๐ Isosceles Triangle Theorem
The Isosceles Triangle Theorem states that if two sides of a triangle are congruent (equal in length), then the angles opposite those sides are also congruent.
- ๐ Definition: If a triangle has two sides of equal length, it is an isosceles triangle.
- ๐ Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Conversely, if two angles of a triangle are congruent, then the sides opposite those angles are congruent.
- โ๏ธ Example: In triangle $ABC$, if $AB = AC$, then $\angle B = \angle C$.
๐ Equilateral Triangle Theorem
The Equilateral Triangle Theorem states that if a triangle is equilateral (all three sides are congruent), then it is also equiangular (all three angles are congruent), and each angle measures 60 degrees.
- ๐ Definition: If a triangle has three sides of equal length, it is an equilateral triangle.
- ๐ Theorem: If a triangle is equilateral, then it is also equiangular. Each angle in an equilateral triangle measures $60^{\circ}$.
- โ๏ธ Example: In triangle $DEF$, if $DE = EF = FD$, then $\angle D = \angle E = \angle F = 60^{\circ}$.
๐ Isosceles vs. Equilateral: A Side-by-Side Comparison
| Feature |
Isosceles Triangle |
Equilateral Triangle |
| Sides |
Two sides are congruent. |
All three sides are congruent. |
| Angles |
Two angles are congruent (opposite the congruent sides). |
All three angles are congruent and equal to $60^{\circ}$. |
| Relationship |
A general case of triangles with at least two equal sides. |
A special case where all sides and angles are equal. |
| Angle Measurement |
Two angles are equal, but their measure can vary. |
Each angle is exactly $60^{\circ}$. |
๐ก Key Takeaways
- โ๏ธ Isosceles: At least two sides and two angles are congruent.
- โ๏ธ Equilateral: All three sides and all three angles are congruent, with each angle measuring $60^{\circ}$.
- โ๏ธ Relationship: An equilateral triangle is always an isosceles triangle, but an isosceles triangle is not always an equilateral triangle.