Question

which fact helps you prove the isosceles triangle theorem, which states that the base angles of any isosceles triangle have equal measure?
the base angles of an equilateral triangle have equal measure.
if two sides of a triangle are equal, the third side must be equal to the others.
if a triangle is equiangular, then it is equilateral.
if one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Explanation:

• Analyze Option 1: The base angles of an equilateral triangle having equal measure is about equilateral triangles, not directly helping prove the isosceles triangle theorem (which is about triangles with two equal sides), so this is not relevant.
• Analyze Option 2: The statement "If two sides of a triangle are equal, the third side must be equal to the others" is incorrect. In an isosceles triangle, only two sides are equal, the third is different (except in equilateral triangles, but equilateral is a special case of isosceles). So this is wrong.
• Analyze Option 3: "If a triangle is equiangular, then it is equilateral" is about the relationship between equiangular and equilateral triangles, not about proving the isosceles triangle theorem's base angles are equal. So this is not helpful.
• Analyze Option 4: The theorem "If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle" (and its converse) is used in proving the isosceles triangle theorem. By assuming the base angles are unequal, we can show via this theorem that the sides opposite would be unequal, contradicting the definition of an isosceles triangle (two equal sides), thus proving the base angles must be equal.

Answer:

D. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.