BO and CO are external bisector of `angleB` and `angleC` of `triangle`ABC intersecting at O. If `angleA=60^@, angleABC=70^@`, Find `angleBOC`

Answer

To find the angle \( \angle BOC \) in triangle \( ABC \) where \( BO \) and \( CO \) are the external bisectors of angles \( B \) and \( C \) respectively, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- \( \angle A = 60^\circ \)
- \( \angle ABC = 70^\circ \)

### Step 2: Calculate \( \angle C \)
Using the angle sum property of a triangle, which states that the sum of the internal angles in a triangle is \( 180^\circ \):
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the known values:
\[ 60^\circ + 70^\circ + \angle C = 180^\circ \]
Now, solving for \( \angle C \):
\[ \angle C = 180^\circ - 60^\circ - 70^\circ = 50^\circ \]

### Step 3: Determine the Angles at Point O
Since \( BO \) is the external bisector of \( \angle B \), it divides \( \angle B \) into two equal parts:
\[ \angle OBC = \angle OBA = 70^\circ \]

Similarly, since \( CO \) is the external bisector of \( \angle C \):
\[ \angle OCB = \angle OCA = 50^\circ \]

### Step 4: Set Up the Equation for Triangle BOC
Now, we need to find \( \angle BOC \). In triangle \( BOC \), the sum of the angles is also \( 180^\circ \):
\[ \angle BOC + \angle OBC + \angle OCB = 180^\circ \]
Substituting the known values:
\[ \angle BOC + 70^\circ + 50^\circ = 180^\circ \]

### Step 5: Solve for \( \angle BOC \)
Now, we can solve for \( \angle BOC \):
\[ \angle BOC = 180^\circ - 70^\circ - 50^\circ \]
Calculating this gives:
\[ \angle BOC = 180^\circ - 120^\circ = 60^\circ \]

### Final Answer
Thus, the value of \( \angle BOC \) is \( 60^\circ \).

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