Classification of Triangles

When comparing the lengths of a triangle's sides, all three sides may be equal, two sides may be equal, or all three sides may be different lengths. This observation forms the basis of a classification system.

1. In an equilateral triangle, all sides are equal in length. ("Equilateral" is derived from two words: "equi" meaning "equal," and "lateral" meaning "sides.") Since equal sides in a triangle have equal angles opposite to them, all the angles of an equilateral triangle are equal as well, measuring 60 degrees. Equilateral triangles are discussed further in Properties of Equilateral Triangles.

2. An isoceles triangle is a triangle with two equal sides (and consequently two equal angles). The two equal sides are opposite of the two equal angles. Since any two sides of an equilateral triangle are equal, all equilateral triangles are isosceles triangles, but isosceles triangles are not necessarily equilateral. Isosceles triangles are discussed further in Properties of Isosceles Triangles.

3. In a scalene triangle, all sides have different lengths. If a triangle is not isosceles, it is a scalene triangle.

4. A degenerate triangle is a triangle where all vertices are colinear, so the lengths of two sides add up to the length of the third side. Such a triangle doesn't "look" like a triangle. It looks a bit more like a line segment.

Classify a triangle on the basis of sides if it has angles measuring 50, 60, and 70 degrees.

ANSWER

All three angles are different. If any two sides were equal, two of the angles would be equal. So all sides are different in length. Hence, it is a scalene triangle. $_\square$

Classify a triangle with sides 2, 3, and 6 units in length.

ANSWER

These dimensions are all different, which appears to represent a scalene triangle. However, such a triangle cannot exist due to the triangle inequality.

In short, the sum of any two sides of a triangle is always greater than the third side, because the straight line is the shortest path between any two points, so if any other path is drawn between those two points it must be longer than a straight line. $_\square$

Triangles can also be classified by their angles as follows:

1. In an obtuse triangle, one angle has a measure greater than $90^\circ$.
2. In a right triangle, one angle has a measure of exactly $90^\circ$.
3. In an acute triangle, all three angles measure less than $90^\circ$.
4. In an equiangular triangle, all three angles are $60^\circ$. Note that an equiangular triangle is also an equilateral triangle.

Triangles exhibit the following properties:

• The three angles in a triangle always add up to $180^\circ$.
• The three angles of an equilateral triangle are all equal to $60^\circ$.
• An isosceles triangle has a pair of equal angles.

Classify the triangle below.

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ANSWER

Since the three angles in a triangle sum to $180^\circ$, the missing angle in the triangle above is

$$180^\circ-60^\circ-60^\circ=60^\circ.$$

Thus, all three angles are equal to $60^\circ$ and the triangle is equilateral.$\ _\square$

Classify the triangle below.

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All three angles in the triangle above are acute angles, that is, smaller than $90^\circ$. Hence, by definition, the triangle is acute.$\ _\square$

Classify the triangle below.

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ANSWER

One angle $(124^\circ)$ in the triangle above is obtuse, that is, larger than $90^\circ$. Hence, by definition, the triangle is obtuse.$\ _\square$

Which of the following triangles is an isosceles triangle?

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ANSWER

Since the three angles of any triangle always add up to $180^\circ$, the value of the missing angle can be calculated for each triangle. The missing angles for triangles $A$, $B$, $C$, and $D$ are

$$\begin{aligned} 180^\circ-104^\circ-38^\circ&=38^\circ & (A)\\ 180^\circ-90^\circ-55^\circ&=35^\circ & (B)\\ 180^\circ-146^\circ-12^\circ&=22^\circ & (C)\\ 180^\circ-61^\circ-52^\circ&=67^\circ. & (D) \end{aligned}$$

Only triangle $A$ has two equal angles, and therefore the answer is triangle $A$. Observe that triangle $B$ is a right-angled triangle, triangle $C$ is an obtuse triangle, and triangle $D$ is an acute triangle.$\ _\square$

The ratio of the three angles that form a triangle is $1:2:3$. Classify this triangle.

ANSWER

Since the three angles add up to $180^\circ$, the respective values of the three angles are

$$\begin{aligned} 180^\circ\times\frac{1}{6}&=30^\circ\\ 180^\circ\times\frac{2}{6}&=60^\circ\\ 180^\circ\times\frac{3}{6}&=90^\circ. \end{aligned}$$

One of the angles is a right angle, so this triangle is a right-angled triangle.$\ _\square$

In the figure below, the lines $l_1$ and $l_2$ are parallel. Given $\angle BAD=75^\circ$ and $\angle ACE=150^\circ$, classify $\triangle ABC$.

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ANSWER

Since $l_1$ and $l_2$ are parallel,

$$\angle BAD=\angle ABC=75^\circ.$$

Since $\angle ACB=180^\circ-\angle ACE=180^\circ-150^\circ=30^\circ$,

$$\angle BAC=180^\circ-\angle ACB-\angle ABC=180^\circ-75^\circ-30^\circ=75^\circ.$$

Therefore, $\angle BAC=\angle ABC=75^\circ$, which implies that $\triangle ABC$ is an isosceles triangle.$\ _\square$