https://file.aiquickdraw.com/mathbot/file/1729522590432vkxw5e7q.jpg

Let's break down and go through the steps for solving problems **#1** and **#4** from the image you provided.

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### Problem 1:

You are given a pair of intersecting lines, which form angles of 128° and \( x \). The problem asks you to identify if the angles are **vertical** or **adjacent**, and then find the value of \( x \).

#### Steps:
1. **Identify the relationship**: Since the angles are opposite each other, they are **vertical angles**. Vertical angles are always **equal**.
2. **Set up the equation**: The two angles must be equal:
   \[
   x = 128^\circ
   \]
3. **Solution**: Since the angles are equal, \( x = 128^\circ \).

The work shown in the image matches this reasoning.

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### Problem 4:

In this problem, there are two adjacent angles given as \( 4x^\circ \) and \( 2x^\circ \). The sum of these two angles forms a straight line, meaning they are supplementary angles.

#### Steps:
1. **Recognize the relationship**: Adjacent angles that form a straight line are **supplementary**, meaning their sum is 180°.
2. **Set up the equation**: The sum of the two angles \( 4x^\circ + 2x^\circ \) must equal 180°:
   \[
   4x + 2x = 180
   \]
3. **Solve for \( x \)**:
   \[
   6x = 180
   \]
   Divide both sides by 6:
   \[
   x = 30^\circ
   \]

So, \( x = 30^\circ \).

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Both problems involve identifying the relationship between angles (vertical or adjacent) and using basic angle properties like equality and supplementary angles to solve for \( x \).

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Do you have any questions, or would you like further details on any of these steps?

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Here are 5 questions related to this concept to expand your understanding:

1. What is the difference between vertical and adjacent angles?
2. How would you solve for \( x \) if the angles formed a right angle instead of a straight line in problem 4?
3. What property do complementary angles have, and how does it differ from supplementary angles?
4. How do you solve for an unknown angle in a triangle if two angles are given?
5. If two angles are labeled as \( 3x \) and \( 5x \), and they form a straight line, what is the value of \( x \)?

**Tip**: Always double-check if angles are supposed to be equal or if they sum to a particular value, like 90° (right angles) or 180° (straight lines).

This guide walks through the solution for problems involving vertical and supplementary angles, explaining how to find the value of x in different geometric configurations. Suitable for students in Grades 7-9, the solution covers the relationship between angles, including vertical and supplementary angles, with step-by-step calculations.

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