Question

which of these statements is most useful to prove the two triangles are congruent using the asa postulate?
every angle has exactly one bisector.
angle b is congruent to angle d.
segment ac is congruent to segment ac.
the sum of the interior measures of a triangle is 180°.

Explanation:

Step1: Recall ASA Postulate

ASA (Angle - Side - Angle) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Analyze the given triangles

In triangles \(ABC\) and \(ADC\), we can see that \(\angle BAC=\angle DAC = 30^{\circ}\) and \(\angle BCA=\angle DCA=40^{\circ}\). The side \(AC\) is common to both triangles.

Step3: Evaluate each option

• Option 1: "Every angle has exactly one bisector" - This is about angle bisectors in general and does not help in proving the triangles congruent by ASA.
• Option 2: "Angle \(B\) is congruent to angle \(D\)" - To use ASA, we need two angles and the included side. Knowing \(\angle B\cong\angle D\) does not directly help with the ASA postulate as we already have two angles from the diagram.
• Option 3: "Segment \(AC\) is congruent to segment \(AC\)" - This is the reflexive property of congruence. Since \(AC\) is the included side between the two angles (\(\angle BAC\) and \(\angle BCA\) in \(\triangle ABC\); \(\angle DAC\) and \(\angle DCA\) in \(\triangle ADC\)), and we already know \(\angle BAC\cong\angle DAC\) and \(\angle BCA\cong\angle DCA\), by ASA (two angles and the included side), we can prove \(\triangle ABC\cong\triangle ADC\) if \(AC\cong AC\) (which it is by reflexive property).
• Option 4: "The sum of the interior measures of a triangle is \(180^{\circ}\)" - This is about the angle - sum property of triangles and does not help in proving congruence by ASA.

Answer:

Segment \(AC\) is congruent to segment \(AC\) (the third option)