − 280° = 80°. 100° repeats an angle without checking the total, 160° comes from using 180° in place of 360° (180° − 20°), and 260° is 360° − 100°, which uses only one of the three angles.
  15. 15. scalene right triangle
    All three sides are different lengths, so it is scalene. The square corner mark shows a 90° angle, so it is a right triangle. A triangle gets one name from each list.
  16. 16. A square is a rectangle: it has four right angles, which is all a rectangle needs.
    A rectangle is a quadrilateral with four right angles. Nothing in that rule says the sides must be different lengths, so a square counts. Turning a square does not change its angles, and being a rhombus does not stop it from also being a rectangle.
  17. 17. No: 90° + 90° = 180°, which leaves 0° for the third angle.
    The three angles must add to 180°. Two right angles already use all 180°, so there is nothing left for a third corner. A right isosceles triangle has one 90° angle and two 45° angles. Right angles do appear in triangles; just only one at a time.
  18. 18. isosceles: only two sides are equal
    Equilateral means all three sides are equal. Here only two are, so the triangle is isosceles. It is not scalene, because scalene means no equal sides, and the angles do not change a name by sides.
  19. 19. False
    A rhombus needs four equal sides but not right angles. A rhombus that leans over (for example with angles of 60° and 120°) is not a square. The reverse is true: every square is a rhombus.
  20. 20. Yes. Its four sides are still equal and its four angles are still 90°. Turning a shape does not change its side lengths or angles.
    Names come from properties, not from which way a shape is facing. A square standing on a corner still has every property of a square.
  21. 21. It has at least two acute angles.
    The angles add to 180°, so at most one angle can be 90° or more; the other two must be acute. Many triangles have no right angle, an obtuse triangle has only one obtuse angle, and an isosceles triangle can be obtuse (for example 30°, 30°, 120°).
  22. 22. Yes: a rectangle has two pairs of parallel sides, so it has at least one pair.
    Under the "at least one pair" definition, every parallelogram (and so every rectangle) is also a trapezoid. Some books use "exactly one pair", and then a rectangle would not count, so check which definition your class uses. Slanted sides are not part of either definition.
  23. 23. 70°
    180° − 40° = 140° is shared equally by the two base angles: 140° ÷ 2 = 70°. 140° forgets to split between two angles, 40° assumes all three angles match, and 50° comes from 90° − 40°.
  24. 24. 45° each
    180° − 90° = 90° is left for the other two angles. They are equal, so each is 90° ÷ 2 = 45°.
  25. 25. a square
    A rectangle needs four right angles and a rhombus needs four equal sides. Only a square always has both. A parallelogram may have neither; a trapezoid may have neither; and four equal sides alone make a rhombus that might lean.
  26. 26. square, rectangle, rhombus, parallelogram, trapezoid (with the "at least one pair" definition) and quadrilateral
    It has four equal sides and four right angles, so it is a square. Four right angles also make it a rectangle, four equal sides make it a rhombus, and its two pairs of parallel sides make it a parallelogram and, with this lesson's definition, a trapezoid. It has four sides, so it is a quadrilateral.
  27. 27. 40° and 40°
    Two 100° angles would already make 200°, more than 180°, so the 100° angle must be the one that is different. 180° − 100° = 80°, shared equally: 40° and 40°. 80° and 80° forgets to share the 80°, and 50° and 50° halves 100° instead.
  28. 28. 60°
    The angles of a quadrilateral add to 360°. 90° + 90° + 120° = 300°, and 360° − 300° = 60°.
  29. 29. scalene right triangle
    The sides are all different, so it is scalene; the 90° angle makes it right. It is not isosceles, because no two sides (or angles) are equal, and it is not acute or obtuse, because its largest angle is exactly 90°.
  30. 30. a quadrilateral with four equal sides
    Four equal sides make a rhombus, and every rhombus has two pairs of parallel sides, so it is a parallelogram. One pair of parallel sides describes a trapezoid that may not have a second pair. Two right angles or two equal sides alone do not force both pairs of sides to be parallel.

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