Three lines intersect at a point generating six angles . If one of these angles is `90^(@)` , then the number of other distinct angles is :

Options

A. 1 or 2
B. 1 or 3
C. 2 or 3
D. 2 or 4

Answer

To solve the problem step by step, we will analyze the situation involving three intersecting lines that create six angles, with one of those angles being 90 degrees.

### Step 1: Understand the Configuration
Three lines intersect at a single point, creating six angles. We can label these lines as PQ, RS, and TU.

**Hint:** Visualize the intersection of the lines to see how the angles are formed.

### Step 2: Identify the Given Angle
We are given that one of these angles is 90 degrees. Let's denote this angle as ∠POS.

**Hint:** Remember that a right angle measures 90 degrees.

### Step 3: Determine Vertically Opposite Angles
When two lines intersect, the angles opposite each other are equal. Therefore, if ∠POS = 90 degrees, then the angle opposite to it, which we can denote as ∠ROQ, is also 90 degrees.

**Hint:** Use the property of vertically opposite angles to find equal angles.

### Step 4: Analyze Remaining Angles
Now we have two angles (∠POS and ∠ROQ) that are both 90 degrees. This means we have two angles already accounted for. The remaining four angles will be formed by the intersection of the third line (TU) with the other two lines (PQ and RS).

**Hint:** Count how many angles are left after accounting for the known angles.

### Step 5: Identify Distinct Angles
The angles formed by the intersection of the third line with the other two lines can be either equal or different. If the angles are equal, we will have two distinct angles (90 degrees and the other angle). If they are different, we will have three distinct angles (90 degrees and two different angles).

**Hint:** Consider the possibility of equal angles versus different angles when the third line intersects.

### Step 6: Conclusion
Regardless of whether the remaining angles are equal or different, we can conclude that there will be a maximum of two distinct angles formed by the third line intersecting with the other two lines. Therefore, the number of other distinct angles, excluding the two 90-degree angles, is:

- If the angles are equal, we have 1 distinct angle.
- If the angles are different, we have 2 distinct angles.

Thus, the answer is **2 distinct angles**.

**Final Answer:** The number of other distinct angles is **2**.