Important Angle Theorems

1. Complementary and Supplementary Angles

• Complementary Angles: Two angles whose sum is 90°.
  • Example: If Angle A + Angle B = 90°, they are complementary.
  • Calculation: If one angle is 70°, the other is 20° (90 - 70).
• Supplementary Angles: Two angles whose sum is 180°.
  • Example: If Angle A + Angle B = 180°, they are supplementary.
  • Calculation: If one angle is 70°, the other is 110° (180 - 70).

2. Sum of Angles in Polygons

• Triangles: Sum of the three interior angles is always 180°.
  • Calculation: If two angles are 70° and 75°, the third is 35° (180 - 70 - 75).
• General Polygons: Sum of interior angles = (n-2) * 180°, where n is the number of sides.
  • Example: Pentagon (5 sides): (5-2) * 180 = 540°.
  • Quadrilateral (4 sides): (4-2) * 180 = 360°.
• Exterior Angles: Sum of exterior angles of any polygon is always 360°.*

3. Isosceles Triangle Theorem

• If two sides of a triangle are congruent, the opposite angles are congruent.
  • Example: If two sides are equal, the angles opposite these sides are equal.

4. Exterior Angle Theorem

• Exterior angle of a triangle equals the sum of the two opposite interior angles.
  • Example: If exterior angle is 110° and one opposite interior angle is 40°, the other is 70° (110 - 40).

5. Vertical Angle Theorem

• When two lines intersect, opposite angles are equal.
  • Example: Angle A = Angle C, and Angle B = Angle D.

6. Alternate Angles (Parallel Line Theorem)

• Alternate Interior Angles: Angles between two parallel lines cut by a transversal are equal.
  • Example: If Angle A and Angle C are on opposite sides of the transversal, they are equal.
• Alternate Exterior Angles: These are also equal.
  • Example: Angle E = Angle H.

7. Co-Interior Angles (Parallel Line Theorem)

• Interior angles on the same side of a transversal sum to 180°.
  • Example: If Angle A and Angle B are on the same side, A + B = 180°.

8. Corresponding Angles (Parallel Line Theorem)

• Angles in the same relative position at each intersection of the transversal with parallel lines are equal.
  • Example: Top Left angles at both intersections are equal.

9. Angle Subtended by an Arc (Circle Theorem)

• Inscribed angles subtended by the same arc are equal.
  • Example: Angles X and Y subtended by the same arc are equal.

10. Angle at the Center vs. Circumference (Circle Theorem)

• Angle subtended by an arc at the center is double the angle subtended at the circumference.
  • Calculation: If Angle at circumference is X, the angle at center is 2X.

Conclusion

• Practice solving angle problems using these theorems.
• Check yourself with provided problems and compare with correct answers.