4th Grade · 4.G.A.2

Types of triangles in 4th grade

The types of triangles your child learns in 4th grade depend on their angles and their sides. Your child names right, acute and obtuse triangles, finds right triangles in any position and sorts other shapes by their sides and angles.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

What it means, in plain words

To classify triangles by angles, your child looks at the widest angle. A right triangle has one right angle, a square corner. An obtuse triangle has one angle wider than a square corner. An acute triangle has three angles that are all narrower than a square corner. A triangle can have only one right angle or one obtuse angle, never two.

Classifying triangles by their sides gives three more names. An equilateral triangle has all three sides the same length. An isosceles triangle has two sides the same length. A scalene triangle has no two sides the same length. Every triangle gets one name from each list, so one triangle can be both right and isosceles.

Classifying shapes goes beyond the types of triangles. Your child sorts shapes by whether they have parallel sides (sides that never meet), perpendicular sides (sides that meet at a square corner) or angles of a certain kind. Right triangles become a group of their own, and your child learns to find them whichever way they are turned.

Your child should be able to:

Name a triangle as right, acute or obtuse by its angles
Name a triangle as equilateral, isosceles or scalene by its sides
Find the right triangles in a group of shapes, however they are turned
Sort shapes by whether they have parallel sides, perpendicular sides or right angles

Example

A triangle has one square corner and two sides that are each 5 cm long. What types of triangle is it?

It has one right angle, so by its angles it is a right triangle.
Two of its sides are the same length, so by its sides it is isosceles.
It is a right isosceles triangle.

Example

A triangle has sides of 4 cm, 6 cm and 9 cm, and one of its angles is wider than a square corner. What types of triangle is it?

One angle is wider than a square corner, so by its angles it is an obtuse triangle.
No two sides are the same length, so by its sides it is scalene.
It cannot also be a right triangle, because a triangle has only one angle that is a square corner or wider.
It is an obtuse scalene triangle.

Where children usually get stuck

Your child picks out a right triangle only when its square corner sits at the bottom, and passes over one balanced on its point.
Book pictures of right triangles usually rest on a flat side with the square corner at the bottom left. The name gets tied to the picture, not to the corner.
What helps. Use the corner of a sticky note as a tester. Check every corner of each triangle, whichever way the triangle points.
Your child calls a triangle acute because two of its corners are sharp, without checking the third.
Every triangle has at least two acute angles. Only the widest angle decides the type, so a look at the sharp corners gives the wrong name.
What helps. Ask your child to find the widest angle first and compare only that one with a square corner.
Your child says a triangle cannot be right and isosceles at the same time, and picks one name.
Classifying triangles uses two separate lists, one for angles and one for sides. Learned as one list of six names, it looks like each triangle gets only one.
What helps. Make two columns on paper, angles and sides. For each triangle you draw together, write one name in each column.

