Properties of Triangles

Triangles are the most fundamental polygon in all of geometry. Every polygon — no matter how many sides — can be broken down into triangles. This makes the triangle the basic building block of shapes, and understanding triangle properties is essential for everything from calculating areas to engineering stable structures.

Because triangles are so foundational, their properties show up everywhere: architecture, carpentry, surveying, navigation, and dozens of other fields. In this lesson, you will learn how to classify triangles, understand why their angles always add up to $180\degree$, and apply key theorems that make working with triangles predictable and powerful.

Classifying Triangles by Sides

Triangles are first classified by how many of their sides are equal.

Scalene Triangle

A scalene triangle has all three sides of different lengths. Because no two sides are equal, no two angles are equal either. Scalene triangles are the most “general” type — they have no special symmetry.

Isosceles Triangle

An isosceles triangle has exactly two equal sides. The two equal sides are called the legs, and the third side is called the base. The two angles opposite the equal sides (the base angles) are always equal.

Equilateral Triangle

An equilateral triangle has all three sides equal. Since the sides are all the same length, the angles must all be equal too. Each angle in an equilateral triangle is exactly $60\degree$ (because $180\degree \div 3 = 60\degree$).

Classifying Triangles by Angles

Triangles are also classified by the size of their largest angle.

Acute Triangle

An acute triangle has all three angles less than $90\degree$. Every equilateral triangle is acute, but not every acute triangle is equilateral.

Right Triangle

A right triangle has one angle that is exactly $90\degree$. The side opposite the right angle is called the hypotenuse and is always the longest side. Right triangles are the basis of the Pythagorean theorem and trigonometry.

Obtuse Triangle

An obtuse triangle has one angle greater than $90\degree$. A triangle can have at most one obtuse angle, because two angles greater than $90\degree$ would already sum to more than $180\degree$.

Triangle Classification Summary

A triangle always has both a sides classification and an angles classification. For example, a triangle can be an “isosceles right triangle” or a “scalene obtuse triangle.”

Triangle Angle Sum Theorem

The triangle angle sum theorem states that the three interior angles of any triangle always add up to $180\degree$.

$\angle A + \angle B + \angle C = 180\degree$

This is true for every triangle — scalene, isosceles, equilateral, acute, right, or obtuse. No exceptions.

Why Does This Work?

Here is an intuitive way to see it. Imagine cutting out a paper triangle and tearing off all three corners. If you place the three torn corners side by side with their vertices touching, they will always form a perfectly straight line — and a straight line is $180\degree$.

More formally, if you draw a line through one vertex parallel to the opposite side, the three angles at that vertex (two formed by the parallel line and one from the triangle) recreate the three interior angles, and together they span a straight angle of $180\degree$.

Using the Theorem

If you know two angles of a triangle, you can always find the third:

$\text{missing angle} = 180\degree - (\text{first angle} + \text{second angle})$

Exterior Angle Theorem

When one side of a triangle is extended beyond a vertex, the angle formed between the extended side and the adjacent side is called an exterior angle.

The exterior angle theorem states: an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles (also called remote interior angles).

$\text{exterior angle} = \angle A + \angle B$

where $\angle A$ and $\angle B$ are the two interior angles that are not next to the exterior angle.

Why does this work? The interior angle at $B$ and the exterior angle at $B$ are supplementary (they add to $180\degree$). Since $A + B + C = 180\degree$, we know $A + C = 180\degree - B$. But the exterior angle also equals $180\degree - B$. Therefore the exterior angle equals $A + C$.

Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For a triangle with sides $a$, $b$, and $c$:

$a + b > c$ $a + c > b$ $b + c > a$

All three conditions must be true. If even one fails, the three lengths cannot form a triangle.

Quick test: You only need to check whether the two shorter sides add up to more than the longest side. If that passes, the other two inequalities are automatically satisfied.

Properties of Isosceles Triangles

An isosceles triangle has two equal sides, and the angles opposite those equal sides — the base angles — are always equal.

$\text{If } \overline{AB} = \overline{AC}, \text{ then } \angle B = \angle C$

The converse is also true: if two angles of a triangle are equal, then the sides opposite those angles are equal, making the triangle isosceles.

Finding angles in an isosceles triangle: If you know the vertex angle (the angle between the two equal sides), the two base angles are:

$\text{base angle} = \frac{180\degree - \text{vertex angle}}{2}$

If you know one base angle, the vertex angle is:

$\text{vertex angle} = 180\degree - 2 \times \text{base angle}$

Properties of Equilateral Triangles

An equilateral triangle is a special case of an isosceles triangle where all three sides — and therefore all three angles — are equal.

• Every angle is exactly $60\degree$
• Every altitude, median, angle bisector, and perpendicular bisector from a given vertex coincides (they are all the same line segment)
• An equilateral triangle has three lines of symmetry
• The center (centroid, circumcenter, incenter, and orthocenter) is the same point

These symmetry properties make equilateral triangles especially useful in tiling, structural engineering, and design.

