In `DeltaABC, /_A : /_B : /_C = 5:4:1`. What is the value (in degrees) of `/_B + /_C`?

Options

A. 50
B. 70
C. 90
D. 100

Answer

To solve the problem, we need to find the value of \( \angle B + \angle C \) given the ratio of the angles in triangle \( ABC \) as \( \angle A : \angle B : \angle C = 5 : 4 : 1 \).

### Step-by-Step Solution:

1. **Assign Variables Based on Ratios**:
Let the angles be represented in terms of a variable \( x \):
- \( \angle A = 5x \)
- \( \angle B = 4x \)
- \( \angle C = 1x \)

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in a triangle is always \( 180^\circ \).
Therefore, we can set up the equation:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the expressions for the angles:
\[ 5x + 4x + 1x = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ 10x = 180^\circ \]

4. **Solve for \( x \)**:
Divide both sides by 10:
\[ x = \frac{180^\circ}{10} = 18^\circ \]

5. **Find the Angles**:
Now, we can find the individual angles:
- \( \angle A = 5x = 5 \times 18^\circ = 90^\circ \)
- \( \angle B = 4x = 4 \times 18^\circ = 72^\circ \)
- \( \angle C = 1x = 1 \times 18^\circ = 18^\circ \)

6. **Calculate \( \angle B + \angle C \)**:
Now we can find \( \angle B + \angle C \):
\[ \angle B + \angle C = 72^\circ + 18^\circ = 90^\circ \]

### Final Answer:
The value of \( \angle B + \angle C \) is \( 90^\circ \).

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