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.

Test your knowledge on supplementary and complementary angles, angle relationships, and polygon angle properties. This quiz covers essential concepts such as vertical angles and linear pairs to enhance your understanding of geometry.