Worked Examples

Example 1: Find the missing angle

A triangle has two angles measuring $50\degree$ and $70\degree$. Find the third angle.

Step 1: Apply the triangle angle sum theorem:

$\angle A + \angle B + \angle C = 180\degree$

Step 2: Substitute the known angles:

$50\degree + 70\degree + \angle C = 180\degree$

$120\degree + \angle C = 180\degree$

Step 3: Solve:

$\angle C = 180\degree - 120\degree = 60\degree$

Answer: The third angle is $60\degree$. This triangle has angles $50\degree$, $60\degree$, and $70\degree$ — all less than $90\degree$, so it is an acute scalene triangle.

Example 2: Classify a triangle with sides 5, 5, 8

Step 1: Classify by sides. Two sides are equal ($5 = 5$), so this is an isosceles triangle.

Step 2: Find the angles. The equal sides are 5 and 5, and the base is 8. Using the law of cosines to find the angle opposite the base:

$\cos(\theta) = \frac{5^2 + 5^2 - 8^2}{2 \cdot 5 \cdot 5} = \frac{25 + 25 - 64}{50} = \frac{-14}{50} = -0.28$

$\theta = \cos^{-1}(-0.28) \approx 106.3\degree$

Step 3: Since the vertex angle is approximately $106.3\degree$ (greater than $90\degree$), this is an obtuse triangle.

Step 4: The two base angles are equal:

$\text{base angle} = \frac{180\degree - 106.3\degree}{2} \approx 36.9\degree$

Answer: This is an isosceles obtuse triangle with a vertex angle of about $106.3\degree$ and base angles of about $36.9\degree$ each.

Example 3: Find an exterior angle

A triangle has interior angles of $45\degree$, $65\degree$, and $70\degree$. Find the exterior angle at the vertex where the $70\degree$ interior angle is located.

Step 1: The exterior angle at a vertex is supplementary to the interior angle at that vertex:

$\text{exterior angle} = 180\degree - 70\degree = 110\degree$

Step 2: Verify using the exterior angle theorem — the exterior angle equals the sum of the two remote interior angles:

$45\degree + 65\degree = 110\degree \checkmark$

Answer: The exterior angle is $110\degree$.

Example 4: Can sides 3, 4, 8 form a triangle?

Step 1: Apply the triangle inequality theorem. Check if the two shorter sides sum to more than the longest side:

$3 + 4 = 7$

Step 2: Compare to the longest side:

$7 < 8$

Step 3: Since $3 + 4 = 7$, which is less than $8$, the triangle inequality is not satisfied.

Answer: No, sides of length 3, 4, and 8 cannot form a triangle. The two shorter sides are not long enough to “reach” each other around the longest side.

Example 5: Isosceles triangle with a vertex angle of $40\degree$

An isosceles triangle has a vertex angle of $40\degree$. Find the base angles.

Step 1: The vertex angle is between the two equal sides. The two base angles are equal. Use the angle sum theorem:

$40\degree + \angle B + \angle B = 180\degree$

$40\degree + 2\angle B = 180\degree$

Step 2: Solve:

$2\angle B = 180\degree - 40\degree = 140\degree$

$\angle B = 70\degree$

Answer: Each base angle is $70\degree$. The triangle has angles $40\degree$, $70\degree$, and $70\degree$.

Real-World Application: Carpentry — Why Triangles Brace Structures

If you have ever seen diagonal bracing on a wall frame, a gate, or a shelf, you have seen triangles at work. But why triangles instead of rectangles or other shapes?

Triangles are rigid. A rectangle made of four bars with pin joints can collapse into a parallelogram — push on one corner and the whole shape deforms. But a triangle made of three bars with pin joints cannot deform at all. The three fixed side lengths lock the three angles into place. This property is called structural rigidity.

Carpenters use this constantly:

• Wall framing: Diagonal braces turn a rectangular wall frame into two triangles, preventing the wall from racking (leaning sideways)
• Roof trusses: A roof truss is a network of triangles that distributes the roof load to the walls without the members bending
• Gate bracing: A diagonal brace on a garden gate creates a triangle that prevents the gate from sagging

This rigidity comes directly from the triangle properties covered in this lesson. Once the three side lengths are fixed, the triangle inequality guarantees the triangle exists, and the angle sum theorem means the angles are fully determined. There is no “wiggle room” — and that is exactly what makes triangles the strongest shape in construction.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse)
• The triangle angle sum theorem says all three interior angles add to $180\degree$ — always
• The exterior angle theorem says an exterior angle equals the sum of the two non-adjacent interior angles
• The triangle inequality theorem says any two sides must sum to more than the third — if not, no triangle exists
• Isosceles triangles have two equal sides and two equal base angles
• Equilateral triangles have all sides and angles equal (each $60\degree$), with maximum symmetry
• Triangles are structurally rigid, which is why they are the shape of choice for bracing in carpentry and engineering

Return to Geometry for more topics in this section.