Geometry – Quick Reference Sheet
Point, Lines and Planes
- 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.
- Distance Formula ⇒
- Midpoint Formula ⇒
- Slope ⇒
- Equation of a line (point-slope form) ⇒
- Equation of a line (slope-intercept form) ⇒
- 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.
- Regular: a polygon in which sides are congruent (equilateral) and interior angles are congruent (equiangular)
- Convex Polygon: a polygon in which any line segment connecting two endpoints lies on the interior of the polygon.
- Concave Polygon: a polygon in which one or more line segments connecting two endpoints does not lie on the interior of the polygon.
Angle Measures for Regular Polygons
|Sum of Interior
- 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
- 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
- Image: the point, line or figure that results from a transformation
- Pre-Image: the original point, line or figure
- 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
- 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.
- 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.
- 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
- 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.
- 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.
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
- 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.
- Definitions: the sum of each leg squared is equal to the hypotenuse squared.
- 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