Geometry basics: angles and rays

• This section reviews basic geometrical terms and concepts for studying geometry.

Ray (or Half-Line)

• A ray is part of a line extending in one direction from a point called the endpoint.
• The endpoint is labeled as 0.

Angle Definition

• An angle is formed by two rays sharing the same endpoint.
• The rays are the sides of the angle.
• The shared endpoint is the vertex.

Directed Angles

• A directed angle considers the order of the sides.
• The initial side, OA, rotates to the terminal side, OB.
• Rotation direction indicated by an arrow and Greek letters (α, β, δ).
• Counterclockwise rotation is positive.
• Clockwise rotation is negative.
• The amount of rotation is not restricted.
• One complete revolution returns the terminal side to its start.

Angle Measurement: Degree Measure and Radian Measure

• Degree measure assigns 360° to a complete counterclockwise rotation.
• One degree equals sixty minutes (1° = 60').
• One minute equals sixty seconds (1' = 60").
• A right angle measures 90°.
• A small square at the vertex indicates a right angle.
• An acute angle is less than 90° (0° < α < 90°).
• An obtuse angle is greater than 90° but less than 180° (90° < β < 180°).

Radian Measure

• One radian is the angle cutting off an arc of one unit length from the unit circle.
• The circumference of a circle with radius R is 2πR.
• The unit circle's circumference is 2π.
• 2π corresponds to 360°.
• 1 radian corresponds to 360°/2π or approximately 57°17′44.8″.
• 1 degree corresponds to π/180 radians or approximately 0.01745 radians.

Angle Pairs: Definitions

• When two straight lines meet, they form 4 angles.

Vertical Angles

• Vertical angles are opposite angles formed by intersecting lines.
• Vertical angles are equal (α = γ, β = δ).

Adjacent Angles

• Adjacent angles share a vertex and a common ray.
• The sum of adjacent angles is 180° (α + β = 180°).

Complementary Angles

• Two angles are complementary if their sum is 90° (α + β = 90°).

Plane Figures: Triangles

• A triangle is a plane figure bounded by three straight lines.
• An isosceles triangle has two sides of equal length.
• An equilateral triangle has all sides equal in length.
• The sum of the angles in a triangle is 180° (α + β + γ = 180°).
• Pythagorean Theorem applies to right triangles: a² + b² = c².
• The area of a triangle is A = 1/2 * (base * altitude).
• The perimeter of a triangle is p = a + b + c.

Plane Figures: Quadrilaterals

• A quadrilateral is a plane figure bounded by four straight lines.
• A parallelogram is a quadrilateral with opposite sides parallel.
• A rectangle is a parallelogram with all angles as right angles.
• A square is a rectangle with all sides of equal length.
• A rhombus is a parallelogram with all sides equal.
• A trapezium is a quadrilateral with two parallel sides.
• The sum of the angles in any quadrilateral is 360° (α + β + γ + δ = 360°).
• The area of a parallelogram is A = base * altitude = b * h.
• The area of a trapezium is A = 1/2 * (b₁ + b₂) * h.

Circle Definition

• A circle is formed by a point moving so its distance from a fixed point is constant.
• The fixed point is the center.
• The constant distance is the radius (R).
• The bounding line is the circumference.
• A diameter is straight line through the centre joining two points on the circumference
• A chord is straight line joining any two points on the circumference
• Circumference = 2πR.
• Area = πR².
• An arc is a part of the circumference.
• A chord divides the circle into two segments.
• A sector is a part of a circle enclosed by two radii and an arc.

Circle Sectors and Segments

• Radius R, includes angle α radians
• Angle α is subtended at the center by arc of length L.
• A complete revolution is 360° = 2π radians.
• Arc Length: L = Rα.
• Sector Area: A = (L * R)/2 or A = (R² * α)/2
• Segment area calculated by the difference of area of a sector and an isosceles triangle.

Geometric Solids

• Three-dimensional figure consisting of intersecting polygons called faces.
• Two faces meet in a line called an edge.
• Two edges meet at a vertex or corner.
• The surface area is the sum of the areas of the faces.

Parallelepiped

• All faces are rectangles.
• Surface Area = 2(ab + ac + bc).
• Volume = a * b * c.

Right Prism

• Bases in parallel planes, other faces are rectangles.

Square Prism

• Surface Area = 2a² + 4ab
• Volume = a² * b

Triangular Prism

• Surface Area = (a+b+c).h + 2 * (area of base)
• Volume = area of base * h

Right Circular Cylinder

• Bases are circles in parallel planes.
• Surface area = 2πr² + 2πrh = 2πr(r + h)
• Volume = πr²h
• where r = radius of base, h = height

Right Pyramid

• Base is a polygon.
• Faces are triangles with a common vertex (apex).
• The normal from the apex to the base passes through the center of symmetry of the base (for right pyramids).
• Slant edge: distance from apex to a corner of the base.
• Slant height: Height of one of the triangular faces.
• Surface Area = area of base + sum of the areas of the triangles.
• Volume = (area of base * height) / 3

Right Cone

• Like a pyramid with a circular base and curved surface.
• Slant height: Distance from apex to a point on base circumference.
• Surface area = πr² + πrs
• Volume = (πr²h) / 3

Sphere

• All surface points are equidistant from center.
• Surface Area = 4πr²
• Volume = (4/3) πr³

Trigonometry: Similar Triangles

• Two triangles with equal corresponding angles are similar.
• Corresponding sides of similar triangles are proportional.

Right Triangle Trigonometry

• A right triangle has one right angle and two acute angles.
• Two right triangles are similar if an acute angle of one equals an acute angle of the other.
• Trigonometric ratios are ratios of side lengths in a right triangle.

Definitions in a right triangle

• sine = (side opposite the angle)/ hypotenuse
• cosine = (side adjacent to the angle) / hypotenuse
• tangent = (side opposite the angle) / (side adjacent to the angle)
• cotangent = (side adjacent to the angle) / (side opposite the angle)
• cotα = 1/tanα; tangent and cotangent are reciprocal ratios.
• if α + β = 90°, α and β are complementary.
• sinβ= cosα
• cosβ = sinα
• tanβ = cot α

Trigonometric Ratios and Right Triangles

• Trig ratios find side lengths/angle measures in any right triangle given:
  • A side and acute angle
  • Hypotenuse and acute angle
  • Two Sides
  • Hypotenuse and one side

Angle of Elevation

• Considering a point above the observer's location, it's the angle between the horizontal line and line of sight to that point.

Angle of Depression

• Considering a point below the observer's location, it's the angle between the horizontal line and line of sight to that point.

Angle Subtended by an object

• The angle formed when observing an object(PQ) between point, O, is the angle POQ

Trig identities

• tan(α) = sin(α) / cos(α)
• cot(α) = cos(α) / sin(α)
• sin²(α) + cos²(α) = 1
• 1 + tan²(α) = 1/cos²(α)

Trigonometric ratios

• Trigonometric ratios of angles between 90 and 180°.
• If θ is in the supplement of α then
  • sin α= sin (180°- α)
  • cos α= - cos (180°- α)
  • tan α= - tan (180°- α)

The Law of Sines (sine rule):

• This rule applies in anytriangle ABC with sides a b, c and corresponding angles α β and re
  • a /sin α = b /sin β = c /sin re