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Give **two** simplified expressions for BC^DBC^D.

To solve for two simplified expressions of angle \( \angle BCD \), we need to use the given information and the properties of the cyclic quadrilateral formed by points \( A, B, C, D \), which is inscribed in a circle. Let's analyze the diagram and the provided information:

### Given:
- \( \angle CBE = 3x + 42^\circ \)
- \( \angle DEA = x + 70^\circ \)

### Step 1: Relationship Between Opposite Angles in a Cyclic Quadrilateral
In a cyclic quadrilateral, the opposite angles sum to \( 180^\circ \). Therefore, the angles \( \angle CBE \) and \( \angle DEA \) are related by the equation:
\[
\angle CBE + \angle DEA = 180^\circ
\]
Substituting the given expressions:
\[
(3x + 42^\circ) + (x + 70^\circ) = 180^\circ
\]
Simplifying this equation:
\[
4x + 112^\circ = 180^\circ
\]
\[
4x = 180^\circ - 112^\circ
\]
\[
4x = 68^\circ
\]
\[
x = 17^\circ
\]

### Step 2: Find \( \angle CBE \) and \( \angle DEA \)
Now that we know \( x = 17^\circ \), we can substitute this value into the expressions for \( \angle CBE \) and \( \angle DEA \):
\[
\angle CBE = 3x + 42^\circ = 3(17^\circ) + 42^\circ = 51^\circ + 42^\circ = 93^\circ
\]
\[
\angle DEA = x + 70^\circ = 17^\circ + 70^\circ = 87^\circ
\]

### Step 3: Identify Two Expressions for \( \angle BCD \)
Since \( B, C, D, E \) lie on the circle, angles subtended by the same chord are equal. Therefore, \( \angle BCD = \angle BEC \). Using the known angles:
1. \( \angle BCD = 180^\circ - \angle CBE = 180^\circ - 93^\circ = 87^\circ \)
2. \( \angle BCD = \angle DEA = 87^\circ \)

Thus, the two simplified expressions for \( \angle BCD \) are:
1. \( 180^\circ - \angle CBE \)
2. \( \angle DEA \)

Would you like further details or clarifications? Here are some related questions:

1. How do we derive the angle relationships in a cyclic quadrilateral?
2. What are other properties of angles in a circle?
3. How do chords in a circle affect the angles they subtend?
4. What is the sum of interior angles in any quadrilateral?
5. Can we apply these properties to non-cyclic quadrilaterals?

**Tip**: Remember, in cyclic quadrilaterals, opposite angles always add up to \( 180^\circ \).

Give two simplified expressions for BC^D.

Learn how to derive two simplified expressions for angle BCD using the properties of cyclic quadrilaterals and chord theorems. This problem covers essential concepts in geometry and circle theorems, suitable for Grades 10-12.

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