Triangles

Triangles are the simplest polygon but extraordinarily rich in properties and relationships. This chapter explores congruence—when two triangles have identical shape and size—and the rules that guarantee congruence without measuring all sides and angles. You'll discover that triangles can be compared efficiently using just a few measurements, why triangles are rigid structures (unlike squares which can collapse), and how triangle inequalities ensure that certain side lengths cannot form valid triangles. These properties underpin structural engineering, navigation, and computer graphics.

Triangle Basics: Definitions and Classification

A triangle is a polygon with three sides, three angles, and three vertices. The vertices are usually labeled A, B, C and the opposite sides as a, b, c.

Angle sum property: The sum of all interior angles in any triangle is 180°.

Classification by angles:

• Acute triangle: All angles less than 90°
• Right triangle: One angle equals 90° (the side opposite the right angle is called the hypotenuse)
• Obtuse triangle: One angle greater than 90°

Classification by sides:

• Scalene triangle: All three sides have different lengths
• Isosceles triangle: Two sides have equal length (the equal sides are called legs, the third is the base)
• Equilateral triangle: All three sides have equal length (all angles are 60°)

Real-world structures: Triangles are used extensively in construction because they're rigid—once the three side lengths are fixed, the triangle's shape cannot change. This is why roof trusses are triangular.

Congruence of Triangles: When Triangles Are Identical

Two triangles are congruent if they have the same shape and size. We write ∠ABC ≅ ∠DEF.

If triangles are congruent:

• Corresponding angles are equal
• Corresponding sides are equal

The challenge: To verify congruence, must we check all six measurements (three sides and three angles)? Fortunately, no! Several rules guarantee congruence with fewer checks.

Congruence Rules (Criteria)

SSS (Side-Side-Side):

• If three sides of one triangle equal the three sides of another triangle, the triangles are congruent
• You don't need to check any angles—equal sides guarantee equal angles!
• This is why triangle frames are rigid: fixing the three side lengths completely determines the triangle

SAS (Side-Angle-Side):

• If two sides and the included angle (the angle between them) of one triangle equal two sides and the included angle of another triangle, the triangles are congruent
• The angle must be between the two sides

ASA (Angle-Side-Angle):

• If two angles and the included side (the side between them) of one triangle equal two angles and the included side of another triangle, the triangles are congruent

AAS (Angle-Angle-Side):

• If two angles and a non-included side of one triangle equal two angles and the corresponding non-included side of another triangle, the triangles are congruent
• The side doesn't need to be between the angles

RHS (Right angle-Hypotenuse-Side):

• For right triangles: If the hypotenuse and one other side of a right triangle equal the hypotenuse and corresponding side of another right triangle, the triangles are congruent
• This is a special case specific to right triangles

Important non-rule: AAA (three equal angles) does NOT guarantee congruence! Triangles can have the same angles but different sizes (similar but not congruent).

Properties of Isosceles and Equilateral Triangles

Isosceles triangle properties:

• The base angles (angles opposite the equal sides) are equal
• Conversely, if two angles in a triangle are equal, the sides opposite those angles are equal
• The angle bisector from the vertex angle is also the perpendicular bisector of the base

Equilateral triangle properties:

• All three angles are 60°
• All angle bisectors, medians, altitudes, and perpendicular bisectors coincide
• It's the most "balanced" triangle shape

Triangle Inequalities: What Sides CAN Form a Triangle

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

If a triangle has sides of length a, b, and c:

• a + b > c
• b + c > a
• a + c > b

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

• Check: 3 + 4 = 7, which is not greater than 8
• No! The sides cannot form a triangle

Example: Can sides of length 3, 4, and 5 form a triangle?

• Check: 3 + 4 = 7 > 5 ✓, 3 + 5 = 8 > 4 ✓, 4 + 5 = 9 > 3 ✓
• Yes! These form a triangle (the famous 3-4-5 right triangle)

Why this matters: This inequality ensures that three line segments can actually "close" into a triangle. If one side is too long, the other two cannot reach each other.

Medians, Altitudes, and Angle Bisectors

Median: A line segment from a vertex to the midpoint of the opposite side

• Every triangle has three medians
• The three medians intersect at the centroid, which divides each median in a 2:1 ratio

Altitude: A perpendicular line segment from a vertex to the opposite side (or its extension)

• Every triangle has three altitudes
• The three altitudes intersect at the orthocenter

Angle bisector: A line segment from a vertex that bisects the angle

• Every triangle has three angle bisectors
• The three angle bisectors intersect at the incenter (center of the inscribed circle)

Connecting to Related Topics

Understanding triangles prepares you for:

• chapter-08-quadrilaterals: Quadrilaterals can be divided into triangles
• chapter-09-circles: Triangles can be inscribed in or circumscribed around circles
• chapter-10-herons-formula: Calculating triangle areas without measuring height

Key Formulas and Theorems

• Angle sum: Interior angles sum to 180°
• Congruence criteria: SSS, SAS, ASA, AAS, RHS
• Isosceles property: Equal sides have equal opposite angles
• Triangle inequality: Sum of any two sides > third side
• Pythagorean theorem (for right triangles): a² + b² = c²
• Triangle area: (1/2) × base × height

Socratic Questions

1. Why does SSS (Side-Side-Side) congruence work? Why can't two different shaped triangles have the same three side lengths?

1. If two triangles have all three angles equal (AAA), are they congruent? Why or why not? Can you sketch two different triangles with the same angles?

1. The triangle inequality ensures that a + b > c. What happens geometrically when a + b = c? Can you visualize what happens as you approach this limiting case?

1. An isosceles triangle has two equal sides. Why must the angles opposite those equal sides also be equal? Can you prove this using congruence rules?

1. Why are triangles chosen for structural strength in architecture and engineering? How do congruence rules ensure that a triangular frame maintains its shape?