The angles of a quadrilateral are 5x-30, 4x+60, 60-x and 3x+61.find the smallest of these angles​

1. Analyze the question

We are given the four interior angles of a quadrilateral in terms of $x$. We need to find the value of $x$ first, then substitute it back into each expression to determine which one yields the smallest numerical value.

2. Explain the concept

The sum of the interior angles of any quadrilateral is always 360°. Therefore, we can set up an equation by adding all four given expressions and equating them to 360: $(5x - 30) + (4x + 60) + (60 - x) + (3x + 61) = 360$

3. Show the solution

Step 1: Solve for $x$ Group the like terms: $(5x + 4x - x + 3x) + (-30 + 60 + 60 + 61) = 360$ $(11x) + (151) = 360$

Subtract 151 from both sides: $11x = 360 - 151$ $11x = 209$

Divide by 11: $x = \frac{209}{11}$ $x = 19$

Step 2: Calculate each angle Now, substitute $x = 19$ into each expression:

• Angle 1: $5x - 30 = 5(19) - 30 = 95 - 30 = \mathbf{65^\circ}$
• Angle 2: $4x + 60 = 4(19) + 60 = 76 + 60 = \mathbf{136^\circ}$
• Angle 3: $60 - x = 60 - 19 = \mathbf{41^\circ}$
• Angle 4: $3x + 61 = 3(19) + 61 = 57 + 61 = \mathbf{118^\circ}$

Step 3: Identify the smallest angle Comparing the results ($65^\circ, 136^\circ, 41^\circ, 118^\circ$), the smallest angle is 41°, which corresponds to the expression $60 - x$.

4. Identify the answer

The expression $60 - x$ corresponds to option C.

5. Tip

When dealing with geometry problems involving "sum of angles," always write down the geometric rule first (e.g., "Sum of angles in a quadrilateral = 360°"). This helps you avoid silly mistakes when setting up your algebraic equation. Also, always double-check your arithmetic when adding the constants!