Geometric Theorems

Why This Matters

Geometric theorems aren't just random rules to memorize—they're the logical backbone of everything you'll do in Honors Geometry. These theorems connect to bigger ideas you're being tested on: proving relationships between shapes, establishing congruence and similarity, understanding circle properties, and applying algebraic reasoning to geometric figures. When you see a proof or a multi-step problem on an exam, you're really being asked to identify which theorem applies and why.

Here's the key insight: theorems are organized by what they help you prove. Some establish that triangles are identical (congruence), others show triangles have the same shape but different sizes (similarity), and still others reveal hidden relationships in circles or parallel lines. Don't just memorize the theorem names—know what concept each one demonstrates and when to reach for it. Master the "why" behind each theorem, and you'll crush both multiple choice and proofs.

---

Triangle Congruence: Proving Triangles Are Identical

These theorems let you prove two triangles are exactly the same size and shape without measuring all six parts. The key principle: you only need three specific pieces of information to lock in congruence.

SSS (Side-Side-Side) Congruence

• Three pairs of equal sides—if all three sides of one triangle match all three sides of another, the triangles must be congruent
• No angle information needed—the sides alone determine the triangle's shape completely
• Best for coordinate geometry proofs where you can calculate distances using the distance formula

SAS (Side-Angle-Side) Congruence

• Two sides and the included angle—the angle must be between the two sides you're comparing
• "Included" is critical—using a non-included angle gives you the ambiguous SSA case, which doesn't prove congruence
• Most common in proofs involving shared sides or vertical angles

ASA (Angle-Side-Angle) Congruence

• Two angles and the included side—the side must be between the two angles
• Works because the third angle is determined—if two angles match, the third must also match (angles sum to $180°$)
• Useful when parallel lines create equal angles in a figure

AAS (Angle-Angle-Side) Congruence

• Two angles and a non-included side—the side can be anywhere, not just between the angles
• Logically equivalent to ASA—knowing two angles means you know all three, so the "included" distinction disappears
• Often appears in problems with parallel lines where alternate interior angles are equal

---

Triangle Similarity: Same Shape, Different Size

Similarity theorems prove triangles have identical angles and proportional sides. The underlying principle: similar triangles are scaled versions of each other, making them powerful for indirect measurement.

AA (Angle-Angle) Similarity

• Only two angle pairs needed—if two angles match, the third automatically matches ($180°$ rule)
• Most efficient similarity test—requires less information than any congruence theorem
• Foundation for trigonometry and solving for unknown lengths using proportions

SAS Similarity

• Two proportional sides with an equal included angle—note this is proportion, not equality
• Ratio must be consistent—if one pair of sides has ratio $2:1$, the other pair must also be $2:1$
• Don't confuse with SAS Congruence—similarity uses proportions, congruence uses equal lengths

SSS Similarity

• All three side pairs proportional—every side of one triangle is the same multiple of the corresponding side
• Calculate the scale factor by dividing corresponding sides; all ratios must match
• Useful for proving similarity when no angle measures are given

---

Triangle Properties: Internal Relationships

These theorems reveal how segments, angles, and points within triangles relate to each other. The core idea: special points and lines in triangles follow predictable patterns.

Isosceles Triangle Theorem

• Equal sides mean equal base angles—the angles opposite the congruent sides are always congruent
• Converse is also true—if two angles are equal, the sides opposite them are equal
• Creates a "two-for-one" relationship that simplifies many proofs and calculations

Exterior Angle Theorem

• Exterior angle equals the sum of remote interior angles—not adjacent interior angles, but the two "far" ones
• Useful formula: if the exterior angle is $x$, then $x = \text{angle}_1 + \text{angle}_2$ for the non-adjacent angles
• Quick way to find missing angles without setting up a full $180°$ equation

Triangle Inequality Theorem

• Sum of any two sides must exceed the third—written as $a + b > c$ for all three combinations
• Tests whether lengths can form a triangle—if any combination fails, no triangle exists
• Common exam question type: "Which set of lengths could be sides of a triangle?"

