In the given the length of arcs AB and BC are in the ratio `3:2.`
If `angle AOB = 96 ^(@)` find
` angle ABC`

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Given Information
We are given that the lengths of arcs AB and BC are in the ratio of 3:2, and the angle AOB is 96 degrees. We need to find the angle ABC.

### Step 2: Relate the Angles to the Arcs
Since the lengths of the arcs are in the ratio of 3:2, the angles subtended by these arcs at the center of the circle (angle AOB and angle BOC) will also be in the same ratio.

Let:
- Angle AOB = 3x
- Angle BOC = 2x

From the problem, we know that:
\[ 3x = 96^\circ \]

### Step 3: Solve for x
To find x, we can set up the equation:
\[ x = \frac{96^\circ}{3} = 32^\circ \]

### Step 4: Find Angle BOC
Now we can find angle BOC:
\[ BOC = 2x = 2 \times 32^\circ = 64^\circ \]

### Step 5: Analyze Triangle OBC
In triangle OBC, since OB = OC (both are radii of the circle), angles OBC and OCB are equal. Let:
- Angle OBC = Angle OCB = y

Using the angle sum property of triangles:
\[ OBC + OCB + BOC = 180^\circ \]
\[ y + y + 64^\circ = 180^\circ \]
\[ 2y + 64^\circ = 180^\circ \]
\[ 2y = 180^\circ - 64^\circ \]
\[ 2y = 116^\circ \]
\[ y = 58^\circ \]

### Step 6: Analyze Triangle OAB
In triangle OAB, we know:
- Angle AOB = 96 degrees
- Angles OAB and OBA are equal (let's call them z).

Using the angle sum property again:
\[ OAB + OBA + AOB = 180^\circ \]
\[ z + z + 96^\circ = 180^\circ \]
\[ 2z + 96^\circ = 180^\circ \]
\[ 2z = 180^\circ - 96^\circ \]
\[ 2z = 84^\circ \]
\[ z = 42^\circ \]

### Step 7: Find Angle ABC
Finally, angle ABC can be found by adding angles OAB and OBC:
\[ ABC = OAB + OBC = 42^\circ + 58^\circ = 100^\circ \]

### Conclusion
Thus, the angle ABC is:
\[ \text{Angle ABC} = 100^\circ \]

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