Triangles Classification

Why This Matters

Triangle classification isn't just about memorizing categories—it's the foundation for nearly everything else you'll encounter in geometry. When you understand why triangles are grouped by angles and sides, you unlock the ability to apply theorems strategically, recognize which formulas work in which situations, and prove geometric relationships with confidence. These classifications connect directly to congruence proofs, similarity reasoning, coordinate geometry, and trigonometric applications that appear throughout your coursework.

You're being tested on your ability to identify triangle types, apply the correct theorems based on those types, and prove relationships using congruence and similarity criteria. The exam loves asking you to connect classification to calculation—for example, recognizing a right triangle so you can deploy the Pythagorean theorem, or spotting an isosceles triangle to set up equal angles. Don't just memorize the definitions—know what each classification enables you to do.

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Classification by Angles

How a triangle's angles relate to 90° determines its fundamental behavior and which theorems apply. The largest angle dictates the classification, and this directly affects side relationships and area calculations.

Acute Triangles

• All three angles measure less than 90°—this creates the most "balanced" triangle shape where no single angle dominates
• The orthocenter lies inside the triangle—unlike obtuse triangles, all altitudes intersect within the figure
• Satisfies $a^2 + b^2 > c^2$ for the longest side $c$—this inequality test distinguishes acute from right and obtuse triangles

Right Triangles

• Contains exactly one 90° angle—the side opposite this angle is the hypotenuse, always the longest side
• The Pythagorean theorem applies: $a^2 + b^2 = c^2$—this is your primary tool for finding missing side lengths
• The orthocenter lies at the right angle vertex—the two legs themselves serve as altitudes

Obtuse Triangles

• Contains exactly one angle greater than 90°—the other two angles must be acute (angle sum = 180°)
• Satisfies $a^2 + b^2 < c^2$ for the longest side $c$—use this to identify obtuse triangles from side lengths alone
• The orthocenter lies outside the triangle—altitudes from acute vertices must be extended to intersect

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Classification by Sides

Side relationships determine symmetry properties and constrain angle measures. Equal sides force equal opposite angles, which is the key principle connecting these classifications.

Equilateral Triangles

• All three sides are equal, forcing all angles to equal 60°—this is the only triangle with three lines of symmetry
• Every equilateral triangle is also equiangular—the terms are interchangeable for triangles specifically
• Centroid, orthocenter, circumcenter, and incenter all coincide—maximum symmetry creates a single triangle center

Isosceles Triangles

• At least two sides are equal, with base angles (opposite the equal sides) also equal—this is the Isosceles Triangle Theorem
• One line of symmetry passes through the vertex angle and the midpoint of the base—the altitude, median, and angle bisector from the vertex angle are identical
• Includes equilateral triangles as a special case—"at least two" means all three sides can be equal

Scalene Triangles

• All three sides have different lengths, producing three different angle measures—no symmetry exists
• No two altitudes, medians, or angle bisectors coincide—each segment type produces three distinct lines
• Most general triangle type—theorems that work for scalene triangles work for all triangles

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Special Right Triangles

These triangles have fixed angle measures that produce predictable side ratios. Memorizing these ratios eliminates the need for trigonometry in many calculations.

45-45-90 Triangles

• Angles measure 45°, 45°, and 90°—this is the only right triangle that is also isosceles
• Side ratio is $1 : 1 : \sqrt{2}$—the two legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg
• Formed by cutting a square along its diagonal—this geometric origin explains the $\sqrt{2}$ ratio

30-60-90 Triangles

• Angles measure 30°, 60°, and 90°—formed by cutting an equilateral triangle in half along an altitude
• Side ratio is $1 : \sqrt{3} : 2$—short leg : long leg : hypotenuse, where the short leg is opposite 30°
• The hypotenuse is exactly twice the short leg—this quick relationship often appears in exam problems

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Fundamental Theorems

These theorems establish the rules governing triangle existence and measurement. They apply regardless of classification and form the backbone of triangle problem-solving.

Triangle Inequality Theorem

• The sum of any two sides must exceed the third side—written as $a + b > c$, $a + c > b$, and $b + c > a$
• Determines whether a triangle can exist with given side lengths—if any inequality fails, no triangle is possible
• The difference of two sides is less than the third—this corollary helps establish bounds: $|a - b| < c < a + b$

Pythagorean Theorem

• For right triangles only: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse—the converse also holds for proving right angles
• Extends to classify all triangles: $a^2 + b^2 > c^2$ (acute), $a^2 + b^2 = c^2$ (right), $a^2 + b^2 < c^2$ (obtuse)
• Foundation for distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$—coordinate geometry relies on this connection

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Congruence and Similarity

These criteria establish when triangles are identical (congruent) or proportionally scaled (similar). Congruence preserves all measurements; similarity preserves only angles and ratios.

Congruence Criteria (SSS, SAS, ASA, AAS)

• SSS (Side-Side-Side): three pairs of equal sides guarantee congruence—no angle information needed
• SAS (Side-Angle-Side): two sides and the included angle must match—the angle must be between the two sides
• ASA and AAS: two angles plus one side (included for ASA, non-included for AAS)—knowing two angles determines the third

Similarity Criteria (AA, SAS, SSS)

• AA (Angle-Angle): two pairs of equal angles guarantee similarity—the third angle is automatically equal
• SAS Similarity: two sides in proportion with the included angle equal—ratios replace equality
• SSS Similarity: all three sides in the same proportion—if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, triangles are similar

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Triangle Segments and Centers

Special segments create intersection points called triangle centers, each with unique properties. The type of segment determines which center it creates.

Altitude, Median, and Angle Bisector

• Altitude: perpendicular from a vertex to the opposite side—creates right angles and is used in area calculations ($\text{Area} = \frac{1}{2}bh$)
• Median: connects a vertex to the midpoint of the opposite side—divides the triangle into two regions of equal area
• Angle bisector: divides an angle into two equal parts—creates proportional segments on the opposite side (Angle Bisector Theorem)

Centroid, Orthocenter, and Circumcenter

• Centroid: intersection of medians, located at $\frac{2}{3}$ of the distance from each vertex—this is the center of mass
• Orthocenter: intersection of altitudes—position varies: inside (acute), at vertex (right), outside (obtuse)
• Circumcenter: intersection of perpendicular bisectors—equidistant from all three vertices, center of the circumscribed circle

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Area Formulas

Multiple formulas exist because different information is available in different problems. Choose the formula that matches your known quantities.

Base-Height Formula

• $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$—requires a perpendicular height, not a slant side
• Any side can serve as the base—the corresponding height is the perpendicular distance to the opposite vertex
• Most efficient when height is given or easily calculated—often combined with special right triangle ratios

Heron's Formula

• $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter
• Uses only side lengths—no angle or height information required, ideal for coordinate geometry problems
• Verify triangle inequality first—if $s - a$, $s - b$, or $s - c$ is negative or zero, no valid triangle exists

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Quick Reference Table

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Self-Check Questions

1. Given side lengths 7, 10, and 12, how would you determine whether this triangle is acute, right, or obtuse without drawing it?

2. Which two triangle centers can lie outside the triangle, and what type of triangle causes this to happen?

3. Compare and contrast SSS for congruence versus SSS for similarity—what's the key difference in what you're checking?

4. A 30-60-90 triangle has a hypotenuse of 14. What are the lengths of the two legs, and which leg is opposite which angle?

5. You know all three side lengths of a triangle but not its height. Which area formula should you use, and what preliminary calculation does it require?