Question

for the given diagram which of the following is true?
options:

• $\triangle abc \cong \triangle acd$ by asa and $ac = ac$
• $\triangle acd \cong \triangle abc$ by sas and $ac = ca$
• $\triangle abc \cong \triangle acd$ by asa and $ac = ca$
• $\triangle abc \cong \triangle adc$ by asa and $ac = ac$

Explanation:

Step1: Analyze the triangles

We have two triangles, \( \triangle ABC \) and \( \triangle ADC \). The common side is \( AC = AC \) (reflexive property). The angles at \( A \) are marked equal (so \( \angle BAC=\angle DAC \)) and the angles at \( C \) are marked equal (so \( \angle BCA = \angle DCA \)).

Step2: Identify the congruence criterion

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. Here, \( \angle BAC=\angle DAC \), \( AC = AC \) (included side), and \( \angle BCA=\angle DCA \). So the triangles \( \triangle ABC \) and \( \triangle ADC \) are congruent by ASA, and \( AC = AC \).

Now let's check each option:

• Option 1: \( \triangle ABC\cong\triangle ACD \) (incorrect triangle naming, should be \( \triangle ADC \)) and ASA with \( AC = AC \) (triangle naming error).
• Option 2: SAS is incorrect, we have two angles and included side, not two sides and included angle. Also triangle naming \( \triangle ACD\cong\triangle ABC \) (incorrect order and criterion).
• Option 3: \( \triangle ABC\cong\triangle ACD \) (incorrect triangle naming, should be \( \triangle ADC \)).
• Option 4: \( \triangle ABC\cong\triangle ADC \) by ASA and \( AC = AC \) (correct triangle naming, correct criterion and side equality).

Answer:

D. \( \triangle ABC \cong \triangle ADC \) by ASA and \( AC = AC \)