Question

proving the converse of the isosceles triangle theorem
given: in triangle abc, ∠b ≅ ∠c
prove: ( overline{ab} cong overline{ac} )
complete the paragraph proof.
we are given that ∠b ≅ ∠c. assume segment ab is not congruent to
if ab > ac, then ( mangle c > mangle b ) by the
. if ab < ac, then ( mangle c < mangle b ) by the converse of the triangle parts relationship theorem.
but by the definition of congruent, we know the measure of angle b equals the measure of
by the given statement. therefore, we have a contradiction: ab = ac, and ( overline{ab} cong overline{ac} ).

Explanation:

Step1: First Blank

We aim to prove \( \overline{AB} \cong \overline{AC} \). By the method of contradiction, we assume \( AB \) is not congruent to \( AC \). So the first blank should be \( \overline{AC} \).

Step2: Second Blank

The theorem that relates the length of a side of a triangle to the measure of its opposite angle is the "Triangle Parts Relationship Theorem" (also known as the "Larger Angle Opposite Larger Side" theorem). If \( AB > AC \), the angle opposite \( AB \) is \( \angle C \) and the angle opposite \( AC \) is \( \angle B \), so \( m\angle C > m\angle B \) by this theorem. So the second blank is "Triangle Parts Relationship Theorem" (or "Larger Angle Opposite Larger Side" theorem).

Step3: Third Blank

We are given that \( \angle B \cong \angle C \), so by the definition of congruent angles, \( m\angle B = m\angle C \). So the third blank is \( \angle C \).

Answer:

First blank: \( \overline{AC} \)
Second blank: Triangle Parts Relationship Theorem (or Larger Angle Opposite Larger Side theorem)
Third blank: \( \angle C \)