Centroid Theorem

• Medians intersect at the centroid with a $2:1$ ratio from vertex to midpoint
• The centroid divides each median so the segment from vertex to centroid is twice as long as centroid to midpoint
• Center of mass—if you balanced a triangular plate on a pin, the centroid is where it balances

Midpoint Theorem (Midsegment Theorem)

• Midsegment is parallel to the third side and exactly half its length
• Connects midpoints of two sides—creating a smaller similar triangle inside
• Scale factor is always $\frac{1}{2}$—the inner triangle formed by all three midsegments has $\frac{1}{4}$ the area

---

Right Triangles: The Pythagorean Foundation

Right triangles get special treatment because the $90°$ angle creates unique relationships. The principle: the right angle constrains the triangle in ways that produce elegant formulas.

Pythagorean Theorem

• $a^2 + b^2 = c^2$ where $c$ is always the hypotenuse (longest side, opposite the right angle)
• Works only for right triangles—if a triangle satisfies this equation, it must contain a $90°$ angle (converse)
• Foundation for distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ is just Pythagorean Theorem in disguise

Thales' Theorem

• Diameter subtends a right angle—any point on a semicircle forms a $90°$ angle with the diameter's endpoints
• Converse helps find circle centers—if you have a right angle, the hypotenuse could be a diameter
• Connects circles and right triangles—powerful for problems involving both

---

Parallel Lines: Angle Relationships

When parallel lines are cut by a transversal, predictable angle pairs emerge. The key insight: parallel lines preserve angle measures across the transversal.

Alternate Interior Angles Theorem

• Alternate interior angles are equal when formed by parallel lines and a transversal
• "Alternate" means opposite sides of the transversal; "interior" means between the parallel lines
• Converse proves parallelism—if alternate interior angles are equal, the lines must be parallel

Parallel Lines and Transversals (Corresponding Angles)

• Corresponding angles are equal—same position at each intersection point
• Same-side interior angles are supplementary—they add to $180°$, not equal
• Use angle relationships to find all eight angles once you know just one

---

Circle Theorems: Arcs, Angles, and Tangents

Circle theorems connect angles to arcs and establish relationships between lines and circles. The underlying principle: an angle's position relative to the circle determines its relationship to the intercepted arc.

Inscribed Angle Theorem

• Inscribed angle equals half the intercepted arc—vertex on the circle, sides are chords
• Central angle equals the full arc—so inscribed angle is half the central angle for the same arc
• Inscribed angles intercepting the same arc are equal—regardless of where the vertex sits on the circle

Angle Bisector Theorem

• Bisector divides opposite side proportionally to the adjacent sides
• Formula: if the bisector from angle $A$ hits side $BC$ at point $D$, then $\frac{BD}{DC} = \frac{AB}{AC}$
• Combines angle properties with proportional reasoning—useful in similarity problems

Tangent-Secant Theorem

• Tangent squared equals secant times external segment—$(\text{tangent})^2 = (\text{whole secant}) \times (\text{external part})$
• Power of a point concept—all tangent-secant relationships from the same external point follow this pattern
• Watch your segments—the secant length includes both the part inside and outside the circle

---

Quick Reference Table

---

Self-Check Questions

1. Which two congruence theorems both require an "included" element, and what's the difference between them?

2. A problem gives you two triangles with two pairs of equal angles. Which theorem proves they're similar, and why don't you need the third angle?

3. Compare the Centroid Theorem and the Midsegment Theorem: both involve midpoints, but what different relationships do they describe?

4. If you're given a triangle inscribed in a circle with one side as the diameter, which theorem tells you something about the triangle's angles? What does it guarantee?

5. You need to prove two lines are parallel using angle relationships. Name two different angle pair types you could use, and explain how their relationships differ (equal vs. supplementary).