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Find the measure of DCA

A. 145
B. 159
C. 92
D. 113                         

To find the measure of angle \( \angle DCA \), we need to use the properties of circles and angles.

Here is the given information:
- \( \angle DAB = 91^\circ \)
- \( \angle ACB = (17x - 3)^\circ \)
- \( \angle ABD = (3x + 19)^\circ \)

Since \( \angle DAB \) and \( \angle DCA \) are supplementary angles (they are angles on a straight line), their sum should be \( 180^\circ \).

First, let’s solve for \( x \).

1. From the image:
   \[
   \angle ACB + \angle ABD = \angle DAB
   \]
   Substitute the given angles:
   \[
   (17x - 3) + (3x + 19) = 91
   \]
   Simplify and solve for \( x \):
   \[
   20x + 16 = 91
   \]
   \[
   20x = 75
   \]
   \[
   x = \frac{75}{20}
   \]
   \[
   x = 3.75
   \]

2. Now substitute \( x \) back into the angles:
   \[
   \angle ACB = 17x - 3 = 17(3.75) - 3 = 63.75 - 3 = 60.75^\circ
   \]
   \[
   \angle ABD = 3x + 19 = 3(3.75) + 19 = 11.25 + 19 = 30.25^\circ
   \]

Since we know the relationship \( \angle DAB = \angle ACB + \angle ABD \), we can validate our calculations:

\[
60.75^\circ + 30.25^\circ = 91^\circ 
\]

This verification is correct.

Now, since \( \angle DAB = 91^\circ \) and \(\angle DCA\) are supplementary:

\[
\angle DCA = 180^\circ - 91^\circ = 89^\circ
\]

Hence, the correct answer is:

C. 89

If you have any further questions or need more details, please let me know. Here are 5 more questions you might want to ask:

1. How do you find the measure of inscribed angles in a circle?
2. What are the properties of supplementary angles?
3. Can you explain how to solve equations involving multiple variables?
4. How do you verify your solution in geometry problems?
5. What are the different types of angles formed in circles?

**Tip:** Always double-check your calculations and ensure your understanding of angle relationships in geometric figures.

Discover how to find the measure of angle DCA using circle geometry principles and supplementary angles. This problem involves calculating angles in a triangle and applying the angle sum property. Suitable for high school students studying geometry.

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