Review of the Assumptions
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.)
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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