Geometry Final

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60 Terms

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Point

Location with no size or dimension

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Line

Series of points extending infinitely in both directions — 1-dimensional

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Minimum number of points for a plane

At least 3 non-collinear points (not all in a straight row)

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Postulate

"Through any two points there is exactly 1 line." (accepted without proof)

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Theorem

"Vertical angles are congruent." (proved from postulates and previous theorems)

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Ray

Starts at a point and goes forever in 1 direction

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Angle

Formed by two rays with a shared endpoint (vertex)

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Construct a line

Connect two points with a straight-edge

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Construct an angle

Draw two rays with a shared starting point

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Construct a parallel line

Use compass and straight-edge (copy angle)

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Construct a perpendicular line

Draw a circle to create intersection points and connect

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Bisect a segment

Compass from endpoints — intersection forms midpoint

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Bisect an angle

Compass to cut angle in half

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Construct a triangle/polygon/hexagon within a circle

Draw circle, then mark points evenly with compass

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Undefined terms

Base for all geometry (point, line, plane)

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Defined terms

Explained using undefined terms (ray, segment, angle)

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Prime (')

Transformation's result (image)

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Preimage

The original figure

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Translation

Moves without turning or reflecting

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Reflection

Flipped over a mirror line

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Rotation

Turning about a point

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Dilation

Enlarging or shrinking, but retaining shape

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Congruence

Figures are congruent if their sizes and shape are identically matching.

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CPCTC

After proving triangles congruent, their corresponding components are congruent

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Similarities/Differences (translation, reflection, rotation)

All preserve shape and size, but reflection reverses orientation.

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Proof of congruence

Combine rigid transformation with congruence criteria (SSS, SAS, etc.).

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Vertical Angles

Equal.

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Corresponding Angles

Equal (with parallel lines).

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Alternate Interior

Equal (with parallel lines).

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Alternate Exterior

Equal (with parallel lines).

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Consecutive/Same-side Angles

Supplementary (with parallel lines).

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Supplementary Angles

Add to 180º.

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Complementary Angles

Add to 90º.

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Equidistant

Equal distance from something (like lines or points).

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Transversal

Crosses two lines.

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Identifying Parallel Lines

If corresponding/alt. interior/alt. exterior are equal or consecutive are supplementary — lines are parallel.

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Associative Property

(a + b) + c = a + (b + c).

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Reflexive Property

a = a.

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Substitution Property

If a = b, then a can replace b.

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Transitive Property

If a = b and b = c, then a = c.

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Proof Types

Paragraph: Explain in words; Two column: List statements & reasons side by side; Flowchart: Visual representation of logical flow.

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Vertical Angles Theorem

If two lines intersect, their vertical pairs are equal (proof by intersection, linear pairs adding to 180º).

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Properties of Parallelograms

Opposite sides are congruent; Opposite angles are congruent; Diagonals bisect each other.

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k (similar figures)

scale factor (image side/preimage side)

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Preimage vs Image

Preimage = original; Image = after transformation

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Constructing similar figures

dilate by scale factor with center of dilation

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Not a dilation

if scale factors differ in directions

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Identify congruent angles

matching corner to corner

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Finding side lengths

apply scale factor

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How to dilate a figure

(x, y) → (k∙x, k∙y)

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Determining if two triangles are similar

AA, SSS (with ratios), or SAS (with ratio + included angle)

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SAS similarity postulate

if two pairs of sides are in ratio and included angle is equal

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SSS similarity postulate

if all side ratios are equal

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Triangle Proportionality Thm

a segment parallel to a side divides the other two side proportions

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Converse Triangle Proportional Thm

if segments cut proportions, lines are parallel

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Right Triangle Similarity Thm

Altitude forms two triangles, each is similar to the original

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Converse Right Triangle Similar Thm

if relationships match, then triangles are similar

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Pythagorean Thm

a² + b² = c² (right-angled)

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Proving base angle congruence (isosceles trapezium)

base angles are equal due to reflection or diagonal properties

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Proof (kite)

its long diagonal is perpendicular to its short diagonal; adjacent sides are congruent by reflection or by triangle congruence