To prove triangles \( \Delta CDE \) and \( \Delta CDQ \) congruent by the Angle-Side-Angle (ASA) criterion, you need two angles and the included side of each triangle to be congruent.
From the given options:
- a. \( DE \cong DQ \): This is a side, not an angle, so it does not help apply ASA directly.
- b. \( CE \cong CQ \): This is also a side, not an angle.
- c. \( \angle DCE \cong \angle DCQ \): This is one of the angles you're looking for.
- d. \( \angle E \cong \angle Q \): This is another angle you're looking for.
To utilize the ASA criterion, you would need to show that:
- \( \angle DCE \cong \angle DCQ \) (option c).
- \( \angle E \cong \angle Q \) (option d).
- The included side \( CD \) is common to both triangles, which is congruent by definition.
Therefore, options c and d together would provide the necessary angle information. If you have these two angles and the side \( CD \) as the included side to apply ASA, then your triangles \( \Delta CDE \) and \( \Delta CDQ \) would be congruent.
Thus, the correct answer to prove \( \Delta CDE \) and \( \Delta CDQ \) congruent by ASA is c and d together.
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