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For problem 25, the diagram shows two intersecting lines, with one angle given as \( 60^\circ \) and the other expression involving \( x \) as \( 7x - 3 \).

These angles appear to be *vertical angles*, which means they are equal to each other. Therefore, we can set up the equation:

\[
7x - 3 = 60
\]

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

\[
7x - 3 = 60 \quad \text{(Add 3 to both sides)}
\]
\[
7x = 63 \quad \text{(Divide both sides by 7)}
\]
\[
x = 9
\]

So, the value of \( x \) is **9**.

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For question 26, you are asked to select the statements that can be used to introduce angles in a proof. Here are the most relevant ones:

- **Linear Pair Theorem**: This theorem states that if two angles form a linear pair, they are supplementary (sum to \( 180^\circ \)).
- **Vertical Angle Theorem**: This theorem states that vertical angles (angles opposite each other when two lines intersect) are congruent.
- **Angle Addition Postulate**: This postulate says that if a point lies on the interior of an angle, the sum of the two smaller angles equals the larger angle.

The following are likely not directly useful at the top of a proof for introducing angles:
- **Definition of Bisector**: This refers to something cutting an angle or segment in half, not necessarily for introducing angle measures.
- **Definition of Right Angle**: This simply states that a right angle measures \( 90^\circ \).
- **Definition of Complementary**: This refers to two angles summing to \( 90^\circ \), but may not directly apply here.
- **Definition of Congruency**: This means that angles or segments are equal in measure, but it's not a tool for proving new relationships initially.

Let me know if you'd like to dive deeper into any specific topic!

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Would you like further clarification or have any additional questions?

Here are five related questions you might explore:

1. What are some common uses of the Vertical Angle Theorem in geometry?
2. How does the Linear Pair Theorem help in solving for unknown angle measures?
3. Can you explain how the Angle Addition Postulate is applied in geometric proofs?
4. What are complementary and supplementary angles, and how do they differ?
5. How does the definition of congruency apply to angles in parallel line proofs?

**Tip:** Always look for relationships like vertical angles or linear pairs when solving for unknown angles in intersecting lines.

Use the diagram below to find the value of x: 7x - 3 = 60, vertical angles

Learn how to solve for x using the Vertical Angle Theorem in a geometry problem involving two intersecting lines. We explain the step-by-step process, starting with the equation 7x - 3 = 60.

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