Overview
This lecture covers the essential properties of triangles and quadrilaterals, focusing on angle relationships, types of triangles, and solving geometric problems involving parallel lines.
Properties of Triangles
- The sum of the interior angles of any triangle is always 180°.
- A scalene triangle has all sides and all angles unequal.
- An isosceles triangle has two equal sides and the angles opposite these sides are also equal (called base angles).
- An equilateral triangle has all sides equal, and each interior angle is 60°.
Properties of Quadrilaterals
- Quadrilaterals are four-sided shapes; key types include squares, rectangles, parallelograms, rhombi, and trapezia.
- In a square, all sides and angles are equal.
- In a rectangle, opposite sides are equal and all angles are 90°.
- A parallelogram has two pairs of opposite, equal, and parallel sides. Opposite angles are equal; adjacent angles are supplementary (sum to 180°).
- A rhombus is a parallelogram with all sides equal; diagonals bisect the angles.
- A trapezium (trapezoid) has only one pair of parallel sides; it cannot be derived from parallelograms or related shapes.
Parallel Line Concepts in Geometry
- When two lines are parallel, alternate angles formed by a transversal are equal.
- Co-interior (consecutive) angles between parallel lines add up to 180°.
- Vertically opposite angles are equal when two lines intersect.
- Adjacent supplementary angles on a straight line add up to 180°.
Worked Examples & Problem-Solving Strategies
- Use properties of isosceles triangles and parallel lines to identify equal angles.
- Use the sum of triangle angles to find unknown angles when two angles are given.
- In quadrilaterals with parallel lines, apply alternate and co-interior angle rules to solve for unknowns.
- For rhombi and parallelograms, use symmetry and properties to distribute or calculate angles.
- In complex figures, analyze which angle relations (alternate, co-interior, vertically opposite) are available.
Key Terms & Definitions
- Isosceles triangle — a triangle with two sides and their opposite angles equal.
- Equilateral triangle — a triangle with all sides and angles equal (each angle 60°).
- Scalene triangle — a triangle with all sides and angles unequal.
- Parallelogram — a quadrilateral with two pairs of parallel, equal sides.
- Rhombus — a parallelogram with all sides equal and diagonals bisecting angles.
- Trapezium (trapezoid) — a quadrilateral with one pair of parallel sides.
- Alternate angles — angles on opposite sides of a transversal that are equal when lines are parallel.
- Co-interior angles — angles inside parallel lines on the same side of a transversal; sum to 180°.
- Vertically opposite angles — angles across from each other at a crossing point of lines; always equal.
Action Items / Next Steps
- Review and memorize key triangle and quadrilateral properties.
- Practice solving geometric problems involving parallel lines, triangles, and various quadrilaterals.
- Complete assigned exercises on calculating unknown angles using these properties.