REM Geometry

From GEOMETRY 1.0 SS - LANCELIN.

Line

Denoted by 2 points or by a lowercase letter.

Line Segment

A part of a line that is bounded by 2 distinct end points.

Ray

A part of a line that has a fixed starting point and extends infinitely in 1 direction.

Collinear Points

Points that are on the same line.

Segment Addition Postulate

If points A, X, and B are collinear points and point X is between points A and B, then AX + XB = AB.

Ruler Postulate

The point of a line can be placed in a one-to-one correspondence with the real numbers such that the distance between 2 distinct points is the absolute value of the difference between the corresponding real numbers.

Angle Addition Postulate

If point X is in the interior of ∠AMB then m∠AMX + m∠XMB = m∠AMB.

Corresponding Angles

2 congruent angles that are in the same position on parallel lines in relation to a transversal.

Midpoint of a Segment

Divides a segment into 2 congruent segments.

Segment Bisector

A line, ray, or line segment that cuts a line segment into 2 equal parts.

Right Triangle

A triangle containing an interior right angle (which measures 90°).

Isoceles Triangle

A triangle containing at least 2 equal length sides and 2 equal interior angle measures; Equilateral triangles can also be isosceles triangles, but not every isosceles triangle is an equilateral.

Equidistant

At equal distances.

Angle Bisector

A line, ray, or line segment that cuts an angle into 2 equal parts.

Reflectional Symmetry

When reflecting a pattern through a line is used to create symmetry.

Rotational Symmetry

When rotating a pattern around a point is used to create symmetry.

Translational Symmetry

When sliding a pattern is used to create symmetry.

Tessellation

An arrangement of shapes closely fitted together in a repeated pattern without gaps or overlaps.

Complementary Angles

2 angles with measures that sum to 90 degrees.

Congruent Complements Theorem

States that complements (complementary angles) of congruent angles are congruent.

Corollary

A theorem connected by a short proof to an existing theorem, which can include converses of theorems; It is a simple deduction from a theorem or postulate that requires only a few simple statements in addition to the proof of the original theorem or postulate.

Corollary to Congruent Complements Theorem

States that complements (complementary angles) of the same angle are congruent.

Supplementary Angles

2 angles with measures that sum to 180°.

Congruent Supplements Theorem

States that supplements (supplementary angles) of congruent angles are congruent.

Corollary to Congruent Supplements Theorem

States that supplements (supplementary angles) of the same angle are congruent.

Linear Pair

2 adjacent angles formed by two intersecting lines.

Linear Pair Postulate

States that if 2 pairs form a linear pair, then they are supplementary.

Vertical Angles

Opposite angles that share the same vertex. They are formed by intersecting lines, and their angle measures are equal.

Vertical Angles Theorem

States that if 2 lines intersect, then vertical angles are congruent.

Transversal

A line that intersects 2 or more lines on the same plane at different points.

Corresponding Angles

Angles that are in the same position on parallel lines in relation to the transversal when parallel lines are cut by a transversal.

Corresponding Angles Theorem

States that if 2 parallel lines are intersected by a transversal, then the corresponding angles are congruent.

Alternate Exterior Angles Theorem

States that if 2 parallel lines are intersected by a transversal, then alternate exterior angles are congruent.

Alternate Interior Angles Theorem

States that if 2 parallel lines are intersected by a transversal, then alternate interior angles are congruent.

Consecutive Interior Angles Theorem

States that if 2 parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.

Consecutive Exterior Angles Theorem

States that if 2 parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are supplementary.

Transformations

Can be described using words or can be described algebraically.

The Origin

Referenced to (0, 0) on the coordinate plane.

Translating to the Right

Add to the x-coordinate (x,y) → (x + #, y).

Translating to the Left

Subtract from the x-coordinate (x, y) → (x - #, y).

Translating Up

Add to the y-coordinate (x, y) → (x, y + #).

Translating Down

Subtract from the y-coordinate (x, y) → (x, y - #).

Rotations of 90 Degrees Clockwise

(x, y) → (y, -x).

Rotations of 180 Degrees Clockwise

(x, y) → (-x, -y).

Rotations of 270 Degrees Clockwise

(x, y) → (-y, x).

Rotations of 90 Degrees Counterclockwise

(x, y) → (-y, x).

Rotations of 180 Degrees Counterclockwise

(x, y) → (-x, -y).

Rotations of 270 Degrees Counterclockwise

(x, y) → (y, -x).

Reflections Across the X-Axis

The y-coordinate changes to its opposite (x, y) → (x, -y)..

Reflections Across the Y-Axis

The x-coordinate changes to its opposite (x, y) → (-x, y).

Reflections Across y = x

The coordinates switch places (x, y) → (y, x).

Reflections Across y = -x

The coordinates switch places and become opposites (x, y) → (-y, -x).

Isometry

A transformation that preserves distance and angle measure.

Definition of Congruence in Terms of Rigid Motions

States that 2 figures are congruent if and only if there exists 1 or more rigid motions that will map 1 figure onto the other.

Side-Side-Side (SSS) Congruence Postulate

States that 2 triangles are congruent if the 3 sides of 1 triangle are congruent to the 3 sides of the other, respectively.

Side-Angle-Side (SAS) Congruence Postulate

States that if 2 sides and the angle in between them of 1 triangle are congruent to 2 sides and the angle in between them of a second triangle, respectively, then the 2 triangles are congruent.

Angle-Side-Angle (ASA) Congruence Theorem

States that if 2 angles and an included side of 1 triangle are congruent respectively to 2 angles and an included side of a second triangle, then the 2 triangles are congruent.

Angle-Angle-Side (AAS) Congruence Theorem

States that if 2 angles and a side opposite 1 of them are congruent to 2 angles and the corresponding side of another triangle, then the triangles are congruent.

Hypotenuse-Leg (HL) Theorem

States that 2 right triangles are congruent if the hypotenuse and a leg of 1 triangle are congruent to the hypotenuse and a leg of the other.

CPCTC

Abbreviation for “Corresponding Parts of Congruent Triangles are Congruent.”

CPCPC

Abbreviation for “Corresponding Parts of Congruent Polygons are Congruent.”