Transcribed Image Text:### Triangle Angle Calculation
The diagram depicts a triangle labeled as ABC. The angles at each vertex are given as follows:
- Angle at vertex A is \((4x - 4)^\circ\)
- Angle at vertex B is \((x - 44)^\circ\)
- Angle at vertex C is \(\left(\frac{1}{2} x - 52\right)^\circ\)
#### Explanation of the Diagram
- Vertex A has the angle \((4x - 4)^\circ\), which is indicated near the vertex with a proper labeled angle arc.
- Vertex B has the angle \((x - 44)^\circ\), similarly indicated near the vertex with an angle arc.
- Vertex C has the angle \(\left(\frac{1}{2} x - 52\right)^\circ\).
### Objective
The angles within the triangle sum up to \(180^\circ\). Thus, to find the value of \(x\):
1. **Write the equation representing the sum of the angles:**
\[
(4x - 4) + (x - 44) + \left(\frac{1}{2} x - 52\right) = 180
\]
2. **Combine like terms:**
\[
4x - 4 + x - 44 + \frac{1}{2} x - 52 = 180
\]
3. **Combine the \(x\) terms:**
\[
4x + x + \frac{1}{2} x = 5.5x
\]
4. **Combine the constant terms:**
\[
-4 - 44 - 52 = -100
\]
So the equation becomes:
\[
5.5x - 100 = 180
\]
5. **Solve for \(x\):**
\[
5.5x = 280
\]
\[
x = \frac{280}{5.5}
\]
\[
x \approx 50.91
\]
Thus, substituting \(x\) back into the original angles will give the exact measurements of the angles at vertices A, B, and C.
### Checking
To ensure accuracy, substitute \(x\) back into each angle expression to verify their sum is \(180^\circ\).
- Angle A: \(4(50.91) - 4
Figure in plane geometry formed by two rays or lines that share a common endpoint, called the vertex. The angle is measured in degrees using a protractor. The different types of angles are acute, obtuse, right, straight, and reflex.
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![### Triangle Angle Calculation
The diagram depicts a triangle labeled as ABC. The angles at each vertex are given as follows:
- Angle at vertex A is \((4x - 4)^\circ\)
- Angle at vertex B is \((x - 44)^\circ\)
- Angle at vertex C is \(\left(\frac{1}{2} x - 52\right)^\circ\)
#### Explanation of the Diagram
- Vertex A has the angle \((4x - 4)^\circ\), which is indicated near the vertex with a proper labeled angle arc.
- Vertex B has the angle \((x - 44)^\circ\), similarly indicated near the vertex with an angle arc.
- Vertex C has the angle \(\left(\frac{1}{2} x - 52\right)^\circ\).
### Objective
The angles within the triangle sum up to \(180^\circ\). Thus, to find the value of \(x\):
1. **Write the equation representing the sum of the angles:**
\[
(4x - 4) + (x - 44) + \left(\frac{1}{2} x - 52\right) = 180
\]
2. **Combine like terms:**
\[
4x - 4 + x - 44 + \frac{1}{2} x - 52 = 180
\]
3. **Combine the \(x\) terms:**
\[
4x + x + \frac{1}{2} x = 5.5x
\]
4. **Combine the constant terms:**
\[
-4 - 44 - 52 = -100
\]
So the equation becomes:
\[
5.5x - 100 = 180
\]
5. **Solve for \(x\):**
\[
5.5x = 280
\]
\[
x = \frac{280}{5.5}
\]
\[
x \approx 50.91
\]
Thus, substituting \(x\) back into the original angles will give the exact measurements of the angles at vertices A, B, and C.
### Checking
To ensure accuracy, substitute \(x\) back into each angle expression to verify their sum is \(180^\circ\).
- Angle A: \(4(50.91) - 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41774409-65e0-43ad-a43d-34f80707331e%2Fb7d0e618-20bb-4dba-b888-97c9a86d0ad7%2Fqw9hct8_processed.jpeg&w=3840&q=75)
