If the angles of triangle ABC are `2x^@ , (3x - 12)^@, and (5x – 18)^@` , then find the largest angle?

Options

A. `98^@`
B. `87^@`
C. `80^@`
D. `107^@`

Answer

To find the largest angle in triangle ABC with angles given as \(2x^\circ\), \((3x - 12)^\circ\), and \((5x - 18)^\circ\), we will follow these steps:

### Step 1: Write down the angles
The angles of the triangle are:
- Angle A = \(2x^\circ\)
- Angle B = \((3x - 12)^\circ\)
- Angle C = \((5x - 18)^\circ\)

### Step 2: Use the triangle angle sum property
According to the triangle angle sum property, the sum of the angles in a triangle is \(180^\circ\). Therefore, we can set up the equation:
\[ 2x + (3x - 12) + (5x - 18) = 180 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 2x + 3x + 5x - 12 - 18 = 180 \]
This simplifies to:
\[ 10x - 30 = 180 \]

### Step 4: Solve for \(x\)
Add \(30\) to both sides:
\[ 10x = 210 \]
Now, divide by \(10\):
\[ x = 21 \]

### Step 5: Find the angles
Now that we have the value of \(x\), we can find the measures of the angles:
- Angle A = \(2x = 2(21) = 42^\circ\)
- Angle B = \(3x - 12 = 3(21) - 12 = 63 - 12 = 51^\circ\)
- Angle C = \(5x - 18 = 5(21) - 18 = 105 - 18 = 87^\circ\)

### Step 6: Identify the largest angle
Now we compare the angles:
- Angle A = \(42^\circ\)
- Angle B = \(51^\circ\)
- Angle C = \(87^\circ\)

The largest angle is \(87^\circ\).

### Final Answer
The largest angle in triangle ABC is \(87^\circ\).

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