In a triangle ABC, `angleA = 90^@, angleC = 55^@`, AD is perpendicular to BC. What is the value of `angleBAD`?

Options

A. `35^@`
B. `60^@`
C. `45^@`
D. `55^@`

Answer

To find the value of angle BAD in triangle ABC, where angle A = 90° and angle C = 55°, we can follow these steps:

### Step 1: Determine angle B
In a triangle, the sum of all angles is 180°. Since angle A is 90° and angle C is 55°, we can find angle B as follows:

\[ \text{angle B} = 180° - \text{angle A} - \text{angle C} \]
\[ \text{angle B} = 180° - 90° - 55° = 35° \]

### Step 2: Analyze triangle ADC
Since AD is perpendicular to BC, triangle ADC is also a right triangle. We know that angle ACD is 90° (because AD is perpendicular to BC).

### Step 3: Find angle DAC
In triangle ADC, we can find angle DAC using the fact that the sum of angles in a triangle is 180°:

\[ \text{angle DAC} = 180° - \text{angle ACD} - \text{angle A} \]
\[ \text{angle DAC} = 180° - 90° - 55° = 35° \]

### Step 4: Find angle BAD
Now, we can find angle BAD. Since angle BAD and angle DAC are complementary (they add up to angle B), we can write:

\[ \text{angle BAD} + \text{angle DAC} = \text{angle B} \]
\[ \text{angle BAD} + 35° = 35° \]
\[ \text{angle BAD} = 35° - 35° = 0° \]

However, this is incorrect as we have made an error in understanding the angles.

### Correct Calculation of angle BAD
From triangle ABC, we know:
- angle A = 90°
- angle C = 55°
- angle B = 35°

Since AD is perpendicular to BC, angle BAD is equal to angle C:

\[ \text{angle BAD} = \text{angle C} = 55° \]

### Final Answer
Thus, the value of angle BAD is:

\[ \text{angle BAD} = 55° \]