Table of Contents
Understanding Triangles
A triangle is one of the simplest and most important shapes in geometry. It is a polygon with exactly three sides and three angles. In this chapter, we focus on ideas that are specific to triangles: their basic properties, how their sides and angles are related, and some special facts that are used again and again in later geometry.
(Remember that general ideas about points, lines, and angles are treated in the parent chapter, and specific topics like triangle classification, congruence, and the Pythagorean theorem each have their own chapters.)
Basic Structure of a Triangle
A triangle is formed by three non‑collinear points (points not on a single straight line), joined pairwise by three line segments.
- The three points are called the vertices.
- The three line segments are the sides.
- The three corners formed at the vertices are the angles of the triangle.
It is common to name a triangle by its vertices, for example $\triangle ABC$. Then:
- Side $a$ is opposite vertex $A$.
- Side $b$ is opposite vertex $B$.
- Side $c$ is opposite vertex $C$.
So when you see $a, b, c$ in a triangle, they usually refer to the lengths of the sides, and $A, B, C$ to the angles at the corresponding vertices.
Angle Sum of a Triangle
One of the most fundamental properties of triangles is how their interior angles add up.
Angle Sum Theorem (for triangles in the plane)
For any triangle in a flat (Euclidean) plane, the sum of its three interior angles is always $180^\circ$:
$$
A + B + C = 180^\circ.
$$
This is true no matter what the shape or size of the triangle is, as long as you are in ordinary flat geometry.
This fact is used to:
- Find a missing angle when two angles are given.
- Check whether three angle measures could make a triangle (they must add to $180^\circ$, and each must be positive).
For example, if a triangle has angles of $50^\circ$ and $60^\circ$, then the third angle is
$$
180^\circ - 50^\circ - 60^\circ = 70^\circ.
$$
Triangle Inequality
Not every triple of numbers can be the lengths of the sides of a triangle. The triangle inequality describes exactly which lengths can form a triangle.
Triangle Inequality Theorem
In any triangle with side lengths $a, b, c$:
- The sum of the lengths of any two sides is greater than the length of the remaining side:
$$
a + b > c, \quad b + c > a, \quad c + a > b.
$$
If any of these are not satisfied, then no triangle with those side lengths exists (in the plane).
This is useful for:
- Checking if three given lengths can form a triangle.
- Estimating the range in which a third side must lie, given the other two.
For example, if you have sides $5$ and $7$, then for a third side $x$:
- $5 + 7 > x \Rightarrow x < 12$,
- $5 + x > 7 \Rightarrow x > 2$,
- $7 + x > 5$ (always true if $x>0$).
So $2 < x < 12$ for the three lengths to form a triangle.
Naming and Types of Angles in a Triangle
Each angle in a triangle is less than $180^\circ$, but triangles can be described by how large their angles are:
- Acute triangle: All three interior angles are less than $90^\circ$.
- Right triangle: Exactly one angle is $90^\circ$.
- Obtuse triangle: Exactly one angle is greater than $90^\circ$.
This classification by angles helps decide which formulas or theorems apply. For instance, right triangles are closely connected to the Pythagorean theorem and trigonometry (covered in later sections), while acute and obtuse triangles involve different relationships between side lengths.
Exterior Angles of a Triangle
An exterior angle of a triangle is formed by extending one side of the triangle beyond a vertex. The angle outside the triangle that is adjacent to an interior angle is an exterior angle.
If $\angle A$ is an interior angle at vertex $A$, and you extend side $AB$ beyond $B$, the angle formed between this extension and side $AC$ is an exterior angle at $A$.
There are two important facts:
- Linear Pair with Interior Angle
An interior angle and its adjacent exterior angle form a straight line, so they add to 0^\circ$:
$$
\text{exterior angle at } A + A = 180^\circ.
$$ - Exterior Angle Theorem
An exterior angle of a triangle is equal to the sum of the two non‑adjacent (remote) interior angles.
If at vertex $A$ you form an exterior angle $E_A$, and the other two interior angles are $B$ and $C$, then:
$$
E_A = B + C.
$$
This is very useful when solving for missing angles, especially in multi‑triangle diagrams.
Relationship Between Sides and Angles
In any triangle, there is a direct relationship between the lengths of sides and the sizes of opposite angles:
- The longest side is opposite the largest angle.
- The shortest side is opposite the smallest angle.
- If two sides are equal, their opposite angles are equal (and conversely).
Symbolically:
- If $a > b$, then the angle opposite $a$ is larger than the angle opposite $b$:
$$
a > b \quad \Rightarrow \quad A > B.
$$
This helps you:
- Compare angles when you know the side lengths.
- Compare side lengths when you know the angles.
- Recognize when information is inconsistent (for example, a diagram with a long side drawn opposite a clearly smaller angle cannot be correct).
These relationships will later connect to more advanced results like the Law of Sines and Law of Cosines.
Perimeter of a Triangle
The perimeter of a triangle is the total length around it. For a triangle with side lengths $a, b, c$, the perimeter $P$ is
$$
P = a + b + c.
$$
This is used in practical problems like fencing a triangular plot of land or framing.
In word problems, side lengths might be expressed in terms of variables (for example, $x$, $x+2$, $2x+1$). Then the perimeter is a simple algebraic expression combining those terms.
Altitudes, Medians, and Angle Bisectors (Overview)
Triangles have several special line segments drawn from vertices to the opposite sides. Detailed properties and their intersections belong in more advanced sections, but it is useful here to introduce their basic meanings:
- An altitude is a segment from a vertex perpendicular to the opposite side (or its extension). It represents a height of the triangle.
- A median is a segment from a vertex to the midpoint of the opposite side.
- An angle bisector is a segment from a vertex that splits the angle at that vertex into two equal angles.
These special segments lead to several important points in a triangle:
- The concurrency point of medians is called the centroid.
- The concurrency point of perpendicular bisectors of the sides is the circumcenter.
- The concurrency point of angle bisectors is the incenter.
- The concurrency point of altitudes is the orthocenter.
The detailed properties and uses of these points are developed later, but knowing the vocabulary is helpful when first working with triangle diagrams.
Area of a Triangle (Basic Formula)
The area of a triangle is the amount of space inside it. A basic formula, using a base and its corresponding height, is:
$$
\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height}.
$$
If you choose one side as the base, the corresponding height (altitude) is the perpendicular distance from the opposite vertex to the line containing that side.
For example, if a triangle has a base of length $b$ and corresponding height $h$, then
$$
\text{Area} = \frac{1}{2} b h.
$$
Other area formulas for triangles (such as those involving all three sides or trigonometric functions) are introduced later, but this simple one is the most common starting point.
Summary of Key Triangle Facts
- A triangle has three sides, three vertices, and three angles.
- The sum of the interior angles is $180^\circ$.
- The triangle inequality: the sum of any two sides is greater than the third.
- A triangle’s sides and angles correspond in size: longer sides face larger angles.
- An exterior angle equals the sum of the two remote interior angles.
- Perimeter is the sum of the side lengths, $P = a + b + c$.
- Area can be found by $\dfrac{1}{2} \times \text{base} \times \text{height}$.
- Special segments—altitudes, medians, and angle bisectors—play crucial roles in deeper triangle geometry.
These basic properties form the foundation for the more specific topics you will study next: types of triangles, triangle congruence, and the Pythagorean theorem.