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A classmate sorted 4 shapes into the cell labeled "has parallel sides and has perpendicular sides." One shape does not belong in this cell. Which shape should be moved?
A
B
C
D
Show the answer and how to solve it
The scalene triangle should be moved. It has no parallel sides and no right angles, so it fails both requirements for this cell.
1. 1First, read the cell's full label: 'has parallel sides and has perpendicular sides.' Every shape in this cell must have BOTH — at least one pair of parallel sides AND at least one right angle (perpendicular sides).
2. 2Check each shape for parallel sides. The scalene triangle has NO parallel sides. The rectangle has 2 pairs of parallel sides. The tilted square has 2 pairs of parallel sides. The right trapezoid has 1 pair of parallel sides (top and bottom). Three shapes pass; the scalene triangle does not.
3. 3Check each shape for right angles. The scalene triangle has NO right angles. The rectangle has 4 right angles. The tilted square still has 4 right angles — turning a shape does not change its angles. The right trapezoid has 2 right angles where the vertical side meets the top and bottom. Three shapes pass; the scalene triangle does not.
4. 4The scalene triangle fails both tests — it has no parallel sides and no right angles. It does not belong in this cell. Move the scalene triangle.
Common mistake. A common mistake is thinking a triangle might have parallel sides because two of its sides look like they could line up. A scalene triangle has three sides of different lengths — no parallel pairs and no right angles.
The idea underneath. A shape's parallel and perpendicular properties depend on its actual sides and angles, not on which corner is on top. Tilting or turning a shape never adds or removes its right angles.
2Warm-up
Drag each shape into the bin that matches its properties.
Drag each item into the group where it belongs, or tap the item, then tap the group.
Has parallel sides but no right angle
Drag or tap items here
Has a right angle but no parallel sides
Drag or tap items here
Has both parallel sides and a right angle
Drag or tap items here
Has neither parallel sides nor a right angle
Drag or tap items here
Keyboard instructions: focus an item and press Enter or Space to pick it up. Move focus to a bin or the bank and press Enter or Space to drop it. Press Escape to cancel.
Show the answer and how to solve it
Rectangle goes in 'Has both parallel sides and a right angle'. Parallelogram goes in 'Has parallel sides but no right angle' because its corners are not right angles. Right triangle goes in 'Has a right angle but no parallel sides' because triangles have no parallel sides.
1. 1There are four bins: 'Has parallel sides but no right angle', 'Has a right angle but no parallel sides', 'Has both parallel sides and a right angle', and 'Has neither parallel sides nor a right angle'. For each shape, ask two questions: Does it have a pair of parallel sides? Does it have a right angle (a corner that is exactly 90 degrees)? Then drop the shape into the bin that matches both answers.
2. 2Look at the rectangle. The top and bottom sides are parallel, and the left and right sides are parallel, so it has parallel sides. Every corner is exactly 90 degrees, so it has right angles. The rectangle has both properties, so it goes in the 'Has both parallel sides and a right angle' bin.
3. 3Look at the parallelogram. The top and bottom sides are parallel, and the two slanted sides are also parallel, so it has parallel sides. But its corners lean to one side — none of them are right angles. The parallelogram has only the parallel-sides property, so it goes in the 'Has parallel sides but no right angle' bin.
4. 4Look at the right triangle. Triangles never have parallel sides, so that property is not met. However, one corner is exactly 90 degrees, which is a right angle. The right triangle has only the angle property, so it goes in the 'Has a right angle but no parallel sides' bin.
Common mistake. A common mistake is thinking a parallelogram has right angles because it looks similar to a rectangle. A parallelogram's corners are slanted and are NOT right angles. Students who make this mistake will incorrectly place the parallelogram in the 'Has both parallel sides and a right angle' bin instead of the 'Has parallel sides but no right angle' bin.
The idea underneath. Check each property independently for each shape, then place the shape in the bin that matches whether both, one, or neither property applies. Triangles can only ever satisfy the angle part of such a rule, because triangles never have parallel sides.
3On level
Some shapes have a pair of perpendicular sides — sides that meet to form a right angle. Select all the shapes below that have at least one pair of perpendicular sides.
A
B
C
D
Show the answer and how to solve it
The rectangle and the right triangle both have perpendicular sides, because each has at least one pair of sides that meet at a right angle. Tilting a shape does not change whether its sides are perpendicular.
1. 1Perpendicular sides meet at a right angle ( $90^{\circ}$ ) — like the corner of a square or the letter L. Look at each shape and check whether any two sides meet at a right angle.
2. 2The equilateral triangle has three equal angles, each $60^{\circ}$ . None of them is a right angle, so it has no perpendicular sides.
3. 3The rectangle has four right angles at its corners. Even though it is tilted, the corners are still right angles. So the rectangle has perpendicular sides.
4. 4The parallelogram has two pairs of parallel sides, but its angles are about $60^{\circ}$ and $120^{\circ}$ , not $90^{\circ}$ . Parallel and perpendicular are different ideas — this shape has no perpendicular sides.
5. 5The right triangle has one right angle. Even though it is tilted, the right angle is still there. So the right triangle has perpendicular sides.
Common mistake. A common mistake is thinking that perpendicular sides must look horizontal and vertical on the page. Perpendicular just means the sides meet at a right angle — the shape can be tilted and still have perpendicular sides.
The idea underneath. Perpendicular sides meet at a right angle. This property is preserved when a shape is rotated — perpendicular does not require the sides to be horizontal and vertical on the page.
4Stretch
Select all shapes that have an obtuse angle AND at least one pair of parallel sides.
A
B
C
D
E
F
Show the answer and how to solve it
The parallelogram and the trapezoid are correct. Each has at least one obtuse angle AND at least one pair of parallel sides. The other shapes meet only one part of the rule, or neither part.
1. 1The rule has TWO parts joined by AND. A shape fits the rule only if it has BOTH an obtuse angle (a corner wider than a right angle, wider than a square corner) AND at least one pair of parallel sides (two sides that point the same way and never meet). A shape that meets only one part does NOT fit the rule.
2. 2The rectangle has 2 pairs of parallel sides, but all 4 of its corners are right angles — none is obtuse. Only one part of the rule is met, so the rectangle does NOT fit the rule.
3. 3The parallelogram has 2 obtuse angles (the two wider corners) AND 2 pairs of parallel sides (opposite sides are parallel). Both parts of the rule are met, so the parallelogram fits the rule.
4. 4The kite has an obtuse angle at its top vertex, but none of its four sides are parallel — no two sides run in the same direction. Only one part of the rule is met, so the kite does NOT fit the rule.
5. 5The trapezoid has 2 obtuse angles (the two wider top corners) AND 1 pair of parallel sides (the top base and the bottom base are parallel). Both parts of the rule are met, so the trapezoid fits the rule.
6. 6The equilateral triangle has 3 acute angles — every corner is the same small angle, and none is obtuse — and it has no parallel sides. Neither part of the rule is met, so the equilateral triangle does NOT fit the rule.
7. 7The right triangle has one right angle and two acute angles — no obtuse angle — and no pair of its sides is parallel. Neither part of the rule is met, so the right triangle does NOT fit the rule.
8. 8The parallelogram and the trapezoid are the only two shapes that have BOTH an obtuse angle AND at least one pair of parallel sides.
Common mistake. A common mistake is checking only one part of the rule. Some students see parallel sides on the rectangle and pick it, missing that all its corners are right angles — none is obtuse. Others see the obtuse angle on the kite and pick it, forgetting that kites do not have parallel sides. The word AND means BOTH parts must be true at the same time.
The idea underneath. When a rule uses the word AND, both parts must be true at the same time. To use a combined rule, check each part separately for every shape — only shapes that pass BOTH checks fit the rule.