Geometry: Angles and Polygons

Supplementary And Complementary Angles

• Supplementary Angles: Two angles that add up to 180 degrees.
• Complementary Angles: Two angles that add up to 90 degrees.

Angle Relationships

• Adjacent Angles: Angles that share a common side and vertex but do not overlap.
• Vertical Angles: Angles opposite each other when two lines intersect; they are always equal.
• Linear Pair: A pair of adjacent angles that form a straight line; they are supplementary.

Polygon Angle Properties

• Sum of Interior Angles: For an n-sided polygon, the sum is (n - 2) × 180 degrees.
• Measure of Each Interior Angle: For a regular polygon, each interior angle = [(n - 2) × 180] / n degrees.

Vertical Angles

• Always equal when two lines intersect.
• Example: If angle A is 50 degrees, the angle opposite (angle B) is also 50 degrees.

Linear Angles

• Formed when two adjacent angles create a straight line.
• If one angle is 120 degrees, the adjacent angle must be 60 degrees (180 - 120 = 60).

Formula to Calculate Number of Sides in a Polygon

• To find the number of sides (n) given the sum of interior angles (S):
  • n = (S / 180) + 2.

Formula to Calculate Measure of Angles on a Regular Polygon

• For each interior angle of a regular polygon with n sides:
  • Measure of each angle = [(n - 2) × 180] / n.
• For each exterior angle:
  • Measure of each exterior angle = 360 / n.

Conjectures

• Angle Sum Conjecture: The sum of the interior angles of a polygon depends on the number of sides.
• Exterior Angle Theorem: The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles.

Supplementary and Complementary Angles

• Supplementary Angles: Defined as two angles whose measures total 180 degrees, allowing for straight-line formation.
• Complementary Angles: Defined as two angles whose measures total 90 degrees, commonly seen in right-angle scenarios.

Angle Relationships

• Adjacent Angles: Located next to each other, sharing a common vertex and side, without overlapping.
• Vertical Angles: Formed by the intersection of two lines, located opposite each other, always equal in measure.
• Linear Pair: Consists of two adjacent angles that sum up to form a straight line, thereby being supplementary.

Polygon Angle Properties

• Sum of Interior Angles: For any polygon with n sides, the total measure of interior angles is calculated using the formula (n - 2) × 180 degrees.
• Measure of Each Interior Angle: In a regular polygon, each interior angle is given by the formula [(n - 2) × 180] / n degrees.

Vertical Angles

• Always equal when two lines intersect, providing consistent measurements across angles.
• Example: If angle A measures 50 degrees, then the angle directly opposite, angle B, will also measure 50 degrees.

Linear Angles

• Form when two adjacent angles together create a straight line.
• Example calculation: For one angle at 120 degrees, the neighboring angle will measure 60 degrees (180 - 120).

Formula to Calculate Number of Sides in a Polygon

• To find the number of sides (n) from the sum of interior angles (S), use the formula:
  • n = (S / 180) + 2.

Formula to Calculate Measure of Angles in a Regular Polygon

• For each interior angle in a regular polygon with n sides:
  • Measure of each angle = [(n - 2) × 180] / n.
• For each exterior angle:
  • Measure of each exterior angle = 360 / n.

Conjectures

• Angle Sum Conjecture: Indicates that the total sum of interior angles of a polygon is determined by its number of sides.
• Exterior Angle Theorem: States that the measure of any exterior angle is equal to the sum of the two opposite interior angles, establishing a relationship between interior and exterior angles.