Introduction

Geometry is a very organized and logical subject.
So before moving onto the geometry theorems list, let us discuss these foundational pillars—definitions, postulates, and theorems—so that proofs and propositions become clear.
Now let us move through a top-down path from abstract ideas to concrete rules so you can use deduction and reasoning in geometry problems.

Foundational Principles

The purpose of this section is to define the core pillars so later proofs and corollaries make sense in the theorems list.

Definition

Definitions are what we use for explaining things in plain terms so a statement has clear meaning and a concept can be used in a proof.
E.g.: – A circle is the set of all points equidistant from a center point, which gives a precise description useful in propositions about arcs.
Unlike postulates and theorems, definitions do not require proof and instead fix terminology so axioms and lemmas can be applied consistently.

Postulate

A postulate is a basic assumption accepted without proof that forms an axiom for building further results and logical deductions.
E.g.: – Euclid’s parallel postulate asserts that through a point not on a given line there is exactly one parallel line, which is an axiom used in many geometric proofs.
Unlike theorems, postulates are accepted facts rather than proven propositions, and they guide whether certain corollaries or proofs hold.

Theorem

A theorem is a statement that must be proven using definitions, postulates, and logical inference, and the proof may use lemmas or corollaries along the way.
E.g.: – The Pythagorean theorem states a^2 + b^2 = c^2 for a right triangle, and that equality is established by demonstration rather than assumption.
Unlike definitions and postulates, a theorem requires a formal proof or demonstration to move from hypothesis to conclusion.

Prerequisite Vocabulary Glossary

Let us go through all of them to fully understand the geometry theorems list and to make later proofs easier to follow.

Point

A point is a precise location in space with no size used as a vertex or intersection in proofs.

Line

A line is a straight figure that can be extended infinitely in both directions and is used to define parallelism and perpendicularity in propositions.

Ray

A ray is part of a line that starts at an endpoint and extends infinitely in one direction, often used in angle and congruence statements.

Angle

An angle is formed by two rays that share an endpoint, measured in degrees, and plays a central role in congruence, similarity, and trigonometric reasoning.

Angle Theorems

Now let us discuss angle rules so you can reason about congruence and measurement when solving problems involving rays and vertices.

1. Vertical Angles Theorem
   Vertical angles are congruent when two lines intersect because the pairs of opposite angles share equal measures in a proof.
   The equal measures follow from linear pairs and simple deduction in basic geometry.
2. Linear Pair Postulate (Angle Addition)
   If two angles form a linear pair then their measures add to 180°, which is used in proofs about supplementary angles and inference about adjacent rays.
   This postulate often appears as a step in arguments about parallel lines and transversals.
3. Corresponding Angles Theorem
   When a transversal crosses parallel lines, corresponding angles are congruent, and that congruence is often used to prove triangles are similar.
   This theorem links parallelism with angle equality in many geometric demonstrations.
4. Alternate Interior Angles Theorem
   For two parallel lines cut by a transversal, alternate interior angles are congruent and serve as a common lemma in reasoning about parallelism and similarity.
   Recognizing alternate interior angles helps construct proofs about parallel lines and similarity relations.

Triangle Theorems

Let us now proceed to discussing triangle theorems so you can apply sum, similarity, and congruence principles to polygons and trigonometric problems.

5. Triangle Sum Theorem
   The measures of the three interior angles of a triangle sum to 180°, which is fundamental to proofs about angle measures and polygon interior angles.
   This rule is often the first corollary used when solving angle values in triangle problems.
   Example: Suppose a triangle has angles x, y, and z, then x + y + z = 180°.
6. Exterior Angle Theorem
   An exterior angle of a triangle equals the sum of the two remote interior angles, which supports reasoning about supplementary angles and deduction in proofs.
   This theorem frequently appears when using transversals or proving inequalities in triangles.
7. Isosceles Triangle Theorem
   In an isosceles triangle the base angles are congruent when two sides are equal, and that congruence is used in constructions and symmetry arguments.
   Example: If sides AB = AC, then ∠B = ∠C, which is often used as a lemma in larger proofs.
8. Side-Angle-Side (SAS) Congruence Theorem
   If two sides and the included angle of one triangle are equal to two sides and the included angle of another, then the triangles are congruent, supporting rigid motion reasoning and proof by correspondence.
   SAS is a core congruence criterion used to deduce equal sides, angles, and corresponding vertices.
9. Pythagorean Theorem
   In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs, and this algebraic relationship underpins distance, similarity, and coordinate proofs.
   Example: For a right triangle with legs a and b and hypotenuse c, we have a^2 + b^2 = c^2, which is often used as an algebraic step in problem solving.
10. Triangle Inequality Theorem
    The length of any one side of a triangle is less than the sum of the other two sides, which constrains side measures in constructions and optimization problems.
    This inequality is used in proofs about existence and uniqueness of triangles in geometric constructions.

