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The problem asks whether \( \angle ABD \) is a straight angle, given that \( \angle ABC \) and \( \angle CBD \) have the same measure.

We are given:
- \( \angle ABC = \frac{3x}{2} + 2^\circ \)
- \( \angle CBD = 2x - 29.5^\circ \)
- \( \angle ABC = \angle CBD \) (since they are said to have the same measure)

To solve this, we can set the two expressions for \( \angle ABC \) and \( \angle CBD \) equal to each other:
\[
\frac{3x}{2} + 2 = 2x - 29.5
\]

Now, let's solve for \( x \):

1. Subtract \( \frac{3x}{2} \) from both sides:
   \[
   2 + 29.5 = 2x - \frac{3x}{2}
   \]
   This simplifies to:
   \[
   31.5 = 2x - \frac{3x}{2}
   \]

2. To combine the terms on the right-hand side, rewrite \( 2x \) as \( \frac{4x}{2} \):
   \[
   31.5 = \frac{4x}{2} - \frac{3x}{2}
   \]
   Simplifying the right-hand side:
   \[
   31.5 = \frac{x}{2}
   \]

3. Multiply both sides by 2 to solve for \( x \):
   \[
   x = 63
   \]

Now, substitute \( x = 63 \) into either expression to find the measure of \( \angle ABC \) (or \( \angle CBD \)).

For \( \angle ABC \):
\[
\angle ABC = \frac{3(63)}{2} + 2 = 94.5 + 2 = 96.5^\circ
\]

For \( \angle CBD \):
\[
\angle CBD = 2(63) - 29.5 = 126 - 29.5 = 96.5^\circ
\]

Thus, \( \angle ABC = 96.5^\circ \) and \( \angle CBD = 96.5^\circ \).

Finally, to check if \( \angle ABD \) is a straight angle, we add \( \angle ABC \) and \( \angle CBD \):
\[
\angle ABD = 96.5^\circ + 96.5^\circ = 193^\circ
\]

Since \( \angle ABD \) is not equal to \( 180^\circ \), it is not a straight angle.

Would you like more details, or do you have any questions?

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Here are 5 related questions to expand on this problem:
1. What is a straight angle, and how is it defined geometrically?
2. Can two angles add up to more than 180° and still form a straight line?
3. How can you check if two angles are supplementary?
4. What would happen if \( \angle ABC \) were smaller? How would that affect \( \angle ABD \)?
5. What does it mean for two angles to have the same measure?

**Tip**: Remember that a straight angle always measures exactly \( 180^\circ \), and two angles that form a straight line are supplementary.

Given that ∠ABC and ∠CBD have the same measure, where ∠ABC = (3x/2) + 2° and ∠CBD = (2x − 29.5)°, is ∠ABD a straight angle?

Learn how to determine if ∠ABD is a straight angle by solving for x using the given equations for ∠ABC and ∠CBD. Explore key algebraic and geometric concepts, including angle relationships and supplementary angles. Ideal for students in Grades 8-10.

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