The interior angles of a quadrilateral are (x + 15)°, (2x - 45)°, ( x - 30)° and (x + 10)°. Find the value of the least interior angle.

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 identify the smallest angle.

2. Explain the concept

The sum of the interior angles of any quadrilateral is always 360°. The formula for the sum of interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$. For a quadrilateral ($n=4$): $(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$.

3. Show the solution

Step 1: Set up the equation Sum the given angles and equate them to 360°: $(x + 15) + (2x - 45) + (x - 30) + (x + 10) = 360$

Step 2: Simplify the equation Group the $x$ terms and the constant numbers: $(x + 2x + x + x) + (15 - 45 - 30 + 10) = 360$ $5x - 50 = 360$

Step 3: Solve for $x$ $5x = 360 + 50$ $5x = 410$ $x = \frac{410}{5}$ $x = 82$

Step 4: Calculate each angle Now, substitute $x = 82$ into each expression:

1. $(x + 15) = 82 + 15 = 97^\circ$
2. $(2x - 45) = 2(82) - 45 = 164 - 45 = 119^\circ$
3. $(x - 30) = 82 - 30 = 52^\circ$
4. $(x + 10) = 82 + 10 = 92^\circ$

Step 5: Identify the least angle Comparing $97^\circ, 119^\circ, 52^\circ,$ and $92^\circ$, the smallest value is 52°.

4. Identify the answer

The calculated value for the least interior angle is 52°, which corresponds to option D.

5. Tip

When dealing with geometry problems involving variables, always check your final answers by adding them up. If they don't sum to the required total (360° for a quadrilateral), you know you made a calculation error in your algebra!