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CheatSheet - Geometry Theorems

1. Pythagorean Theorem

  • Statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
\[ a^2 + b^2 = c^2 \]

where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.

2. Triangle Sum Theorem

  • Statement: The sum of the interior angles of a triangle is always \(180^\circ\).
\[ \angle A + \angle B + \angle C = 180^\circ \]

3. Exterior Angle Theorem

  • Statement: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
\[ \text{Exterior Angle} = \angle A + \angle B \]

4. Isosceles Triangle Theorem

  • Statement: In an isosceles triangle, the angles opposite the equal sides are congruent.
\[ \text{If } AB = AC, \text{ then } \angle B = \angle C \]

5. Midpoint Theorem

  • Statement: The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
\[ \text{If } D \text{ and } E \text{ are midpoints of sides of triangle ABC, then } DE \parallel BC \text{ and } DE = \frac{1}{2}BC \]

6. Similar Triangles Theorem

  • Statement: If two triangles have corresponding angles equal, then their corresponding sides are proportional, and the triangles are similar.
\[ \triangle ABC \sim \triangle DEF \implies \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \]

7. Alternate Interior Angles Theorem

  • Statement: If two parallel lines are cut by a transversal, the alternate interior angles are congruent.
\[ \angle 1 = \angle 2 \quad (\text{if lines are parallel}) \]

8. Angle Bisector Theorem

  • Statement: The angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides.
\[ \frac{AB}{AC} = \frac{BD}{DC} \]

where \( AD \) is the bisector.

9. Congruence Theorems (Triangles)

  • SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
\[ \triangle ABC \cong \triangle DEF \]

  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent 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 congruent to two angles and the included side of another triangle, the triangles are congruent.

10. Circle Theorems

  • Central Angle Theorem: The measure of the central angle is equal to the measure of its intercepted arc.
\[ \text{Central Angle} = \text{Arc Measure} \]
  • Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
\[ \text{Inscribed Angle} = \frac{1}{2} \times \text{Arc Measure} \]
  • Tangent-Secant Theorem: If a tangent and a secant (or chord) meet at a point on a circle, the square of the tangent is equal to the product of the secant and its external part.
\[ \text{Tangent}^2 = \text{External Part} \times \text{Whole Secant} \]
  • Chord-Chord Theorem: If two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.
\[ AB \times BC = DE \times EF \]

11. Area and Volume Theorems

  • Heron’s Formula (Area of a Triangle):
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]

where \( s = \frac{a + b + c}{2} \) is the semi-perimeter, and \( a, b, c \) are the sides of the triangle.

  • Area of a Regular Polygon:
\[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]

12. Parallelogram Theorems

  • Opposite Sides Theorem: The opposite sides of a parallelogram are congruent.
\[ AB = CD \quad \text{and} \quad AD = BC \]
  • Diagonals Theorem: The diagonals of a parallelogram bisect each other.
\[ AE = EC \quad \text{and} \quad BE = ED \]

13. Tangent to a Circle

  • Statement: A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
\[ \angle O = 90^\circ \]