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 ( angle b cong angle 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: Identify the first blank

The first part is about assuming \( AB \) is not congruent to \( AC \), so we consider the triangle parts relationship. The first blank (after "If \( AB > AC \), then \( m\angle C > m\angle B \) by the") is filled by the "Triangle Parts Relationship Theorem" (specifically the theorem that in a triangle, the larger side is opposite the larger angle). So the reason is the "triangle parts relationship theorem" (the one that relates side lengths to angle measures: larger side ⇨ larger opposite angle).

Step2: Identify the second blank

The second blank (after "If \( AB < AC \), then \( m\angle C < m\angle B \) by the converse of the triangle parts relationship theorem") is just the application of the converse (smaller side ⇨ smaller opposite angle).

Step3: Identify the third blank

The third blank (after "But by the definition of congruent, we know the measure of angle \( B \) equals the measure of by the given statement") is filled by "angle \( C \)" because we are given \( \angle B \cong \angle C \), so their measures are equal.

Answer:

1. First blank: triangle parts relationship theorem (theorem that larger side has larger opposite angle)
2. Second blank: (converse of triangle parts relationship theorem application, so the reason is the converse, but the blank here is part of the statement, so the logic is consistent)
3. Third blank: angle \( C \)

(Note: Since the problem is about filling in the proof blanks for the Converse of the Isosceles Triangle Theorem, the key steps are:

• Assume \( AB

ot\cong AC \). Then either \( AB > AC \) or \( AB < AC \).

• If \( AB > AC \), then \( m\angle C > m\angle B \) (by the triangle angle - side relationship: larger side opposite larger angle).
• If \( AB < AC \), then \( m\angle C < m\angle B \) (by the converse: smaller side opposite smaller angle).
• But we know \( \angle B \cong \angle C \), so \( m\angle B = m\angle C \) (definition of congruent angles). This contradicts the previous two cases, so \( AB \cong AC \).

So the blanks are filled as:

First blank (after "If \( AB > AC \), then \( m\angle C > m\angle B \) by the"): triangle parts relationship theorem (the theorem that in a triangle, the length of a side is related to the measure of its opposite angle: longer side ⇨ larger opposite angle).

Second blank (the structure is correct as is, the logic is the converse application).

Third blank (after "measure of by the given statement"): angle \( C \) (since \( \angle B \cong \angle C \), their measures are equal).)