Here are detailed steps for solving the practice problems:

Problem 1: Triangle Side Length Condition

Problem: Two sides of a triangle measure 5 inches and 9 inches. Which could not be the length of the third side?

1. Use the Triangle Inequality Theorem, which states that the sum of any two sides must be greater than the third side.
2. Calculate:
   • $5 + 9 > x$ ➔ $x < 14$
   • $x + 5 > 9$ ➔ $x > 4$
   • $x + 9 > 5$ ➔ (always true)
3. The possible range for $x$ is $4 < x < 14$.
4. Check the answer choices to find any that do not fit this range.

Solution: Option $b) \,15$, as 15 is outside $4 < x < 14$.

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Problem 2: Solve for $x$ in Figures

Steps:

1. For each figure, apply any relevant triangle properties or angle relationships.
2. Use given values to set up and solve an equation based on the diagram's properties.

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Problem 4: Regular Polygon Angles

Problem: Solve for $x$ and $y$ in a regular polygon.

1. In a regular polygon, each interior angle is the same, and all sides are equal.
2. To find each angle:
   • Use the formula for an interior angle of a regular polygon: $\frac{(n - 2) \times 180}{n}$, where $n$ is the number of sides.
3. Set the calculated angle equal to any provided expressions in the problem to solve for $x$ and $y$.

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Problem 6: Find AE in Triangle with Circumcenter G

1. The circumcenter is the point equidistant from all vertices.
2. Use any given side lengths or properties to determine the missing side $AE$ by setting up an equation.

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Problem 12: Parallelogram Conditions

Problem: Select conditions that prove a quadrilateral is a parallelogram.

1. Go through each condition:
   • Opposite sides congruent
   • Opposite angles congruent
   • Diagonals bisect each other
   • Diagonals are congruent/perpendicular
2. Select only those conditions that are always true for a parallelogram.

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Would you like more detailed explanations for any of these steps?

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Related Practice Questions:

1. How do you calculate the range of a third side in a triangle with two given sides?
2. What are the properties of an angle bisector in a triangle?
3. How do you determine if two triangles are similar using the AA, SSS, and SAS criteria?
4. What steps are needed to find the interior angles of any polygon?
5. How do you calculate the length of a side in a parallelogram using diagonals?

Tip: For triangle problems, always verify conditions with the Triangle Inequality Theorem.