Classify Triangles by Angles and Sides | Grade 5 Math | LevelKite

You already know that a triangle is a polygon with 3 sides. It also has 3 angles, one at each corner. Triangles are not all alike, so you can classify them. To classify means to sort things into groups and give each group a name.

You can classify a triangle in two ways:

by its angles
by its side lengths

In this lesson you never go by how a drawing looks. A drawing can fool your eyes. An angle of 89 degrees and an angle of 90 degrees look almost the same. So use only the facts the problem gives you: the angle sizes and side lengths it states, or the ones you can work out from coordinates. Coordinates are number pairs, such as $(5, 1)$ , that tell where a point is on a grid.

Start with the angles. You measure an angle in degrees, and you write 90 degrees as $90^\circ$ . Three words describe the size of one angle:

A right angle measures exactly $90^\circ$ . It makes a square corner.
An acute angle measures less than $90^\circ$ .
An obtuse angle measures more than $90^\circ$ and less than $180^\circ$ .

A triangle's angle name tells you which kinds of angles it has:

An acute triangle has three acute angles. Example: $50^\circ$ , $60^\circ$ , $70^\circ$ .
A right triangle has one right angle. Example: $90^\circ$ , $30^\circ$ , $60^\circ$ .
An obtuse triangle has one obtuse angle. Example: $120^\circ$ , $25^\circ$ , $35^\circ$ .

To find the angle name, compare every angle with $90^\circ$ . Use angle measures that the problem states or that you can work out from coordinates, never the look of the drawing.

Worked example

Name a triangle by its angles

The angles of a triangle measure $40^\circ$ , $110^\circ$ and $30^\circ$ . Classify the triangle by its angles.

Compare each stated angle with 90°. You do this because the angle name depends on whether each angle is less than, equal to or more than 90°.
$40^\circ < 90^\circ \qquad 110^\circ > 90^\circ \qquad 30^\circ < 90^\circ$
Name each kind of angle. The 40° and 30° angles are less than 90°, so they are acute angles. The 110° angle is more than 90° and less than 180°, so it is an obtuse angle.
$110^\circ > 90^\circ \text{ and } 110^\circ < 180^\circ$
Choose the triangle's name. A triangle with one obtuse angle is an obtuse triangle. It is not an acute triangle, because not all three of its angles are acute.

Answer

The triangle is an obtuse triangle, because its $110^\circ$ angle is an obtuse angle.

Now classify triangles the second way, by their side lengths. The side name tells you how many sides are equal in length.

A scalene triangle has no equal sides. All three lengths are different. Example: 5 cm, 7 cm, 9 cm.
An isosceles triangle has at least two equal sides. Example: 6 cm, 6 cm, 9 cm.
An equilateral triangle has three equal sides. Example: 8 cm, 8 cm, 8 cm.

Look closely at the words at least two. A triangle with three equal sides has at least two equal sides. So every equilateral triangle is also an isosceles triangle. Picture the isosceles triangles as a big group. The equilateral triangles are a smaller group inside it. Scalene triangles sit outside the group, because they have no equal sides.

To find the side name, compare the three side lengths. Use lengths that the problem states or that you can work out from coordinates, never the look of the drawing. When all three sides are equal, say equilateral, because that name tells you the most.

Worked example

Name a triangle by its sides

The sides of a triangle are 7 cm, 4 cm and 7 cm long. Classify the triangle by its side lengths.

Look for equal lengths among the stated sides, because the side name depends on how many sides are equal. The first side and the third side are both 7 cm long.
$7 \text{ cm} = 7 \text{ cm}$
Check whether all three sides are equal, because an equilateral triangle needs three equal sides. The 4 cm side is not equal to the 7 cm sides, so this triangle is not equilateral.
$4 \text{ cm} \neq 7 \text{ cm}$
Choose the name. The two 7 cm sides give the triangle at least two equal sides, so it is an isosceles triangle.

Answer

The triangle is an isosceles triangle, because at least two of its sides are equal: two sides are 7 cm long.

Key concept

Every triangle has an angle name and a side name

A triangle has angles and sides, so it gets a name for each.

