# Geometry – Quick Reference Sheet

Point, Lines and Planes

Definitions:

• Point (•): a position without length, width or depth.
• Line (↔): a series of points that create length, but have no thickness or depth. May be curved, but typically straight with infinite length.
• Parallel Lines (||): two or more lines that never intersect. Both lines will have the same slope.
• Transversal: the line that intersects to parallel lines.
• Perpendicular Lines (⊥): two lines that intersect at a 90° angle. The product of their slopes will equal -1.
• Line segment (-): the line between 2 points.
• Ray (→): a portion of a line that extends infinitely in one directions.
• Angle (∠): a form created when two rays share an endpoint.
• Complimentary Angles: two angles with a sum measurement of 90°.
• Supplementary Angles: two angles with a sum measurement of 180°.
• Vertical Angles: congruent, non-adjacent angles formed by intersecting lines.
• Plane: an infinite series of points or lines with no depth.

Formulas:

• Distance Formula • Midpoint Formula • Slope • Equation of a line (point-slope form) ⇒ • Equation of a line (slope-intercept form) ⇒ Polygons

Definition:

• Polygon: a closed plane figure with a minimum of three sides. Sides are composed of line segments that intersect at endpoints that are non-colinear.

Types:

Angle Measures for Regular Polygons

Shape Number
of Sides
Sum of Interior
Angles
Interior Angle
Measure
Exterior Angle
Measure
Triangle 3 180° 60° 300°
Pentagon 5 540° 108° 252°
Hexagon 6 720° 120° 240°
n-gon n   Circles

Definitions:

• Circle: the set of all points on a plane that are equidistant from the center.
• Radius: a line segment with endpoints on the center and any point on the circle.
• Diameter: a line segment that passes through the center of the circle and has endpoints on the circle
• Chord: a segment that joins any two points on the circle.
• Circumference: the distance around a circle.
• Arc: a part of the circle with endpoints on the circle.
• Secant: a line with two points on the circle.
• Tangent: a line with only one point on the circle. The radius through that point of the circle is perpendicular to the tangent line.
• Central Angle: the angle created by two radii.
• Inscribed Angle: an angle formed by two chords whose vertex lies on the circle,
• Intercepted Arc: an arc between two specific points on a circle.
• Concentric Circles: circles with the same center.

Angle and Arc Relationships

• A Central Angle is measured by the length of the intercepted arc
• An Inscribed Angle is measured by one half the length of the intercepted arc
• Chord and Tangent Angles: angles formed by a chord and a tangent lines are measured by half the length of the arcs they intercept.
• Interior Chord Angles: the angle formed by two intersecting chords within a circle; angle is measured by one half the sum of the intercepted arc lengths.
• Exterior Chord Angles: the angle formed by two chords that intersect outside the circle; angle is measured by one half the difference of the intercepted arc lengths

Formulas

• Equation of a circle: a circle with a center at point (h, k) and a radius of r will have the equation .
• Circumference: the distance around a circle is equal to pi times two times the radius or • Area: the area of a circle is equal to pi times the radius squared or Transformations

Definitions

• Image: the point, line or figure that results from a transformation
• Pre-Image: the original point, line or figure

Types

• Isometry: a transformation in which the image is congruent to the pre-image
• Reflection: points, lines or figures are mirrored or flipped across a point, line or plane
• Rotation: points, lines or figures are turned around a point at a specified angle
• Translation: points, lines or figures are moved a specific distance in a specific direction
• Composition: a series of transformations
• Symmetry: an isometry that maps a figure onto itself

Triangles

Types

• Right Triangle: a triangle with a 90° angle.
• Acute Triangle: a triangle in which all angles are less than 90°.
• Obtuse Triangle: a triangle with one angle greater than 90°.
• Equiangluar Triangle: a triangle in which all angles have the same measure.
• Equilateral Triangle: a triangle constructed of three equal sides.
• Isosceles Triangle: a triangle in which two sides are equal or congruent.
• Scalene Triangle: a triangle in which all sides are different lengths.

Terms

• Altitude: a perpendicular line from the vertex of one angle to the side opposite the angle.
• Median: a line segment connecting the vertext of one angle to the midpoint of the side opposite the angle.
• Incenter: the point at which the angle bisectors of a triangle intersect.
• Circumcenter: the point at which the perpendicular bisectors of each side of the triangle intersect.
• Centroid: the point at which all three medians of a triangle intersect.
• Orthocenter: the point at which all three altitudes of a triangle intersect.

Formulas

• Area: the area of a triangle is equal to one half the base times the height or • Perimeter: the distance around a triangle is equal to the sum of the three sides or Congruence

• Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
• Angle-Side-Angle (ASA): Triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding angles and side of another triangle.
• Angle-Angle-Side (AAS): Triangles are congruent if two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle.
• Side-Angle-Side (SAS): Triangles are congruent if two sides and the included angle of one triangle are congruent to the corresponding sides and angle of another triangle.
• Side-Side-Side (SSS): Triangles are congruent if all three sides of one triangle and the corresponding sides of another triangle are congruent.

Similarity

• Angle-Angle-Angle (AAA): Triangles are congruent if all angles of one triangle are congruent to the corresponding angles of another triangle.
• Corresponding parts of similar triangles are proportionate to each other.

Inequality of Triangles

• Triangle Inequality Theorem: the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
• Sides and Angles: if the measure of one side of a triangle is greater than the measure of a second side, then the angle opposite the first side is greater than the angle opposite the second side.
• Angles and Sides: if the measure of one angle in a triangle is greater than the measure of a second angle, then the side oppostie the first angle is greater than the side opposite the second angle.
• Exterior Angle Inequality Theorem: the measure of an exterior angle on a triangle is greater than the measure of either of the non-adjacent interior angles.
• Hinge Theorem: If two sides of a triangle A are congruent to two sides of triangle B and the angle between the two sides on triangle A is greater than the angle between the two sides on triangle B, then the third side of triangle A is greater than the third side of triangle B.

Polyhedron

Definitions and Formulas

• Polyhedron: a three dimensional figure constructed of polygons.
• Faces: the polygons that form the sides of the polyhedron
• Edges: the line segment formed where two polygons intersect
• Vertices: the point at which the edges intersect to form corners
• Euler’s Formula: the sum of the number of faces and vertices of a polyhedron is equal to two more than the number of edges or • Platonic Solids: the five polyhedron that are constructed of only regular polygons. See Table Below
Polygon Faces Edges Vertices
Tetrahedron Triangle 4 6 4
Cube Square 6 12 8
Octahedron Triangle 8 12 6
Dodecahedron Pentagon 12 30 20
Icosahedron Triangle 20 30 12

Types

• Parallelogram: a four sided figure made of two sets of parallel lines.
• Rhombus: a parallelogram with four equal sides.
• Rectangle: a parallelogram with four right angles.
• Square: a parallelogram with four right angles and four equal sides.
• Trapezoid: a quadrilateral with one set of parallel lines.
• Kite: a quadrilateral with two pairs of congruent sides that are adjacent.

Right Triangle

Pythagorean Theorem

• Definitions: the sum of each leg squared is equal to the hypotenuse squared. • Formula: • Converse of the Pythagorean Theorem:
• If , then triangle ABC is a right triangle
• If , then triangle ABC is an acute triangle
• If , then triangle ABC is an obtuse triangle

Special Right Triangle Trigonometric Ratios • • • • • • 