Review of the Assumptions

New York State Common Core Math Geometry, Module 1, Lesson 33

Worksheets for Geometry

Student Outcomes

• Students examine the basic geometric assumptions from which all other facts can be derived.
• Students review the principles addressed in Module 1.

Review of the Assumptions

Classwork

Review Exercises

We have covered a great deal of material in Module 1. Our study has included definitions, geometric assumptions, geometric facts, constructions, unknown angle problems and proofs, transformations, and proofs that establish properties we previously took for granted.

In the first list below, we compile all of the geometric assumptions we took for granted as part of our reasoning and proof-writing process. Though these assumptions were only highlights in lessons, these assumptions form the basis from which all other facts can be derived (e.g., the other facts presented in the table). College-level geometry courses often do an in-depth study of the assumptions.

The latter tables review the facts associated with problems covered in Module 1. Abbreviations for the facts are within brackets.

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Geometric Assumptions (Mathematicians call these axioms.)

1. (Line) Given any two distinct points, there is exactly one line that contains them.
2. (Plane Separation) Given a line contained in the plane, the points of the plane that do not lie on the line form two sets, called half-planes, such that
   a. Each of the sets is convex.
   b. If 𝑃 is a point in one of the sets and 𝑄 is a point in the other, then 𝑃𝑄 intersects the line.
3. (Distance) To every pair of points 𝐴 and 𝐵 there corresponds a real number dist (𝐴, 𝐵) ≥ 0, called the distance from 𝐴 to 𝐵, so that
   a. dist(𝐴, 𝐵) = dist(𝐵, 𝐴)
   b. dist(𝐴, 𝐵) ≥ 0, and dist(𝐴, 𝐵) = 0 ⟺ 𝐴 and 𝐵 coincide.
4. (Ruler) Every line has a coordinate system.
5. (Plane) Every plane contains at least three noncollinear points.
6. (Basic Rigid Motions) Basic rigid motions (e.g., rotations, reflections, and translations) have the following properties:
   a. Any basic rigid motion preserves lines, rays, and segments. That is, for any basic rigid motion of the plane, the image of a line is a line, the image of a ray is a ray, and the image of a segment is a segment.
   b. Any basic rigid motion preserves lengths of segments and angle measures of angles.
7. (180° Protractor) To every ∠𝐴𝑂𝐵, there corresponds a real number 𝑚∠𝐴𝑂𝐵, called the degree or measure of the angle, with the following properties:
   a. 0° < 𝑚∠𝐴𝑂𝐵 < 180°
   b. Let 𝑂𝐵 be a ray on the edge of the half-plane 𝐻. For every 𝑟 such that 0° < 𝑟° < 180°, there is exactly one ray 𝑂𝐴 with 𝐴 in 𝐻 such that m∠𝐴𝑂𝐵 = 𝑟°.
   c. If 𝐶 is a point in the interior of ∠𝐴𝑂𝐵, then 𝑚∠𝐴𝑂𝐶 + 𝑚∠𝐶𝑂𝐵 = 𝑚∠𝐴𝑂𝐵.
   d. If two angles ∠𝐵𝐴𝐶 and ∠𝐶𝐴𝐷 form a linear pair, then they are supplementary (e.g., 𝑚∠𝐵𝐴𝐶 + 𝑚∠𝐶𝐴𝐷 = 180°).
8. (Parallel Postulate) Through a given external point, there is at most one line parallel to a given line.

Fact/Property

Two angles that form a linear pair are supplementary.

The sum of the measures of all adjacent angles formed by three or more rays with the same vertex is 360°.

Vertical angles have equal measure.

The bisector of an angle is a ray in the interior of the angle such that the two adjacent angles formed by it have equal measure.

The perpendicular bisector of a segment is the line that passes through the midpoint of a line segment and is perpendicular to the line segment.

The sum of the 3 angle measures of any triangle is 180°.

When one angle of a triangle is a right angle, the sum of the measures of the other two angles is 90°.

An exterior angle of a triangle is equal to the sum of its two opposite interior angles.

Base angles of an isosceles triangle are congruent.

All angles in an equilateral triangle have equal measure. [equilat. △]

The facts and properties in the immediately preceding table relate to angles and triangles. In the table below, we review facts and properties related to parallel lines and transversals

If a transversal intersects two parallel lines, then the measures of the corresponding angles are equal.

If a transversal intersects two lines such that the measures of the corresponding angles are equal, then the lines are parallel.

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary

If a transversal intersects two lines such that the same side interior angles are supplementary, then the lines are parallel.

If a transversal intersects two parallel lines, then the measures of alternate interior angles are equal.

If a transversal intersects two lines such that measures of the alternate interior angles are equal, then the lines are parallel.

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