🔺 Triangles Basics

Triangles are the fundamental building blocks of plane geometry. Master these core concepts to unlock most AMC geometry problems.

🎯 Key Concepts

Triangle Congruence

Two triangles are congruent if all corresponding parts are equal. The four main criteria are:

Pro move: Order matters! In $\triangle ABC \cong \triangle DEF$, vertex $A$ corresponds to $D$, $B$ to $E$, and $C$ to $F$.

Triangle Similarity

Two triangles are similar if corresponding angles are equal and corresponding sides are proportional.

Similarity Criteria:

• AA: Two angles equal (third automatically equal)
• SAS: Two sides proportional + included angle equal
• SSS: All three sides proportional

Key Insight: Similarity ratio $k$ means all lengths scale by $k$, areas by $k^2$.

Special Triangles

Isosceles Triangles

• Definition: Two sides equal
• Key Properties: Base angles equal, altitude bisects base and vertex angle
• AMC Appearance: Often appears in angle chasing problems

Equilateral Triangles

• Definition: All three sides equal
• Key Properties: All angles $60°$, all altitudes/medians/angle bisectors equal
• AMC Appearance: Common in coordinate geometry and area problems

🔍 Micro-Examples

Congruence Example

In $\triangle ABC$, if $AB = AC$ and $\angle B = \angle C$, then $\triangle ABC$ is isosceles by definition, and $\triangle ABC \cong \triangle ACB$ by SAS.

Similarity Example

If $\triangle ABC$ has $\angle A = 60°$ and $\angle B = 80°$, and $\triangle DEF$ has $\angle D = 60°$ and $\angle E = 80°$, then $\triangle ABC \sim \triangle DEF$ by AA.

Isosceles Example

In isosceles $\triangle ABC$ with $AB = AC$, if $D$ is the midpoint of $BC$, then $AD$ is both the median and altitude, and $\angle BAD = \angle CAD$.

⚠️ Common Traps

Pitfall: Assuming congruence from SSA

• Fix: SSA only works for right triangles (HL) or when the angle is opposite the longer side

Pitfall: Confusing similarity and congruence

• Fix: Similarity preserves shape, congruence preserves both shape and size

Pitfall: Wrong order in similarity statements

• Fix: Always match corresponding vertices: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$

🎯 AMC-Style Worked Example

Problem: In triangle $ABC$, points $D$ and $E$ are on sides $AB$ and $AC$ respectively, such that $DE \parallel BC$. If $AD = 3$, $DB = 2$, and $BC = 10$, find $DE$.

Solution: Since $DE \parallel BC$, we have $\triangle ADE \sim \triangle ABC$ by AA (corresponding angles equal).

The similarity ratio is $\frac{AD}{AB} = \frac{3}{3+2} = \frac{3}{5}$.

Therefore, $\frac{DE}{BC} = \frac{3}{5}$, so $DE = \frac{3}{5} \cdot 10 = 6$.

Answer: $DE = 6$

🔗 Related Topics

• Triangle Centers - Special points in triangles
• Special Segments - Medians, altitudes, angle bisectors
• Similarity & Ratios - Advanced similarity applications
• Angle Chasing - Using triangle properties in angle problems

💡 Quick Reference

Congruence Shortcuts

• Right triangles: HL (hypotenuse-leg)
• Isosceles: If two sides equal, base angles equal
• Equilateral: If one side equal to others, all angles $60°$

Similarity Shortcuts

• Parallel lines: Create similar triangles
• Angle bisectors: Often create similar triangles
• Right triangles: Look for shared acute angles

Common Ratios

• 30-60-90: Sides in ratio $1 : \sqrt{3} : 2$
• 45-45-90: Sides in ratio $1 : 1 : \sqrt{2}$
• 3-4-5: Right triangle with sides 3, 4, 5

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