In a `Delta ABC, angle A : angle B: angleC = 2 : 3 : 4` A line CD drawn parallel to AB, then `angleACD` is

Options

A. `80^@`
B. `20^@`
C. `40^@`
D. `60^@`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given ratio of angles
We are given that the angles of triangle ABC are in the ratio:
- Angle A : Angle B : Angle C = 2 : 3 : 4

### Step 2: Assign variables to the angles
Let:
- Angle A = 2k
- Angle B = 3k
- Angle C = 4k

### Step 3: Use the property of triangles
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ 2k + 3k + 4k = 180 \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ 9k = 180 \]

### Step 5: Solve for k
Now, divide both sides by 9 to find the value of k:
\[ k = \frac{180}{9} = 20 \]

### Step 6: Calculate the angles
Now we can find the measures of each angle:
- Angle A = 2k = 2 × 20 = 40 degrees
- Angle B = 3k = 3 × 20 = 60 degrees
- Angle C = 4k = 4 × 20 = 80 degrees

### Step 7: Identify the parallel line and transversal
A line CD is drawn parallel to AB. Since CD is parallel to AB, and AC acts as a transversal, we can use the property of alternate interior angles.

### Step 8: Apply the alternate interior angle theorem
According to the alternate interior angle theorem:
- Angle ACD = Angle BAC

### Step 9: Substitute the known angle
Since we found that:
- Angle BAC = 40 degrees

We can conclude that:
- Angle ACD = 40 degrees

### Final Answer
Therefore, the measure of angle ACD is **40 degrees**.

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