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Point
Location with no size or dimension
Line
Series of points extending infinitely in both directions — 1-dimensional
Minimum number of points for a plane
At least 3 non-collinear points (not all in a straight row)
Postulate
"Through any two points there is exactly 1 line." (accepted without proof)
Theorem
"Vertical angles are congruent." (proved from postulates and previous theorems)
Ray
Starts at a point and goes forever in 1 direction
Angle
Formed by two rays with a shared endpoint (vertex)
Construct a line
Connect two points with a straight-edge
Construct an angle
Draw two rays with a shared starting point
Construct a parallel line
Use compass and straight-edge (copy angle)
Construct a perpendicular line
Draw a circle to create intersection points and connect
Bisect a segment
Compass from endpoints — intersection forms midpoint
Bisect an angle
Compass to cut angle in half
Construct a triangle/polygon/hexagon within a circle
Draw circle, then mark points evenly with compass
Undefined terms
Base for all geometry (point, line, plane)
Defined terms
Explained using undefined terms (ray, segment, angle)
Prime (')
Transformation's result (image)
Preimage
The original figure
Translation
Moves without turning or reflecting
Reflection
Flipped over a mirror line
Rotation
Turning about a point
Dilation
Enlarging or shrinking, but retaining shape
Congruence
Figures are congruent if their sizes and shape are identically matching.
CPCTC
After proving triangles congruent, their corresponding components are congruent
Similarities/Differences (translation, reflection, rotation)
All preserve shape and size, but reflection reverses orientation.
Proof of congruence
Combine rigid transformation with congruence criteria (SSS, SAS, etc.).
Vertical Angles
Equal.
Corresponding Angles
Equal (with parallel lines).
Alternate Interior
Equal (with parallel lines).
Alternate Exterior
Equal (with parallel lines).
Consecutive/Same-side Angles
Supplementary (with parallel lines).
Supplementary Angles
Add to 180º.
Complementary Angles
Add to 90º.
Equidistant
Equal distance from something (like lines or points).
Transversal
Crosses two lines.
Identifying Parallel Lines
If corresponding/alt. interior/alt. exterior are equal or consecutive are supplementary — lines are parallel.
Associative Property
(a + b) + c = a + (b + c).
Reflexive Property
a = a.
Substitution Property
If a = b, then a can replace b.
Transitive Property
If a = b and b = c, then a = c.
Proof Types
Paragraph: Explain in words; Two column: List statements & reasons side by side; Flowchart: Visual representation of logical flow.
Vertical Angles Theorem
If two lines intersect, their vertical pairs are equal (proof by intersection, linear pairs adding to 180º).
Properties of Parallelograms
Opposite sides are congruent; Opposite angles are congruent; Diagonals bisect each other.
k (similar figures)
scale factor (image side/preimage side)
Preimage vs Image
Preimage = original; Image = after transformation
Constructing similar figures
dilate by scale factor with center of dilation
Not a dilation
if scale factors differ in directions
Identify congruent angles
matching corner to corner
Finding side lengths
apply scale factor
How to dilate a figure
(x, y) → (k∙x, k∙y)
Determining if two triangles are similar
AA, SSS (with ratios), or SAS (with ratio + included angle)
SAS similarity postulate
if two pairs of sides are in ratio and included angle is equal
SSS similarity postulate
if all side ratios are equal
Triangle Proportionality Thm
a segment parallel to a side divides the other two side proportions
Converse Triangle Proportional Thm
if segments cut proportions, lines are parallel
Right Triangle Similarity Thm
Altitude forms two triangles, each is similar to the original
Converse Right Triangle Similar Thm
if relationships match, then triangles are similar
Pythagorean Thm
a² + b² = c² (right-angled)
Proving base angle congruence (isosceles trapezium)
base angles are equal due to reflection or diagonal properties
Proof (kite)
its long diagonal is perpendicular to its short diagonal; adjacent sides are congruent by reflection or by triangle congruence