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Lesson Plans and Worksheets for Geometry

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More Lessons for Geometry

Common Core 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.

**Geometric Assumptions (Mathematicians call these axioms.)**

- (Line) Given any two distinct points, there is exactly one line that contains them.
- (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 π is a point in one of the sets and π is a point in the other, then ππ intersects the line. - (Distance) To every pair of points π΄ and π΅ there corresponds a real number dist (π΄, π΅) β₯ 0, called the distance from
π΄ to π΅, so that

a. dist(π΄, π΅) = dist(π΅, π΄)

b. dist(π΄, π΅) β₯ 0, and dist(π΄, π΅) = 0 βΊ π΄ and π΅ coincide. - (Ruler) Every line has a coordinate system.
- (Plane) Every plane contains at least three noncollinear points.
- (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. - (180Β° Protractor) To every β π΄ππ΅, there corresponds a real number πβ π΄ππ΅, called the degree or measure of the
angle, with the following properties:

a. 0Β° < πβ π΄ππ΅ < 180Β°

b. Let ππ΅ be a ray on the edge of the half-plane π». For every π such that 0Β° < πΒ° < 180Β°, there is exactly one ray ππ΄ with π΄ in π» such that mβ π΄ππ΅ = πΒ°.

c. If πΆ is a point in the interior of β π΄ππ΅, then πβ π΄ππΆ + πβ πΆππ΅ = πβ π΄ππ΅.

d. If two angles β π΅π΄πΆ and β πΆπ΄π· form a linear pair, then they are supplementary (e.g., πβ π΅π΄πΆ + πβ πΆπ΄π· = 180Β°). - (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 360Β°.

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 180Β°.

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

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. β³]

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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