The isosceles triangle theorem is an essential concept in geometry. This theorem states that if two sides of a triangle are congruent (that is, they have the same length), then the angles opposite those sides are congruent as well.
Here is a step-by-step explanation:
1. Identify the Isosceles Triangle:
- An isosceles triangle is one that has at least two sides that are of equal length. Let’s consider a triangle [tex]\( \triangle ABC \)[/tex] with sides [tex]\( AB \)[/tex] and [tex]\( AC \)[/tex] being equal ([tex]\( AB = AC \)[/tex]).
2. Isosceles Triangle Theorem:
- According to the isosceles triangle theorem, the angles opposite these congruent sides must also be equal. In [tex]\( \triangle ABC \)[/tex]:
- The angle opposite side [tex]\( AB \)[/tex] is [tex]\( \angle C \)[/tex].
- The angle opposite side [tex]\( AC \)[/tex] is [tex]\( \angle B \)[/tex].
- Therefore, [tex]\( \angle B = \angle C \)[/tex].
3. Check the Options:
- Option A: The angles are congruent.
- Option B: The angles are complementary. Complementary angles are two angles whose measures add up to 90°.
- Option C: The angles are supplementary. Supplementary angles are two angles whose measures add up to 180°.
- Option D: The angles are proportional. This would mean that the angles have a constant ratio, which isn't necessarily true for angles in isosceles triangles in this context.
4. Correct Answer:
- The angles opposite the two congruent sides are equal in measure, which matches the definition of congruent angles.
Therefore, the correct answer is:
OA. congruent.
