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

Under what conditions are similar figures congruent?
How do I know which method to use to prove two triangles congruent?
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?

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.

Centroid - The point of concurrency of the medians of a triangle.

Circumcenter - The point of concurrency of the perpendicular bisectors of the sides of a triangle.

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

Congruent - Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal.

Incenter - The point of concurrency of the angle bisectors of a triangle.

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.

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

Median of a Triangle - A segment is a median of a triangle if and only if its endpoints are a vertex of the triangle and the midpoint of the side opposite the vertex.

Orthocenter - The point of concurrency of the altitudes of a triangle.

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.

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

Rigid Motions - Motions or transformations that preserve sizes and shapes. Examples of rigid motions are reflections, translations, and rotations.

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

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