In a triangle ABC, the side BC is extended up to D. Such that `CD = AC`, if `/_BAD = 109^@` and `/_ACB = 72^@` then the value of `/_ABC `is

Options

A. `35^@`
B. `60^@`
C. ` 40^@`
D. `45^@`

Answer

To solve the problem step by step, we will analyze the triangle ABC with the given conditions.

### Step 1: Draw the Diagram
Draw triangle ABC and extend side BC to point D such that CD = AC. Mark the angles as given: ∠BAD = 109° and ∠ACB = 72°.

**Hint:** Always start by visualizing the problem with a diagram. It helps in understanding the relationships between the angles and sides.

### Step 2: Identify Isosceles Triangle
Since CD = AC, triangle ACD is isosceles. Therefore, the angles opposite to equal sides are equal. Let the angles ∠CAD and ∠CDA both be θ.

**Hint:** Recall that in an isosceles triangle, the angles opposite the equal sides are equal.

### Step 3: Set Up the Equation for Angles in Triangle ACD
The sum of angles in triangle ACD is:
\[ \angle CAD + \angle ACD + \angle CDA = 180° \]
Substituting the known values:
\[ θ + 72° + θ = 180° \]
This simplifies to:
\[ 2θ + 72° = 180° \]

**Hint:** Use the property that the sum of angles in a triangle is always 180°.

### Step 4: Solve for θ
Rearranging the equation gives:
\[ 2θ = 180° - 72° \]
\[ 2θ = 108° \]
\[ θ = 54° \]

**Hint:** Isolate the variable to find its value.

### Step 5: Analyze Triangle ABD
Now, consider triangle ABD. We know:
- ∠BAD = 109°
- ∠CAD = θ = 54°

Let ∠ABC = α (the angle we want to find). The sum of angles in triangle ABD gives us:
\[ ∠BAD + ∠ABC + ∠CAD = 180° \]
Substituting the known values:
\[ 109° + α + 54° = 180° \]

**Hint:** Again, apply the triangle angle sum property.

### Step 6: Solve for α
Rearranging the equation gives:
\[ α = 180° - 109° - 54° \]
\[ α = 180° - 163° \]
\[ α = 17° \]

**Hint:** Carefully perform the arithmetic to ensure accuracy.

### Final Answer
Thus, the value of ∠ABC is:
\[ \boxed{17°} \]