Question

1. complete this flowchart proof of the converse of the isosceles triangle conjecture. given: △new with ∠w ≅ ∠e \\(\overline{ns}\\) is an angle bisector show: △new is an isosceles triangle flowchart proof 1 \\(\overline{ns}\\) is an angle bisector given 2 ∠1 ≅ ∠ ? definition of ? 3 ∠w ≅ ∠e ? 4 \\(\overline{ns}\\) ≅ \\(\overline{ns}\\) ? 5 △ ? ≅ △ ? ? 6 \\(\overline{wn}\\) ≅ \\(\overline{ne}\\) ? 7 △new is isosceles ?

Explanation:

Step1: Analyze Step 2

An angle bisector divides an angle into two congruent angles. So, since \(\overline{NS}\) is an angle bisector, \(\angle 1\cong\angle 2\) (by the definition of an angle bisector).

Step2: Analyze Step 3

\(\angle W\cong\angle E\) is given in the problem statement.

Step3: Analyze Step 4

\(\overline{NS}\cong\overline{NS}\) is the reflexive property of congruence (a segment is congruent to itself).

Step4: Analyze Step 5

We have two angles and a side: \(\angle 1\cong\angle 2\), \(\angle W\cong\angle E\), and \(\overline{NS}\cong\overline{NS}\). So, by the AAS (Angle - Angle - Side) congruence criterion, \(\triangle WNS\cong\triangle ENS\).

Step5: Analyze Step 6

Since \(\triangle WNS\cong\triangle ENS\), their corresponding sides \(\overline{WN}\) and \(\overline{NE}\) are congruent (by the definition of congruent triangles: corresponding parts of congruent triangles are congruent, CPCTC).

Step6: Analyze Step 7

A triangle with two congruent sides is an isosceles triangle (by the definition of an isosceles triangle: a triangle with at least two congruent sides).

Filling in the blanks:

• Step 2: \(\angle 1\cong\angle\boldsymbol{2}\), Definition of \(\boldsymbol{\text{angle bisector}}\)
• Step 3: \(\angle W\cong\angle E\), \(\boldsymbol{\text{Given}}\)
• Step 4: \(\overline{NS}\cong\overline{NS}\), \(\boldsymbol{\text{Reflexive Property of Congruence}}\)
• Step 5: \(\triangle\boldsymbol{WNS}\cong\triangle\boldsymbol{ENS}\), \(\boldsymbol{\text{AAS (Angle - Angle - Side) Congruence Criterion}}\)
• Step 6: \(\overline{WN}\cong\overline{NE}\), \(\boldsymbol{\text{CPCTC (Corresponding Parts of Congruent Triangles are Congruent)}}\)
• Step 7: \(\triangle NEW\) is isosceles, \(\boldsymbol{\text{Definition of Isosceles Triangle}}\)

Answer:

• Step 2: \(\angle 1\cong\angle 2\), Definition of angle bisector
• Step 3: \(\angle W\cong\angle E\), Given
• Step 4: \(\overline{NS}\cong\overline{NS}\), Reflexive Property of Congruence
• Step 5: \(\triangle WNS\cong\triangle ENS\), AAS (Angle - Angle - Side) Congruence Criterion
• Step 6: \(\overline{WN}\cong\overline{NE}\), CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
• Step 7: \(\triangle NEW\) is isosceles, Definition of Isosceles Triangle