Proving triangle congruence using the Angle-Side-Angle (ASA) method with bisected angles.

Let’s solve this step by step.

We are given a quadrilateral ABCD with diagonal AC drawn. We’re told:

- AC bisects ∠BAD → that means it splits angle BAD into two equal angles.
- AC bisects ∠BCD → that means it splits angle BCD into two equal angles.

We need to mark the sides (or rather, identify which parts are congruent) to prove that triangles ABC and ADC are congruent using ASA — Angle-Side-Angle.

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Step 1: Understand what ASA means

ASA = Angle - Side - Angle
To use ASA, we need:
- Two angles in one triangle equal to two angles in the other triangle.
- The side between those two angles must also be equal in both triangles.

So for triangles ABC and ADC, we want to show:

∠BAC ≅ ∠DAC
AC ≅ AC (common side)
∠BCA ≅ ∠DCA

Then we can say △ABC ≅ △ADC by ASA.

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Step 2: Use the given info

Given: AC bisects ∠BAD → so ∠BAC = ∠DAC
That gives us our first pair of equal angles.

Also given: AC bisects ∠BCD → so ∠BCA = ∠DCA
That gives us our second pair of equal angles.

And AC is shared by both triangles → so AC = AC (reflexive property)

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Step 3: Match to ASA

In triangle ABC and triangle ADC:

- ∠BAC ∠DAC (from angle bisector at A)
- AC ≅ AC (same side)
- ∠BCA ≅ ∠DCA (from angle bisector at C)

These are two angles and the included side — perfect for ASA!

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Final Answer:

The marked congruent parts are:

- ∠BAC ≅ ∠DAC (red arcs on left)
- AC ≅ AC (blue tick mark on middle)
- ∠BCA ≅ ∠DCA (yellow arcs on right)

This proves △ABC ≅ △ADC by ASA.