In `Delta ABC`, `angle A + angle B= 65^@ , angle B + angle C = 140^@ ` then find `angle B`.

Options

A. `40^(@)`
B. `25^(@)`
C. `35^@`
D. `20^@`

Answer

To solve the problem, we need to find the value of angle B in triangle ABC given the following equations:

1. \( \angle A + \angle B = 65^\circ \) (Equation 1)
2. \( \angle B + \angle C = 140^\circ \) (Equation 2)

We also know that the sum of the angles in a triangle is always \( 180^\circ \):

3. \( \angle A + \angle B + \angle C = 180^\circ \) (Equation 3)

### Step-by-Step Solution:

**Step 1: Express angle A in terms of angle B using Equation 1.**

From Equation 1:
\[ \angle A = 65^\circ - \angle B \]

**Hint:** Rearranging equations can help isolate one variable.

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**Step 2: Express angle C in terms of angle B using Equation 2.**

From Equation 2:
\[ \angle C = 140^\circ - \angle B \]

**Hint:** Similar to the previous step, isolating variables can simplify the problem.

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**Step 3: Substitute the expressions for angle A and angle C into Equation 3.**

Substituting the expressions for \( \angle A \) and \( \angle C \) into Equation 3:
\[ (65^\circ - \angle B) + \angle B + (140^\circ - \angle B) = 180^\circ \]

**Hint:** Substituting known values into an equation can help simplify it further.

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**Step 4: Simplify the equation.**

Combining like terms:
\[ 65^\circ + 140^\circ - \angle B = 180^\circ \]
\[ 205^\circ - \angle B = 180^\circ \]

**Hint:** Always combine like terms to simplify your equations.

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**Step 5: Solve for angle B.**

Rearranging the equation gives:
\[ -\angle B = 180^\circ - 205^\circ \]
\[ -\angle B = -25^\circ \]
\[ \angle B = 25^\circ \]

**Hint:** When isolating a variable, remember to change the sign when multiplying or dividing by a negative number.

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### Conclusion:

The value of \( \angle B \) is \( 25^\circ \).

**Final Answer:** \( \angle B = 25^\circ \)

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