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 straight - edge 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: Analyze the construction

In the construction, when we set the compass to the length of $\overline{AB}$ and draw a circle centered at $A$, all radii of this circle are equal to the length of $\overline{AB}$. Since $\overline{AC}$ is a radius of the circle centered at $A$, $\overline{AC}\cong\overline{AB}$ because all radii of a circle are congruent.

Step2: Analyze the second - circle

When we draw a circle centered at $B$ with the same compass - opening (equal to the length of $\overline{AB}$), all radii of this circle are equal to the length of $\overline{AB}$. Since $\overline{BC}$ is a radius of the circle centered at $B$, $\overline{AB}\cong\overline{BC}$ because all radii of a circle are congruent.

Step3: Apply the transitive property

Since $\overline{AC}\cong\overline{AB}$ and $\overline{AB}\cong\overline{BC}$, by the transitive property of congruence, $\overline{AC}\cong\overline{BC}$.

Step4: Define an equilateral triangle

$\triangle ABC$ is an equilateral triangle because all three sides ($\overline{AB}$, $\overline{BC}$, and $\overline{AC}$) are congruent.

Answer:

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