Geometric Theorems to Know for Honors Geometry

Geometric theorems are essential tools in understanding shapes, angles, and their relationships. From the Pythagorean Theorem to triangle congruence and similarity, these principles form the foundation of Honors Geometry, helping solve complex problems in a structured way.

1. Pythagorean Theorem

   • States that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
   • Used to determine the length of a side in a right triangle when the lengths of the other two sides are known.
   • Fundamental in various applications, including distance calculation in coordinate geometry.

2. Triangle Congruence Theorems (SSS, SAS, ASA, AAS)

   • SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
   • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
   • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
   • AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and a corresponding non-included side of another triangle, the triangles are congruent.

3. Triangle Similarity Theorems (AA, SAS, SSS)

   • AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
   • SAS (Side-Angle-Side): If two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar.
   • SSS (Side-Side-Side): If the sides of one triangle are in proportion to the sides of another triangle, the triangles are similar.

4. Parallel Lines and Transversals Theorem

   • States that when a transversal crosses two parallel lines, several pairs of angles are formed, including corresponding angles, alternate interior angles, and same-side interior angles.
   • Corresponding angles are equal, and alternate interior angles are equal, which can be used to prove lines are parallel.

5. Angle Bisector Theorem

   • States that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides.
   • Useful for finding lengths in triangles and establishing relationships between sides.

6. Inscribed Angle Theorem

   • States that an inscribed angle is half the measure of the intercepted arc.
   • This theorem helps in solving problems involving circles and angles formed by points on the circle.

7. Thales' Theorem

   • States that if A, B, and C are points on a circle where the line segment AC is the diameter, then the angle ABC is a right angle.
   • This theorem is fundamental in circle geometry and helps establish relationships between angles and arcs.

8. Centroid Theorem

   • States that the centroid (the point where the three medians of a triangle intersect) divides each median into a ratio of 2:1.
   • The centroid is the center of mass of the triangle and is important in various applications in geometry.

9. Midpoint Theorem

   • States that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
   • This theorem is useful for proving properties of triangles and parallelograms.

10. Triangle Midsegment Theorem

    • Similar to the Midpoint Theorem, it states that the midsegment of a triangle is parallel to one side and half the length of that side.
    • This theorem aids in understanding the relationships between different segments in triangles.

11. Isosceles Triangle Theorem

    • States that in an isosceles triangle, the angles opposite the equal sides are also equal.
    • This theorem is essential for solving problems involving isosceles triangles and their properties.

12. Exterior Angle Theorem

    • States that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
    • This theorem is useful for finding unknown angles in triangles.

13. Triangle Inequality Theorem

    • States that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
    • This theorem is fundamental in determining whether a set of lengths can form a triangle.

14. Tangent-Secant Theorem

    • States that if a tangent and a secant are drawn from a point outside a circle, the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part.
    • This theorem is useful in solving problems involving circles and tangent lines.

15. Alternate Interior Angles Theorem

    • States that if two parallel lines are cut by a transversal, then each pair of alternate interior angles is equal.
    • This theorem is crucial for proving lines are parallel and solving angle relationships in geometric figures.