Foundations of Geometric Shapes

Point- Points are the basis for all geometric shapes. There are many different definitions for a point, and many mathematicians leave points undefined in order use them as an axiom (an unproven starting point). On a coordinate plane (a graph), a point is an ordered pair of x and y coordinates that give you an exact location.

Line Segment- If you take two points and place an infinite amount of points between them you get a line segment.

Line- A line is like a line segment except there are no endpoints. A line goes on forever in two directions.

Ray- A ray has a beginning point, but continues in only one direction forever


Plane- If you take all the points between two distinct lines or line segments you have a plane.

This website has a ton of graphics, activities, and explanations that could be useful when explaining to students or even having them explore on their own.

Following are the Common Core State Standards for Geometry. Each red heading is a domain, or group of related standards, followed by a theme and then the standards themselves.


Experiment with transformations in the plane

  • G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  • G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
  • G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  • G-CO.4. Develop definitions of rotations, reflections, and translations

    in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  • G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Angle- the measurement of the space between two rays or lines that diverge from a common point. Angles are measured in degrees and are labeled based on three main components: two legs and the vertex. The vertex is the common point that the lines or rays share. The legs of the angle refer to the rays on either side of the vertex. There are also six specific types of angles, as demonstrated in the picture below.


Circle- a round plane figure whose edges consists of points which are all the same distance from a fixed center point called the vertex. The distance around the circle is called the circumference. The distance from one side of the circle to the opposite side through the vertex is called the diameter. Half of the diameter is called the radius.

Perpendicular Line- when two lines are perpendicular, their intersection forms a 90 degree angle. The slope of the lines are opposite reciprocals of each other. See picture below for example.

Parallel Line- two lines are parallel when their slopes are exactly the same. It then logically follows that the two lines will never cross.

Understand congruence in terms of rigid motion

  • G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
  • G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
  • G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Rigid motion is defined as a transformation consisting of translations, reflections, and rotations that leave a given arrangement unchanged. Or in other words, the size and shape of the object is the same after the transformation as it was before the transformation. The picture below shows an example.

The red polygon is the original shape. The polygon was shifted left (blue) and then rotated (purple), but is still the same size and shape as the red polygon. Notice that the lengths of the sides have not changed and neither have the angles at each of the vertices. Therefore, we can conclude that the polygons are congruent.

This website is great for helping students to begin writing proofs. The page provides a "fill in the blank" proof setup and goes in steps until the proof is complete.

Prove Geometric Theorems

  • G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
  • G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
  • G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

triangles 2.PNG

Make geometric constructions

  • G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
  • G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.