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The problem shows a parallelogram \( ABCD \), with given expressions for angles \( \angle ABC \) and \( \angle CDA \). In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to \( 180^\circ \)).

We are given:
\[
\angle ABC = (12x + 3)^\circ
\]
\[
\angle CDA = (10x + 13)^\circ
\]
Since these two angles are adjacent, they must sum to \( 180^\circ \). Therefore, we can set up the equation:
\[
(12x + 3) + (10x + 13) = 180
\]

Let's solve for \( x \):

\[
12x + 3 + 10x + 13 = 180
\]
\[
22x + 16 = 180
\]
\[
22x = 180 - 16
\]
\[
22x = 164
\]
\[
x = \frac{164}{22} = 7.45
\]

Now, substitute \( x = 7.45 \) back into the expression for \( \angle ABC \) to find its actual measure:

\[
\angle ABC = 12(7.45) + 3 = 89.4 + 3 = 92.4^\circ
\]

Thus, the measure of \( \angle ABC \) is approximately **92.4°**.

Would you like further details or clarifications?

Here are five related questions to expand on this topic:
1. How can you determine the other interior angles of the parallelogram using these results?
2. How do we confirm that the opposite angles in a parallelogram are equal?
3. What is the relationship between the diagonals of a parallelogram, and how might they affect the angle measures?
4. What would happen to the angle measures if this were a rectangle instead of a parallelogram?
5. Can you explain why adjacent angles in a parallelogram must be supplementary?

**Tip:** In any quadrilateral, the sum of all interior angles is always \( 360^\circ \).

Given that the measure of ∠ABC = (12x + 3)° and the measure of ∠CDA = (10x + 13)°, find the actual measure of ∠ABC. Round to two decimal places, if necessary. Do not include the units.

This problem involves solving for the measure of an angle in a parallelogram using algebraic expressions for adjacent angles. Learn how to apply the properties of parallelograms and algebraic manipulation to find the solution.

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