Theorems & Formulas

Chapter 1

1. Betweenness of Points: If points A, B and C are colinear and AB + BC = AC, then point B is between points A and C.

2. Definition of Congruent Segments: When a segment is divided at its midpoint the resulting 2 segments are congruent.

3. Distance Formula: √( (x2-x1)² + (y2-y1)² )

4. Midpoint Formula: (x1 + x2) ÷ 2, (y1 + y2) ÷ 2

5. Definition of Congruent Angles: Angles with the same measures are known as congruent angles.

Chapter 2

1. Law of Detachment: If p --> q is a true statement, and p is true then q is true.

2. Law of Syllogism: If p --> q and q --> r are true, then p --> r is also true.

3. 2 Points 1 Line Postulate: Through any 2 points there is exactly one line.

4. 3 Points 1 Plane Postulate: Through any 3 non-colinear points there is exactly one plane.

5. Line in a Plane Postulate: If 2 points lie in a plane, then the entire line containing those points, lies in the plane.

6. 2 Lines Intersection Postulate: If 2 lines intersect each other, then their point of intersection is exactly one point.

7. 2 Planes Intersection Postulate: If 2 planes intersect each other, then their intersection is exactly one line.

8. Reflexive Property: A line or angle is equal to itself. AB = AB

9. Symmetric Property: If AB = CD, then CD must be equal to AB

10. Transitive Property: If AB = CD and CD = XY then AB = XY

11. Segment Addition Theorem: Adding 2 parts of a segment equal the segment. In a segment AC, AB + BC = AC

12. Angle Addition Theorem: Addition of 2 angles of a bigger angle equal the angle. In a right angle, 30° + 60° = 90°

13. Supplementary Theorem: If 2 angles form a linear pair then they are supplementary angles.

14. Complement Theorem: If non-common sides of 2 adjacent angles form a right angle, then the 2 angles are complementary angles.

15. Congruent Comp/ Supp Theorem: If 2 angles are complementary or supplementary to the same angle then they are congruent.

Chapter 3

1. Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

2. Alternate Interior Angle Theorem: If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

3. Alternate Exterior Angle Theorem: f two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

4. Consecutive Interior Angle Theorem: If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.

5. Perpendicular Transversal Theorem: In a plane , if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Chapter 4

1. Pythagorean Theorem: a² + b² = c² (Right Angle)

2. Pythagorean Theorem: a² + b² > c² (Acute Angle)

3. Pythagorean Theorem: a² + b² < c² (Obtuse Angle)

4. Triangle Angle Sum Theorem: The sum of measures of all angles in a triangle must be 180°.

5. Exterior Angle Theorem: The measure of an exterior angle is equal to the sum of the measures of the 2 remote interior angles.

6. Third Angles Theorem: If 2 angles of a triangle are congruent to 2 angles of a second triangle then the third angle is automatically congruent to the last angle of the second triangle.

7. SSS Theorem: If 3 sides of one triangle are congruent to the 3 sides of other triangle then the triangles are congruent.

8. SAS Theorem: If 2 sides and the including angle of a triangle are congruent to 2 sides and the included angle angle of another triangle then both the triangles are congruent.

9. ASA Theorem: If 2 angles and an included side of a triangle are congruent to the 2 angles and an included side of another triangle then both triangles are congruent.

10. AAS Theorem: If 2 angles and the non-included side of a triangle are congruent to the 2 angles and the non-included side of a triangle then both triangles are congruent.

11. Isosceles Triangle Theorem: If 2 sides of a triangle are congruent then the angles opposite those sides are congruent.

Chapter 5

1. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment then it is equidistant from the endpoints of a segment,

2. Circumcenter Theorem: The perpendicular bisectors of a triangle intersect at a point called the Circumcenter that is equidistant from the vertices of a triangle.

3. Angle Bisector Theorem: If a point is on the bisector of an angle then it is equidistant from the sides of a triangle.

4. Incenter Theorem: Incenter is a point of concurrency of all angle bisectors which is equidistant from the sides of a triangle.

5. Centroid Theorem: It is always inside a triangle and is a point of concurrency of all medians of a triangle. It is 2/3 of distance from each vertex to the midpoint of other side of the triangle.

6. Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is always greater than the measures of the 2 remote interior angles.

