Geo Enriched test

SECTION A — FOUNDATIONAL CONCEPTS & REASONING

1. Building Blocks of Geometry

Points, Lines, Rays, Segments

  • A point marks a location; no size.

  • A line extends forever in both directions.

  • A ray has a starting point and extends infinitely in one direction.

  • A segment has two endpoints.

Diagram Markings

  • Tick marks → congruent segments

  • Arrows → parallel lines

  • Right-angle box → perpendicular

  • Curved arcs → angle congruence

2. Types of Reasoning

Inductive Reasoning

Finding a pattern and predicting a rule.

Example:
3, 6, 9, 12 → conjecture: multiply by 3 or add 3.

Deductive Reasoning

Using established rules, theorems, or definitions to prove statements.

Example:
If two angles form a linear pair, then they are supplementary.
This is ALWAYS true, no patterns required.

3. Conditional Logic

  • Conditional: If p, then q.

  • Converse: If q, then p.

  • True/false questions will appear.

Example:
Conditional: If a figure is a square, then it has four equal sides. (true)
Converse: If a figure has four equal sides, then it is a square. (false—could be a rhombus)

SECTION B — ANGLES & BASIC PROOF TOOLS

1. Angle Relationships

Vertical Angles

Opposite angles are equal.
Useful in early proof steps.

Linear Pairs

Two adjacent angles forming a straight line; sum = 180°.

Complementary / Supplementary

  • Complementary → 90°

  • Supplementary → 180°

2. Angle Addition & Segment Addition

Angle Addition Postulate

If point D lies in ∠ABC:

m∠ABD+m∠DBC=m∠ABCm∠ABD+m∠DBC=m∠ABC

Segment Addition

If B is between A and C:
AB + BC = AC

This always becomes an algebra problem on the final.

3. Parallel Line Angle Relationships

When two parallel lines are cut by a transversal:

Congruent pairs:

  • Corresponding

  • Alternate interior

  • Alternate exterior

Supplementary pairs:

  • Same-side interior

  • Same-side exterior

  • Linear pairs

Example Problem:
If ∠1 = 120° (corr.), find ∠5.
∠5 = 120°

SECTION C — TRIANGLES & CONGRUENCE

1. Types and Classification

  • Acute, right, obtuse

  • Equilateral, isosceles, scalene

Key isosceles fact: base angles are congruent.

2. Triangle Angle Sum

Always:

m∠A+m∠B+m∠C=180∘m∠A+m∠B+m∠C=180∘

3. Triangle Exterior Angle Theorem

Exterior angle = sum of two remote interior angles.

4. Congruence Shortcuts

The four valid congruence theorems:

  • SSS

  • SAS

  • ASA

  • AAS

Remember:
SSA ❌ does not prove congruence
AAA ❌ only proves similarity

5. CPCTC

“Corresponding Parts of Congruent Triangles are Congruent.”
Used after proving triangles congruent to show:

  • a side is equal

  • an angle is equal

  • a segment is bisected

  • a line is perpendicular

6. Triangle Inequality Theorem

For any triangle:

  • sum of any two sides > third side

Expect a problem:
“Determine if 3 lengths form a triangle.”

7. Right Triangles

Pythagorean Theorem

a2+b2=c2a2+b2=c2

Classifying by sides

  • If a² + b² = c² → right

c² → acute

  • < c² → obtuse

SECTION D — TRANSFORMATIONS & COORDINATE GEOMETRY

1. Rigid Motions

Transformations preserving size and shape:

  • Translations

  • Reflections

  • Rotations

These prove congruence.

2. Transformation Rules

Translation

(x,y)→(x+a,y+b)(x,y)→(x+a,y+b)

Reflection

  • Across x-axis: (x, –y)

  • Across y-axis: (–x, y)

  • Across line y = x: (y, x)

Rotation (counterclockwise)

  • 90° CCW: (x, y) → (–y, x)

  • 180°: (x, y) → (–x, –y)

  • 270° CCW: (x, y) → (y, –x)

Expect multi-step transformations.

3. Midpoint & Distance

Midpoint

(x1+x22,y1+y22)(2x1​+x2​​,2y1​+y2​​)

Distance

(x2−x1)2+(y2−y1)2(x2​−x1​)2+(y2​−y1​)2​

Slope

m=y2−y1x2−x1m=x2​−x1​y2​−y1​​

Parallel slopes → equal
Perpendicular slopes → negative reciprocals

SECTION E — QUADRILATERALS & PARALLELOGRAMS

1. Parallelogram Properties

  • Opposite sides → parallel & congruent

  • Opposite angles → congruent

  • Consecutive angles → supplementary

  • Diagonals → bisect each other

Expect proofs here.

2. Special Parallelograms

Rectangle

  • Right angles

  • Diagonals congruent

Rhombus

  • All sides congruent

  • Diagonals perpendicular

  • Diagonals bisect angles

Square

  • All rectangle + rhombus properties

Isosceles Trapezoid

  • Legs congruent

  • Base angles congruent

  • Diagonals congruent

3. Coordinate Classification of Quadrilaterals

You must use:

  • Slope (parallel/perpendicular)

  • Distance (congruent sides)

  • Midpoint (diagonals bisect)

Examples:

  • Proving a rhombus → all sides equal (distance)

  • Proving rectangle → diagonals congruent (distance)

SECTION F — POLYGONS & AREA

1. Interior Angle Sum

(n−2)180(n−2)180

2. Each Interior Angle (regular polygon)

(n−2)180nn(n−2)180​

3. Exterior Angle Sum

Always 360° for any polygon.

4. Area of Regular Polygons

A=12aPA=21​aP

5. Quadrilateral Areas

  • Triangle → ½bh

  • Rectangle → lw

  • Square → s²

  • Trapezoid → ½(b₁ + b₂)h

  • Parallelogram → bh

SECTION G — ALGEBRA REVIEW

1. Factoring

  • GCF

  • Trinomials

  • Difference of squares

2. Solving Quadratics

  • Factor

  • Zero Product Property

3. Systems of Equations

  • Substitution

  • Elimination

4. Multi-Step Equations with Fractions

Clear denominators first.