Properties of Triangles

Why This Matters

Triangles are the building blocks of geometry—every polygon can be divided into triangles, and nearly every proof or problem you'll encounter relies on triangle properties in some way. You're being tested on your ability to recognize which property applies in a given situation: Is this a question about angle relationships? Side lengths? Proving two triangles are the same shape? The properties in this guide form an interconnected web of tools, and strong geometry students know how to reach for the right one.

Understanding triangle properties means understanding why they work, not just what they say. The angle sum property connects to parallel lines and transversals. The triangle inequality explains what makes a valid triangle in the first place. Congruence and similarity criteria give you shortcuts for proving relationships without measuring everything. Don't just memorize formulas—know what concept each property demonstrates and when to apply it.

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Angle Relationships

The angles in a triangle follow predictable rules based on the fundamental property that interior angles sum to 180°. These relationships let you find unknown angles quickly.

Sum of Interior Angles

• All triangle interior angles sum to exactly $180°$—this holds for scalene, isosceles, equilateral, acute, right, and obtuse triangles without exception
• Use this to find missing angles by setting up equations: if two angles are known, subtract their sum from $180°$
• Foundation for polygon angle formulas—the formula $(n-2) \times 180°$ for $n$-sided polygons comes directly from dividing polygons into triangles

Exterior Angle Theorem

• An exterior angle equals the sum of the two remote interior angles—the angles that aren't adjacent to it
• Exterior angle is always greater than either remote interior angle alone, which helps with inequality proofs
• Quick angle-finding tool—often faster than using the $180°$ sum when you need a specific exterior angle relationship

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Side Length Rules

Not every combination of three lengths can form a triangle. These properties govern what's geometrically possible and how sides relate to each other.

Triangle Inequality Theorem

• The sum of any two sides must be greater than the third side—test all three combinations: $a + b > c$, $a + c > b$, and $b + c > a$
• Determines if a triangle can exist—if any combination fails, those three lengths cannot form a triangle
• Finding possible ranges—given two sides, the third side must be between their difference and their sum (exclusive)

Pythagorean Theorem

• In right triangles only: $a^2 + b^2 = c^2$ where $c$ is the hypotenuse (longest side, opposite the right angle)
• Converse proves right triangles—if three sides satisfy $a^2 + b^2 = c^2$, the triangle must be a right triangle
• Foundation for distance formula—the formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ is the Pythagorean theorem in coordinate form

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Proving Triangles Congruent

Congruent triangles have identical side lengths and angle measures. These criteria let you prove congruence without measuring all six parts.

SSS (Side-Side-Side)

• Three pairs of equal sides guarantee congruence—no angle information needed
• Most straightforward criterion—if you can show all three sides match, you're done
• Often used with distance formula in coordinate geometry to prove triangles congruent

SAS (Side-Angle-Side)

• Two sides and the included angle (between them) prove congruence—the angle must be between the two known sides
• "Included" is critical—SSA (angle not between the sides) does NOT prove congruence (the ambiguous case)
• Common in proofs with shared sides—when triangles share a side, you already have one "S"

ASA and AAS

• ASA: two angles and the included side—the side connects the two known angles
• AAS: two angles and a non-included side—since the third angle is determined by the $180°$ sum, this works too
• Both require two angles—remember that knowing two angles means you know all three (angle sum property)

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Proving Triangles Similar

Similar triangles have the same shape but different sizes—proportional sides and equal angles. These criteria establish similarity efficiently.

AA (Angle-Angle)

• Two pairs of equal angles guarantee similarity—the third pair must also be equal (angle sum property)
• Most commonly used similarity criterion—you only need to find two matching angles
• Parallel lines create AA situations—when a line parallel to one side cuts a triangle, it creates similar triangles

SAS and SSS Similarity

• SAS Similarity: two pairs of proportional sides with equal included angles—note the sides must be in proportion, not equal
• SSS Similarity: all three pairs of sides in the same proportion—the ratio must be consistent across all three pairs
• Proportion setup is key—write ratios carefully: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

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Special Segments and Their Centers

Every triangle has special line segments—medians, altitudes, angle bisectors, and perpendicular bisectors—that intersect at specific points with useful properties.

Medians and the Centroid

• A median connects a vertex to the midpoint of the opposite side—every triangle has exactly three medians
• Medians intersect at the centroid, which divides each median in a $2:1$ ratio (longer segment toward the vertex)
• Centroid is the center of mass—if you cut out a triangle from cardboard, it balances on the centroid

Altitudes and the Orthocenter

• An altitude is perpendicular from a vertex to the opposite side (or its extension)—it measures height
• Altitudes intersect at the orthocenter—inside for acute triangles, outside for obtuse, on the vertex for right triangles
• Used in area calculations—area = $\frac{1}{2} \times \text{base} \times \text{height}$, where height is an altitude

Angle Bisectors and the Incenter

• An angle bisector divides an angle into two equal parts—every triangle has three angle bisectors
• Angle bisectors intersect at the incenter, which is equidistant from all three sides
• Incenter is the center of the inscribed circle—the largest circle that fits inside the triangle touches all three sides

Perpendicular Bisectors and the Circumcenter

• A perpendicular bisector passes through a side's midpoint at $90°$—it's not drawn from a vertex
• Perpendicular bisectors intersect at the circumcenter, which is equidistant from all three vertices
• Circumcenter is the center of the circumscribed circle—the circle that passes through all three vertices

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Quick Reference Table

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Self-Check Questions

1. Which two congruence criteria both require knowing an angle, and what's the key difference between them?

2. A triangle has sides of length 5, 8, and $x$. Using the triangle inequality theorem, what is the range of possible values for $x$?

3. Compare and contrast the incenter and circumcenter: what segments create each, and what circle does each center?

4. Why does SSA fail to prove congruence while SAS succeeds? What role does the word "included" play?

5. If you know two angles of a triangle measure $45°$ and $60°$, explain two different methods to find the third angle (using interior angle sum vs. exterior angle theorem).