Question

ab and draw a circle centered at point a. opening, draw a circle centered at point b. c and use a straightedge to draw $overline{ac}$ and $overline{bc}$. click the arrows to choose an answer from each menu. $overline{ac}congoverline{ab}$ because all choose... are congruent. $overline{ab}congoverline{bc}$ because all choose... are congruent. choose... by the transitive property. $\triangle abc$ is an equilateral triangle because choose... sides are congruent.

Explanation:

Step1: Recall circle - radius property

When a circle is drawn centered at point \(A\) and we consider the line - segment \(AB\) and \(AC\), \(AC\) and \(AB\) are radii of the circle centered at \(A\). Since all radii of a circle are congruent, \(\overline{AC}\cong\overline{AB}\) because all radii of a circle are congruent.

Step2: Recall circle - radius property for the second circle

When a circle is drawn centered at point \(B\), \(AB\) and \(BC\) are radii of the circle centered at \(B\). Since all radii of a circle are congruent, \(\overline{AB}\cong\overline{BC}\) because all radii of a circle are congruent.

Step3: Apply the 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: Define an equilateral triangle

An equilateral triangle is a triangle in which all three sides are congruent. Since \(\overline{AC}\cong\overline{AB}\), \(\overline{AB}\cong\overline{BC}\), and \(\overline{AC}\cong\overline{BC}\), \(\triangle ABC\) is an equilateral triangle because all three sides are congruent.

Answer:

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