Geometry and Trigonometry Notes

Geometry Terms

  • Point: Indicated by a dot and labeled with a letter (e.g., A).

  • Ray: Part of a line that starts at a point and extends infinitely in one direction.

  • Line Segment: A part of a line with two endpoints.

  • Line: Extends infinitely in both directions.

  • Intersecting Lines: Two lines that meet at a point.

  • Perpendicular Lines: Two lines that intersect at right angles.

  • Parallel Lines: Two lines that never intersect.

Trigonometric Ratios (SOH-CAH-TOA)

  • Trigonometric ratios relate angles to the ratios of two sides in a right-angled triangle.

  • Sine (Sin): Opposite / Hypotenuse

    • Sin A = \frac{Opposite}{Hypotenuse}

  • Cosine (Cos): Adjacent / Hypotenuse

    • Cos A = \frac{Adjacent}{Hypotenuse}

  • Tangent (Tan): Opposite / Adjacent

    • Tan A = \frac{Opposite}{Adjacent}

  • Cosecant (Csc): Reciprocal of Sine (Hypotenuse / Opposite)

  • Secant (Sec): Reciprocal of Cosine (Hypotenuse / Adjacent)

  • Cotangent (Cot): Reciprocal of Tangent (Adjacent / Opposite)

Parallel Lines Cut by a Transversal

  • When parallel lines are intersected by a transversal, specific angle relationships are formed.

  • Adjacent Angles on a Straight Line: Supplementary (add up to 180 degrees).

    • Example: <1 + <2 = 180°

  • Vertical Angles: Always equal.

    • Example: <1 = <4, <2 = <3

  • Corresponding Angles: Equal; they coincide if one set of lines is placed directly over the other.

    • Example: <1 = <5

  • Alternate Interior Angles: Lie on opposite sides of the transversal and between the parallel lines; they are equal.

  • Alternate Exterior Angles: Lie on opposite sides of the transversal and outside the parallel lines.

    • Consecutive Interior Angles: Lie on the same side of the transversal and between the parallel lines; they are supplementary (add up to 180 degrees).

Trigonometry

  • Trigonometry deals with the relationships between the angles and the lengths of the sides of triangles.

  • Angles in trigonometry are often represented by Greek letters:

    • \theta (theta)

    • \beta (beta)

    • \alpha (alpha)

    • \phi (phi)

Pythagorean Theorem

  • In a right-angled triangle:

    • c^2 = a^2 + b^2

    • a^2 = c^2 - b^2

    • b^2 = c^2 - a^2

    • Where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.

Labeling Sides of a Right Triangle

  • Hypotenuse: The longest side, opposite the right angle.

  • Opposite: The side opposite the angle under consideration.

  • Adjacent: The side next to the angle under consideration (not the hypotenuse).

Trigonometric Ratios Applied to Right Triangles

  • Define the relationship between sides and angles.

    • Sin = \frac{Opposite}{Hypotenuse}

    • Cos = \frac{Adjacent}{Hypotenuse}

    • Tan = \frac{Opposite}{Adjacent}

Geometry

  • Focuses on angles, triangles, and parallelograms.

Angle Relationships

  • In a quadrilateral:

    • a + b + c + d = 360°

  • Angles on one side:

    • a + b = 180 (supplementary)

  • Corresponding angles are equal.

  • Alternate angles are equal.

  • Interior angles add up to 180 degrees.

Triangles

  • Equilateral Triangle: All three sides are equal.

  • Isosceles Triangle: Two sides are equal.

  • Scalene Triangle: No equal sides.

  • Right Triangle: Contains a right angle.