Geometry — Isosceles Triangle

About this lesson

This unit focuses on the properties and applications of isosceles and equilateral triangles in geometry. It begins by defining an isosceles triangle as having two congruent sides (legs), a vertex angle between them, and base angles. The key theorem states that base angles of an isosceles triangle are congruent, and its converse says that if two angles are congruent, the opposite sides are congruent. The unit also covers equilateral triangles (all sides congruent) and their corollaries: an equilateral triangle is equiangular, and an equiangular triangle is equilateral. Additionally, it introduces the axis of symmetry for line segments and triangles, explaining that an isosceles triangle has one axis of symmetry (through the vertex perpendicular to the base), an equilateral triangle has three, and a scalene triangle has none. Corollaries relate the median, angle bisector, and perpendicular from the vertex in an isosceles triangle. Finally, the unit covers the 30°-60°-90° triangle, stating that the side opposite the 30° angle is half the hypotenuse. Throughout, examples and exercises involve finding unknown angles and side lengths, proving triangles are isosceles, and solving real-life problems like tent heights and tower designs.