Question

step 1
set the compass to the length of \\(\overline{ab}\\) and draw a circle centered at point a.

step 2
without changing the compass opening, draw a circle centered at point b.

step 3
label one of the points of intersection of the two circles c and use a straightedge to draw \\(\overline{ac}\\) and \\(\overline{bc}\\).

click the arrows to choose an answer from each menu.
\\(\overline{ac} \cong \overline{ab}\\) because all choose... are congruent.
\\(\overline{ab} \cong \overline{bc}\\) because all choose... are congruent.
choose... by the transitive property.
\\(\triangle abc\\) is an equilateral triangle because choose... sides are congruent.

Explanation:

Step1: Analyze \( \overline{AC} \cong \overline{AB} \)

In Step 1, the compass is set to \( \overline{AB} \) and a circle centered at \( A \) is drawn. So \( AC \) is a radius of the circle centered at \( A \), and \( AB \) is also a radius of that circle. All radii of a circle are congruent. So the first "Choose..." should be "radii of a circle".

Step2: Analyze \( \overline{AB} \cong \overline{BC} \)

In Step 2, the compass (with the same opening as \( AB \)) is used to draw a circle centered at \( B \). So \( BC \) is a radius of the circle centered at \( B \), and \( AB \) is a radius of that circle (since the compass opening is \( AB \)). All radii of a circle are congruent. So the second "Choose..." should be "radii of a circle".

Step3: Transitive Property

If \( \overline{AC} \cong \overline{AB} \) and \( \overline{AB} \cong \overline{BC} \), then by the transitive property of congruence, \( \overline{AC} \cong \overline{BC} \).

Step4: Equilateral Triangle

A triangle with three congruent sides is equilateral. Since \( \overline{AC} \cong \overline{AB} \), \( \overline{AB} \cong \overline{BC} \), and \( \overline{AC} \cong \overline{BC} \), all three sides are congruent, so \( \triangle ABC \) is equilateral.

Answer:

1. \( \overline{AC} \cong \overline{AB} \) because all \(\boldsymbol{\text{radii of a circle}}\) are congruent.
2. \( \overline{AB} \cong \overline{BC} \) because all \(\boldsymbol{\text{radii of a circle}}\) are congruent.
3. \(\boldsymbol{\overline{AC} \cong \overline{BC}}\) by the transitive property.
4. \( \triangle ABC \) is an equilateral triangle because \(\boldsymbol{\text{three}}\) sides are congruent.