Geometric Foundations, Constructions and Proof Introduction

"beyond a reasonable doublt" proofs introductory image with judge, gavel, paper, briefcase Prove it! You always hear people talk about "proving" things to be true. They usually mean they will tell you why they are pretty sure it is true. In a courtroom, a lawyer might say they will prove someone's guilt or innocence. Their standard is "beyond a reasonable doubt". What does it mean to really prove something? Is "pretty sure" enough? If I have shown it "beyond a reasonable doubt", do I know for sure it is true? In this unit, you will discover how with simple arguments you can prove mathematical concepts to be true with no doubt at all! Using facts you already know to be true, you will build arguments to prove new ideas are also true. You will also make geometric constructions using a compass and straight-edge and be able to prove your construction works.

Essential Questions

How do I prove geometric theorems involving lines, angles, triangles, and parallelograms?
In what ways can I use congruent triangles to justify many geometric constructions?
How do I make geometric constructions?
How do we use coordinates to compute the perimeters of polygons?
How do we use coordinates to compute areas of triangles and rectangles?

Key Terms

Adjacent Angles - Angles in the same plane that have a common vertex and a common side, but no common interior points.

Alternate Exterior Angles - Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two parallel lines. These angles are on opposite sides of the transversal and are outside the two parallel lines. Alternate exterior angles are equal.

Alternate Interior Angles - Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two parallel lines. These angles are on opposite sides of the transversal and are in between the two parallel lines. Alternate interior angles are equal.

Angle - Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ∠ABC or ∠B denote angles with vertex B.

Bisector - A bisector divides a segment or angle into two congruent parts.

Complementary Angles - Two angles whose measures add up to 90 degrees.

Corresponding Angles - The angles that occupy the same relative position at each intersection, when two lines are intersected by a third line. When the two lines are parallel, the corresponding angles are congruent.

Distance Formula = $LaTeX: \sqrt{\langle x_2-x_1\rangle^2+\langle y_2-y_1\rangle^2}$

Intersecting Lines - When two or more lines, in a plane, cross each other. Unless two lines are coincidental (overlapping), parallel, or skew, they will intersect at one point.

Intersection - The point at which two or more lines intersect (or cross).

Inscribed Polygon - A polygon is inscribed in a circle if and only if each of its vertices lie on the circle.

Line Segment - a part of a line bounded by two distinct endpoints

Linear Pair - Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line.

Parallel Lines - Two lines are parallel if they lie in the same plane and they do not intersect. Parallel lines have the same slope.

Perpendicular Bisector - A line or segment that passes through the midpoint of a second line or segment, forming a right angle.

Perpendicular - Two lines are perpendicular if they intersect and form a right angle.

Remote Interior Angles of a Triangle - The two interior angles of a triangle that are non-adjacent to the exterior angle.

Slope formula - Using two points on a line, ( LaTeX: x_1,y_1 ) and ( LaTeX: x_2,y_2 ), the formula for the slope is.

Supplementary Angles - Two angles whose measures add up to 180 degrees.

Vertical Angles - A pair of opposite angles formed by two intersecting lines. They are always congruent.

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