Geometry as Shape Exam Review Notes

Foundations of Geometry

• Euclid: Known for writing the book "The Elements," which organized all the geometric information known in his time.

• Point: Defined as an exact location in space.

• Line: Defined as an infinite set of points with one dimension ($1D$). It is denoted as $\overleftrightarrow{AB}$.

• Plane: Defined as an infinite set of points with two dimensions ($2D$).

• Collinear Points: Two or more points that lie on the same line.

• Coplanar Points: Points that lie on the same plane.

• Line Segment: A finite part of a line with two endpoints. It is denoted as $\overline{AB}$.

• Ray: A "half line," which is infinite on one side and finite (has an endpoint) on the other. It is denoted as $\overrightarrow{AB}$.

Relationships Between Lines

• Intersecting Lines: Lines that meet or cross at a specific point.

• Perpendicular Lines: Lines that meet to form a right angle, measuring exactly $90^{\circ}$.

• Parallel Lines: Lines located in the same plane that never meet, regardless of how far they are extended.

• Skew Lines: Lines located in different planes that do not meet.

Measuring and Classifying Angles

• Angle Classifications by Measure:

  • Acute Angle: An angle measuring less than $90^{\circ}$ ($x < 90^{\circ}$). Examples from the text: $33^{\circ}$, $88^{\circ}$, $62^{\circ}$.

  • Right Angle: An angle measuring exactly $90^{\circ}$.

  • Obtuse Angle: An angle measuring more than $90^{\circ}$ but less than $180^{\circ}$ ($90^{\circ} < x < 180^{\circ}$). Examples from the text: $124^{\circ}$, $172^{\circ}$, $110^{\circ}$, $95^{\circ}$, $159^{\circ}$, $178^{\circ}$.

  • Straight Angle: An angle measuring exactly $180^{\circ}$.

  • Reflex Angle: An angle measuring greater than $180^{\circ}$. Examples from the text: $310^{\circ}$, $304^{\circ}$, $250^{\circ}$, $195^{\circ}$.

• Angle Relationships:

  • Complementary Angles: Two angles whose measures sum to $90^{\circ}$. For example, the complement of $42^{\circ}$ is $48^{\circ}$, since $90 - 42 = 48$.

  • Supplementary Angles: Two angles whose measures sum to $180^{\circ}$. For example, the supplement of $105^{\circ}$ is $75^{\circ}$, since $180 - 105 = 75$.

  • Vertical Angles: Pairs of opposite angles formed by intersecting lines. They are always congruent.

  • Adjacent Angles: Two angles that share a common vertex and a common side but have no common interior points.

  • Linear Pairs: Adjacent angles formed by intersecting lines whose non-common sides are opposite rays. Linear pairs are always adjacent angles.

Geometric Logic and Curves

• Van Hiele Levels of Understanding Geometry:

  1. Level 0/1: Reasoning by Resemblance: Recognition of shapes based on physical appearance.

  2. Level 2: Reasoning by Attributes: Analyzing shapes based on their specific properties and characteristics.

  3. Level 3: Formal Reasoning: Understanding logical deductions and how properties are related.

  4. Level 4: Axiomatic System: Understanding the formal foundations of geometry.

• Simple Closed Curve: A curve that starts and ends at the same point without crossing itself.

• Polygon: A simple closed curve made up entirely of line segments.

  • Is a circle a polygon? No, because it is not made of line segments; it is a curved shape.

Triangles and Special Centers

• Triangle Interior Angles: The sum of the interior angles of any triangle is exactly $180^{\circ}$.

  • A triangle cannot have two right angles because the sum would be $180^{\circ}$ before the third angle is added ($90 + 90 = 180$), leaving no degrees for a third vertex.

  • A triangle cannot have two obtuse angles for the same reason (the sum would exceed $180^{\circ}$).

• Significant Lines and Centers of a Triangle:

  • Median: A line from a vertex to the midpoint of the opposite side.

  • Centroid: The point where the three medians of a triangle meet.

  • Angle Bisector: A line that divides an angle of the triangle into two congruent angles.

  • Incenter: The point where the three angle bisectors meet.

  • Perpendicular Bisector: A line segment that is perpendicular to a side of the triangle at its midpoint. It does not necessarily pass through the vertex.

  • Circumcenter: The point where the three perpendicular bisectors meet.

  • Altitude (Height): A line from a vertex to the opposite side that is perpendicular to that side.

  • Orthocenter: The point where the three altitudes meet.

Quadrilaterals: Parallelograms and Beyond

• Properties of a Parallelogram:

  1. Opposite sides are parallel ($AB \parallel DC$ and $AD \parallel BC$).

  2. Opposite sides are congruent ($\overline{AB} \cong \overline{DC}$ and $\overline{AD} \cong \overline{BC}$).

  3. Opposite angles are congruent.

  4. Consecutive interior angles are supplementary (sum to $180^{\circ}$).

  5. Diagonals bisect each other.

  6. Each diagonal divides the parallelogram into two congruent triangles.

• Special Parallelograms:

  • Rectangle: A parallelogram with four right angles and congruent diagonals.

