Overview
This lecture covers the top 10 essential angle theorems, including definitions, properties, and examples, plus a short self-test at the end.
Complementary and Supplementary Angles
- Complementary angles sum to 90°; together, they form a right angle.
- Supplementary angles sum to 180°; together, they form a straight line.
- To find an unknown complementary/supplementary angle, subtract the known angle from 90° or 180°, respectively.
Sum of Angles in Triangles and Polygons
- The sum of the interior angles of any triangle is 180°.
- The sum of interior angles of any polygon: (n – 2) × 180°, where n = number of sides.
- The sum of exterior angles of any polygon is always 360°.
Isosceles Triangle Theorem
- If two sides of a triangle are congruent, the angles opposite those sides are congruent.
Exterior Angle Theorem
- An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Vertical Angle Theorem
- When two lines intersect, the opposite (vertical) angles are equal in measure.
Parallel Line Angle Theorems
- Alternate Interior Angles: Equal when parallel lines are cut by a transversal (Z-pattern).
- Alternate Exterior Angles: Equal when parallel lines are cut by a transversal.
- Co-interior (Consecutive Interior) Angles: Add up to 180° (C-pattern).
- Corresponding Angles: Equal when in the same relative position at each intersection (F-pattern).
Angle Subtended by an Arc (Circle Theorems)
- Inscribed angles subtended by the same arc are equal.
- Any angle inscribed in a semicircle is 90°.
Angle at the Center vs. Circumference (Circle Theorem)
- The angle at the center of a circle subtended by an arc is double the angle at the circumference subtended by the same arc.
Key Terms & Definitions
- Complementary Angles — Two angles whose sum is 90°.
- Supplementary Angles — Two angles whose sum is 180°.
- Isosceles Triangle — A triangle with two equal sides and two equal opposite angles.
- Transversal — A line that cuts across two or more (usually parallel) lines.
- Inscribed Angle — An angle formed by two chords in a circle sharing an endpoint.
- Arc — A part of the circumference of a circle.
Action Items / Next Steps
- Attempt the three practice questions on finding missing angles as a self-test.
- Review and memorize each theorem and associated patterns (Z, C, F) for parallel line theorems.