Circle Theorems

Now let us move onto circle theorems so you can connect arcs, chords, tangents, and central angles with proofs about measure and congruence.

11. Central Angle Theorem
    A central angle has measure equal to the measure of its intercepted arc, which links angle measure with arc length in circle geometry and trigonometry.
    This relationship is used when converting between arc measures and sector areas in proofs.
12. Inscribed Angle Theorem
    An inscribed angle measures half the measure of its intercepted arc, and that proportionality is key in deducing chord congruence and angle measures.
    This theorem often produces corollaries about equal arcs and equal chords.
13. Tangent-Secant Theorem
    The power of a point relation states that the square of the tangent length equals the product of the secant external segment and its whole, which is an algebraic corollary used in coordinate proofs.
    This theorem connects tangency, secants, and algebraic measures in circle geometry.
14. Chord Theorem (Equal Chords and Arcs)
    Equal chords in the same circle subtend equal arcs and equal central angles, which is a useful proposition when proving congruence or symmetry in circle constructions.
    Recognizing equal chords leads to equal arc measures and congruent isosceles triangles in proofs.
15. Thales’ Theorem
    If a triangle is inscribed in a circle with a diameter as one side, the angle opposite the diameter is a right angle, which is a classic demonstration linking circles and right triangles.
    This theorem is often used as a construction step in proofs involving perpendicularity and similarity.

Parallelogram Theorems

Let us now proceed to discussing parallelogram theorems so you can use vector-like reasoning, congruence, and parallelism in quadrilateral problems.

16. Opposite Sides Theorem
    In a parallelogram opposite sides are congruent and parallel, which follows from parallelism and is used to prove equality of lengths and vectors.
    This property is a staple in proofs about midpoints and translations.
17. Opposite Angles Theorem
    In a parallelogram opposite angles are congruent, a fact used when reasoning about supplementary angles and interior angle sums.
    This congruence is often a corollary when proving that a quadrilateral is a parallelogram.
18. Diagonals Bisect Each Other Theorem
    In a parallelogram the diagonals bisect each other, which gives midpoint relationships useful in coordinate proofs and congruence arguments.
    This bisecting property is essential in problems involving area or centroid-like reasoning.
19. Converse Parallelogram Tests
    If both pairs of opposite sides of a quadrilateral are congruent or if diagonals bisect each other, then the quadrilateral is a parallelogram, which is used to prove shape classification by deduction.
    These converses provide practical tests in construction and proof strategies.

Summary

We leave you with this thought: theorems are the proved backbone of geometry and algebraic reasoning, and understanding definitions, postulates, and vocabulary like point, line, angle, and vertex equips you to follow logical proofs and derive corollaries.
Now that you have seen the pillars, building blocks, and specific theorems by shape, consider how lemmas and proof techniques—such as contradiction, induction, and construction—extend these results to new propositions and problem solving.

FAQs

What is a theorem?
A theorem is a proven statement that follows from definitions, postulates, and prior theorems, and its proof is a demonstration using logic and inference.

How is a proof different from a postulate?
A proof is a logical sequence of statements that establishes a theorem, whereas a postulate is an assumed axiom used as a starting point in proofs without demonstration.

What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle the sum of the squares of the legs equals the square of the hypotenuse (a^2 + b^2 = c^2), and it is a key algebraic relation used in distance and coordinate proofs.

What does the Vertical Angles Theorem say?
The Vertical Angles Theorem says that when two lines intersect the opposite (vertical) angles are congruent, a simple congruence result often used in angle-chasing proofs.

How can I use these theorems in problem solving?
Start with definitions and postulates, apply relevant theorems as lemmas, and build a logical proof using congruence, similarity, or algebraic relations to reach the conclusion, which reinforces rigorous reasoning and deduction.