Classification of Triangles
Triangles - Classification by Sides
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.
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.
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.
In a scalene triangle, ALL sides having DIFFERENT lengths. If a triangle is not isosceles, it is a scalene triangle.
Classify a triangle on the basis of sides if it has angles measuring 50, 60, and 70 degrees.
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.
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 - Classification By Angles
The angles of a triangle can also be used to classify the shape.
- The three angles in a triangle always add up to \(180^\circ\).
- The three angles of an equilateral triangle are all equal to \(60^\circ\).
- Two angles of an isosceles triangle are equal.
- One angle of a right-angled triangle is \(90^\circ\).
- All angles of an acute-angled triangle are acute angles, smaller than \(90^\circ\).
- One angle of an obtuse-angled triangle is obtuse, meaning larger than \(90^\circ\) and smaller than \(180^\circ\).
Classify the triangle below.
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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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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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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\[\begin{align} 180^\circ-104^\circ-38^\circ&=38^\circ & (A)\\ 180^\circ-90^\circ-55^\circ&=35^\circ & (B)\\ 180^\circ-146^\circ-12^\circ&=22^\circ & (C)\\ 180^\circ-61^\circ-52^\circ&=67^\circ. & (D) \end{align}\]
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.
Since the three angles add up to \(180^\circ\), the respective values of the three angles are\[\begin{align} 180^\circ\times\frac{1}{6}&=30^\circ\\ 180^\circ\times\frac{2}{6}&=60^\circ\\ 180^\circ\times\frac{3}{6}&=90^\circ. \end{align}\]
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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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.\(\ _\square\)