7. Triangle Inequality BetweennessTheorem: Sum of lengths of the 2 smallest sides of a triangle must be greater than the third side of the triangle.

8. Hinge Theorem: If 2 sides of a triangle are congruent are congruent to 2 sides of another triangle and the included angle of the first is larger than the included angle of the second triangle then the third side of the first triangle is longer than the third side of the second triangle.

Chapter 7

1. Cross Products Property: In a proportion, the product of the means is equal to the product of the extremes.

2. Perimeters of Similar Polygons: If 2 polygons are similar then their perimeters are proportional to the scale factor between them.

3. Angle Angle Similarity: If 2 angles of one triangle are congruent to 2 angles of other triangle, then the triangles are similar.

4. SSS Similarity: If the corresponding side lengths of 2 triangles are proportional then the triangles are similar.

5. SAS Similarity: If the lengths of 2 sides of a triangles are proportional to the lengths of the 2 corresponding sides of a triangle and the included angles are congruent then the triangles are similar.

6. Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other sides then it divides the sides into segments of proportional segments.

7. Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to one side of the triangle and its lengths is one half of the side.

8. Parts of Similar Triangles:

• lengths of corresponding altitudes are proportional to lengths of corresponding sides

• lengths of angle bisectors are proportional to the lengths of corresponding sides

• lengths of corresponding medians are proportional to the lengths of corresponding sides

9. Triangle Angle Bisector Theorem: an angle bisector separates the opposite side into two segments that are proportional to the lengths of the other two sides

Chapter 8

1. Finding the Geometric Mean: x^2 = ab and x = √ab

2. Right Angle Geometric Mean Theorems:

• one altitude separates the hypotenuse into 2 segments; the geometric mean of those 2 segments is the length of the altitude

• the length of a leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg

3. Pythagorean Theorem - in all right triangles, a^2 + b^2 = c^2 (a and b are the legs and c is the hypotenuse)

4. Common Pythagorean Triples: [3, 4, 5] [6, 8, 10] [9, 12, 15] [5, 12, 13] [10, 24, 26] [15, 36, 39] [8, 15, 17] [16, 30, 34] [24, 45, 51] [7, 24, 25] [14, 48, 50] [21, 72, 75] (largest number in each triple is the hypotenuse)

5. The Converse of the Pythagorean Theorem: If a^2 + b^2 = c^2, then that triangle is a right triangle

6. Properties of a Special Right Triangle:

• In a 45, 45, 90 triangle - legs (x) are congruent, and the length of the hypotenuse is x√2

• In a 30, 60, 90 triangle - the length of the hypotenuse is 2 times the length of the shortest side, and the length of the longer side is √3 times the length of the shorter side.

7. SOH CAH TOA:

   • • Sine --> Opposite/Hypotenuse

     • Cos --> Adjacent/Hypotenuse

     • Tan --> Opposite/Adjacent

8. Law of Sines:

   • sin (Angle A)/side (a) = sin (Angle B)/side (b) = sin (Angle C)/side (c)

9. Law of Cosines:

   • c² = a² + b² - 2ab cos (Angle C)

Chapter 9

1. Reflection:

   • Across x-axis: (x, y) --> (x, -y)

   • Across y-axis: (x, y) --> (-x, y)

   • Across y=x: (x, y) --> (y, x)

   • Across y=-x: (x, y) --> (-y, -x)

2. Rotation:

• 90 degree rotation: (x, y) --> (-y, x)

• 180 degree rotation: (x, y) --> (-x, -y)

• 270 degree rotation: (x, y) --> (y, -x)

Chapter 10

1. Arc Length: L = x/360 * 2πr (L=length, r=radius)

2. Inscribed Angle Theorem: 2*measure of inscribed angle = Measure of intercepted arc.

3. Inscribed Quadrilateral: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

4. Intersection of Secants: (arc1 + arc2) / 2

5. Intersection of Secants & Tangents: 1/2 * m arc AB

6. Intersections on exterior of circle: 1/2 * (arc DE - arc BC)

7. The product of 2 lengths of the chord equals each other.

8. Square of the tangent equals the measure of products of the secant and its external secant segment.

9. Standard Form of a Circle: r² = (x-h)² + (y-k)²

Chapter 11

1. Area of a Parallelogram: A=bh

2. Area of a Triangle: A=0.5bh

3. Area of a Trapezoid: A = 0.5h (b₁ + b₂)