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It looks like you're solving for angles in a triangle, likely using some relationships between the angles. From the image, there are three angles labeled \(28x^\circ\), \(16x^\circ\), and \(48^\circ\), with an equation you have started working on:

### Step 1: Understanding the problem
In the triangle, based on the exterior angle theorem, the exterior angle (in this case, the \(48^\circ\) angle) is equal to the sum of the two opposite interior angles, which are \(28x^\circ\) and \(16x^\circ\).

So the equation becomes:
\[
48^\circ = (28x^\circ) + (16x^\circ)
\]

### Step 2: Solve the equation
You can combine like terms on the right-hand side:
\[
48^\circ = 28x + 16x
\]
\[
48^\circ = 44x
\]

Now, solve for \(x\) by dividing both sides by 44:
\[
x = \frac{48^\circ}{44}
\]
\[
x = \frac{12}{11}
\]

### Step 3: Plug \(x\) back into the angles
Now that you know \(x = \frac{12}{11}\), substitute this back into the expressions for the angles:
- \(28x^\circ = 28 \times \frac{12}{11} = \frac{336}{11} \approx 30.55^\circ\)
- \(16x^\circ = 16 \times \frac{12}{11} = \frac{192}{11} \approx 17.45^\circ\)

These angles should sum to approximately \(48^\circ\), which is consistent with the exterior angle theorem.

### Converse of the theorem
The converse of the exterior angle theorem states that if an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, then the triangle is valid (follows the exterior angle theorem). In this case, we have verified that the exterior angle (\(48^\circ\)) is indeed the sum of the two opposite interior angles (\(30.55^\circ + 17.45^\circ\)), confirming the theorem.

Would you like more details on this? Here are 5 related questions that might interest you:

1. How do you prove the exterior angle theorem?
2. What happens when the sum of interior angles is not equal to the exterior angle?
3. Can the exterior angle theorem be applied to all polygons?
4. How is the converse of a theorem generally proven?
5. What other relationships between angles exist in a triangle?

**Tip:** Always verify angle relationships using basic geometric principles like the exterior angle theorem when dealing with triangles.

Find the value of x using angle relationships in a triangle with angles (28x)°, (16x)°, and 48°.

Learn how to solve for x in a triangle where the exterior angle is equal to the sum of two opposite interior angles, using the equation 48° = 28x + 16x. Suitable for students in Grades 7-9, this problem explores key concepts in algebra and geometry.

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