Angle name: acute if all three angles are less than $90^\circ$ , right if one angle is exactly $90^\circ$ , obtuse if one angle is more than $90^\circ$ .
Side name: scalene if no sides are equal, isosceles if at least two sides are equal, equilateral if all three sides are equal. Every equilateral triangle is also isosceles.

In this lesson, say the angle name first and then the side name, as in right isosceles triangle. Use only angle measures and side lengths that the problem states or that follow from coordinates on a grid. Never name a triangle from how a drawing looks.

Both names for three triangles, from facts a problem states

Facts the problem states	Angle name	Side name	Both names
All three angles are $60^\circ$ . All three sides are 8 cm long.	acute: all three angles are less than $90^\circ$	equilateral: three equal sides, so also isosceles	acute equilateral triangle
One angle is $120^\circ$ . Two sides are 6 cm long each.	obtuse: $120^\circ$ is more than $90^\circ$ and less than $180^\circ$	isosceles: at least two equal sides	obtuse isosceles triangle
One angle is $90^\circ$ . The sides are 3 cm, 4 cm and 5 cm long.	right: one angle is exactly $90^\circ$	scalene: no equal sides	right scalene triangle

Some problems give no measurements. They place the triangle on a coordinate grid, and the coordinates give you the facts.

Look at the figure. The x-axis runs across the bottom and the y-axis runs up the left side. Each grid square is 1 unit wide and 1 unit tall. The triangle has its vertices, or corner points, at $(1, 1)$ , $(5, 1)$ and $(1, 5)$ . Start where the axes meet: the point $(5, 1)$ is 5 units across and 1 unit up.

Coordinates like these give you two kinds of facts:

A side that lies along a grid line has a length you can count in units.
When one side runs straight across and another runs straight up, they meet in a right angle. The grid is made of squares, and every corner of a square is a right angle.

A slanted side does not lie along a grid line. The grid marks units only along its lines, so you cannot count a slanted side's length. Use that length only when the problem states it.

The triangle with vertices at

(1, 1)

,

(5, 1)

and

(1, 5)

. Its bottom side and its left side lie along grid lines.

Worked example

Give a triangle on a grid both names

The figure shows a triangle with vertices at $(1, 1)$ , $(5, 1)$ and $(1, 5)$ . Each grid square is 1 unit wide and 1 unit tall. Give the triangle both names, one for its angles and one for its sides.

Find the length of the bottom side, from (1, 1) to (5, 1). Both points are 1 unit up, so this side runs straight across along a grid line. It starts 1 unit across and ends 5 units across. On the figure it spans 4 grid squares, and subtracting gives the same count. The side is 4 units long.
$5 - 1 = 4$
Find the length of the left side, from (1, 1) to (1, 5). Both points are 1 unit across, so this side runs straight up along a grid line. It starts 1 unit up and ends 5 units up, so it spans 4 grid squares on the figure. This side is 4 units long too.
$5 - 1 = 4$
Find the angle name. At (1, 1), the side that runs straight across meets the side that runs straight up. Every corner of a grid square is a right angle, so this angle measures 90°. A triangle with one right angle is a right triangle.
$\text{angle at } (1, 1) = 90^\circ$
Find the side name. The bottom side and the left side are both 4 units long, so the triangle has at least two equal sides. It is isosceles. The slanted side is not on a grid line and the problem does not state its length. You do not need it: whatever its length is, the triangle has at least two equal sides.
$4 \text{ units} = 4 \text{ units}$
Put the two names together, because a triangle's full name tells about its angles and its sides. Say the angle name first and then the side name.

Answer

The triangle is a right isosceles triangle. It is right because the angle at $(1, 1)$ measures $90^\circ$ . It is isosceles because at least two of its sides are equal: two sides are 4 units long.

Here is the idea to keep. Every triangle has an angle name and a side name. The angle name is acute, right or obtuse. The side name is scalene, isosceles or equilateral. Isosceles means at least two equal sides, so every equilateral triangle is also isosceles. Check every angle and every side before you choose. Take your facts from what the problem states or from coordinates on a grid, never from how a drawing looks.

Next, you will use sides and angles in the same way to identify quadrilaterals, which are polygons with 4 sides: trapezoids, parallelograms, rectangles, rhombuses and squares.