  • Rhombus: A parallelogram with four congruent sides, perpendicular diagonals, and diagonals that bisect the interior angles.

  • Square: A regular quadrilateral that is a parallelogram, a rectangle, and a rhombus. It possesses all 11 properties listed above.

• Other Quadrilaterals:

  • Trapezoid: A quadrilateral with only one pair of opposite sides parallel.

  • Isosceles Trapezoid: A trapezoid where the non-parallel sides (legs) are congruent.

  • Kite: A quadrilateral with two distinct pairs of adjacent sides that are congruent.

General Polygons

• Polygon Classifications:

  • Regular Polygon: A polygon where all sides are congruent and all interior angles are congruent (equilateral and equiangular).

  • Concave Polygon: A polygon with at least one interior angle greater than $180^{\circ}$ (the shape appears to "cave in").

  • Convex Polygon: A polygon where all interior angles are less than $180^{\circ}$.

• Polygon Naming Convention:

  • 3 sides: Triangle

  • 4 sides: Quadrilateral

  • 5 sides: Pentagon

  • 6 sides: Hexagon

  • 8 sides: Octagon

  • 10 sides: Decagon

  • 12 sides: Dodecagon

• Formulas for Polygons:

  • Sum of Interior Angles: $(n - 2) \times 180^{\circ}$, where $n$ is the number of sides.

    • If $n = 8$ (Octagon): $(8 - 2) \times 180 = 6 \times 180 = 1080^{\circ}$.

    • If $n = 18$: $(18 - 2) \times 180 = 16 \times 180 = 2880^{\circ}$.

  • Sum of Exterior Angles: Always $360^{\circ}$ for any convex polygon.

  • Finding Number of Sides from Interior Angle: If an interior angle is $140^{\circ}$, the exterior angle is $180 - 140 = 40^{\circ}$. Since exterior angles sum to $360$, the number of sides is $\frac{360}{40} = 9$ sides.

Circle Geometry

• Circle Definition: An infinite set of points in a plane that are all the same distance from a fixed point called the center.

• Components of a Circle:

  • Radius: A segment from the center to any point on the circle (e.g., $OA$, $OC$, $OB$).

  • Diameter: A chord that passes through the center ($AoB$).

  • Chord: A segment whose endpoints lie on the circle ($DE$, $AO$ if it connects points).

  • Arc: A portion of the circumference (e.g., $AC$, $CB$, $B E$).

  • Tangent Line: A line that touches the circle at exactly one point ($PQ$).

  • Circumference: The distance around the circle.

Rectangular Coordinate System

• History: Named after René Descartes (Cartesian system).

• Axes:

  • Horizontal Axis: The $x$-axis, also known as the Abscissa.

  • Vertical Axis: The $y$-axis, also known as the Ordinate.

• The Origin: The point where the axes cross, written as $(0, 0)$.

• Distance Formula: To find the distance between $(x_1, y_1)$ and $(x_2, y_2)$, use: $d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

• Example: Distance from $A(4, -6)$ to $B(8, 8)$. $d = \sqrt{(8 - 4)^{2} + (8 - (-6))^{2}} = \sqrt{4^{2} + 14^{2}} = \sqrt{16 + 196} = \sqrt{212} \approx 14.56$

• Midpoint Formula: To find the midpoint between $(x_1, y_1)$ and $(x_2, y_2)$, use: $M = \left( \frac{x_{1} + x_{2}}{2} , \frac{y_{1} + y_{2}}{2} \right)$

• Example: Midpoint between $J(4, 7)$ and $K(8, 3)$. $M = \left( \frac{4 + 8}{2} , \frac{7 + 3}{2} \right) = (6, 5)$

Three-Dimensional Geometry

• Polyhedron: A three-dimensional solid made of flat polygonal faces.

• Euler’s Formula: Relates the number of faces ($F$), vertices ($V$), and edges ($E$): $F + V = E + 2$

• Example from a figure: $6 \text{ faces} + 8 \text{ vertices} = 12 \text{ edges} + 2 \rightarrow 14 = 14$.

• Prism vs. Pyramid:

  • Prism: Has two congruent polygonal bases and rectangular sides (e.g., cube, rectangular prism).

  • Pyramid: Has one polygonal base and triangular sides that meet at a single point.

• Net: A two-dimensional pattern that can be folded to create a three-dimensional object.

• Dihedral Angle: The angle formed by the intersection of two planes (or faces of a polyhedron).

• The 5 Regular Polyhedrons (Platonic Solids):

  1. Tetrahedron: 4 faces (triangles).

  2. Hexahedron (Cube): 6 faces (squares).

  3. Octahedron: 8 faces (triangles).

  4. Dodecahedron: 12 faces (pentagons).

  5. Icosahedron: 20 faces (triangles).