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find angle t, and the angle marked with blue

To solve for the angles \( t \) and the blue-marked angle in this circle-based geometry problem, let's analyze the provided information. Here's what we can observe:

1. **Given Angles and Sides:**
   - The angle on the left is marked as \( 94^\circ \).
   - Two pairs of sides are marked as equal, indicating isosceles triangles.
   - The shape is inscribed in a circle.

### Step-by-Step Solution:

1. **Angle Opposite Equal Sides**:
   - Since the sides opposite to the angles \( t \) and \( 94^\circ \) are equal, this suggests an isosceles triangle in which the base angles are equal.
   - Therefore, if we denote the angle opposite \( 94^\circ \) as \( t \), we conclude \( t = 94^\circ \).

2. **Finding the Blue Angle**:
   - Since we have a quadrilateral inscribed in a circle, we can apply the *cyclic quadrilateral property*, which states that opposite angles sum to \( 180^\circ \).
   - Thus, if one angle is \( 94^\circ \), its opposite angle (the blue-marked angle) must be \( 180^\circ - 94^\circ = 86^\circ \).

### Summary of Results:
- **Angle \( t = 94^\circ \)**.
- **Blue-marked angle = 86^\circ**.

Would you like more details on any part of this solution, or have additional questions?

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Here are 5 related questions to deepen understanding:

1. What are the properties of a cyclic quadrilateral?
2. How do we identify isosceles triangles in circle geometry?
3. Why do opposite angles in a cyclic quadrilateral add up to \( 180^\circ \)?
4. Can these angle properties help in solving other circle-related problems?
5. How does the circle's inscribed angle theorem apply in similar cases?

**Tip:** Always remember that in a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary (sum to \( 180^\circ \)). This property often simplifies solving for unknown angles.

Learn how to find unknown angles in a circle geometry problem involving a cyclic quadrilateral and an isosceles triangle. This problem explores key geometry concepts, including the Cyclic Quadrilateral Theorem and properties of isosceles triangles. Suitable for students in Grades 8-10.

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