Circles - Inscribed angles Worksheets - Free Printable

Let’s solve this step by step.

We are given a circle with points D, C, N, and E on the circumference. There’s a line EC that passes through the center of the circle — so EC is a diameter. That’s important!

Also, we’re told that angle DCN = 37°, and we need to find angle DCE (marked with a “?”).

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Step 1: Use the fact that EC is a diameter

Since EC is a diameter, any angle subtended by it at the circumference is a right angle (90°). This is called the Thales’ Theorem.

So, look at triangle ENC or EDC? Actually, let’s focus on point D.

Angle EDC is an angle subtended by diameter EC → so angle EDC = 90°.

Wait — but we don’t have point connected directly like that. Let’s think differently.

Actually, let’s consider triangle DCN and triangle DCE.

But here’s a better idea:

Look at arc DN. Angle DCN and angle DEN both subtend the same arc DN → so they should be equal? Wait, no — angle DCN is at point C, angle DEN would be at point E.

Alternatively, since EC is a diameter, then angle EDC = 90° (because it's subtended by the diameter).

Yes! So in triangle EDC, angle at D is 90°.

Now, what about angle DNC? Or maybe we can use angles in the same segment.

Wait — another approach:

Angles subtended by the same arc are equal.

Arc DN is subtended by angle DCN (at point C) and also by angle DEN (at point E). So angle DEN = angle DCN = 37°.

Is that right? Let me check.

Point C and point E are both on the circumference, and they both "see" arc DN. So yes — angles subtended by the same arc at the circumference are equal.

So angle DEN = 37°.

Now, in triangle EDC, we know:

- Angle at D = 90° (since EC is diameter)
- Angle at E = 37° (just found)
- Therefore, angle at C (which is angle DCE, the one we want) = 180° - 90° - 37° = 53°

That makes sense!

Let me double-check:

In triangle EDC:
- ∠EDC = 90° (angle in semicircle)
- ∠DEC = DEN = 37° (same arc DN)
- So ∠DCE = 180 - 90 - 37 = 53°

Perfect.

Final Answer:
53°