Overview
This lecture covers foundational properties, definitions, and key theorems related to angles in triangles, focusing on interior and exterior angle sums, special triangles (isosceles, equilateral, right), angle bisector properties, and problem-solving strategies with geometric constructions and congruence.
Triangle Basics and Angle Properties
- A triangle is formed by joining three non-collinear points with line segments.
- The sum of the interior angles of a triangle is always 180°.
- The sum of the exterior angles of a triangle is 360°, applicable to all convex polygons.
- In any triangle, the sum of one interior angle and its adjacent exterior angle is 180°.
- A straight angle (on a line) is 180°, used to prove triangle angle properties.
Special Triangles and Features
Isosceles Triangle
- Two sides are equal, and base angles opposite these sides are also equal.
- The altitude from the apex to the base is also the median and the angle bisector.
- In any triangle, if the altitude and median from a vertex coincide, the triangle is isosceles.
Equilateral Triangle
- All sides and angles are equal (each angle is 60°).
- All altitudes, medians, and angle bisectors coincide and divide the triangle into two congruent halves.
Right Triangle & The "Magnificent Trio"
- One angle is 90°, and the sides adjacent to it are legs; the side opposite is the hypotenuse.
- The largest angle is the right angle; the hypotenuse is the longest side.
- In a right triangle, a median drawn to the hypotenuse equals half the hypotenuse ("magnificent trio").
Key Geometric Strategies and Theorems
- Two Interior–One Exterior Angle Theorem: The sum of two interior angles not adjacent to an exterior angle equals that exterior angle.
- Angle Bisector Intersection: The intersection point of the three angle bisectors is the incenter (center of the inscribed circle).
- Exterior Angle Bisector Theorem: The intersection of two exterior angle bisectors forms a center (excenter).
- If two of the altitude, median, or angle bisector from a vertex coincide, the triangle is isosceles.
- Identical (congruent) shapes have equal angles and sides, aiding in solving for unknowns.
Folding & Symmetry in Geometry
- Folding a triangle over an axis of symmetry creates isosceles triangles and preserves lengths and angles.
- The fold axis acts as an angle bisector and median, often resulting in right or equilateral triangles.
Key Terms & Definitions
- Triangle — A polygon with three sides and three angles formed by three non-collinear points.
- Interior Angle — An angle inside a triangle formed by two sides.
- Exterior Angle — An angle formed by extending one side of the triangle.
- Isosceles Triangle — A triangle with two equal sides and two equal base angles.
- Equilateral Triangle — A triangle with all sides and angles equal (each 60°).
- Right Triangle — A triangle with one 90° angle.
- Median — A line segment from a vertex to the midpoint of the opposite side.
- Angle Bisector — A line dividing an angle into two equal parts.
- Incenter — The intersection point of the angle bisectors, center of the inscribed circle.
- Magnificent Trio — In a right triangle, the median to the hypotenuse equals half the hypotenuse.
Action Items / Next Steps
- Review triangle properties, focusing on angle sum theorems.
- Practice problems involving isosceles, equilateral, and right triangles, especially using symmetry and folding.
- Memorize and apply the two interior–one exterior angle theorem in problem-solving.
- Complete homework problems and question bank sections on triangle angles.