To solve the problem of determining the smallest angle in a quadrilateral with angles in the ratio [tex]\(3: 4: 5: 6\)[/tex], we can follow these steps:
1. Understand the Sum of the Angles in a Quadrilateral:
The sum of all the interior angles in any quadrilateral is always [tex]\(360^\circ\)[/tex].
2. Use the Ratio to Express Each Angle:
Given the ratio [tex]\(3: 4: 5: 6\)[/tex], we have four parts:
- Let each part of the ratio be represented by [tex]\(x\)[/tex].
- Hence, the angles can be represented as [tex]\(3x, 4x, 5x,\)[/tex] and [tex]\(6x\)[/tex].
3. Set Up the Equation for the Sum of the Angles:
According to the properties of a quadrilateral:
[tex]\[
3x + 4x + 5x + 6x = 360^\circ
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Combine the terms on the left-hand side:
[tex]\[
18x = 360^\circ
\][/tex]
Divide both sides by 18:
[tex]\[
x = \frac{360^\circ}{18}
\][/tex]
[tex]\[
x = 20^\circ
\][/tex]
5. Determine Each Angle:
Substitute [tex]\(x = 20^\circ\)[/tex] back into the expressions for each angle:
[tex]\[
3x = 3 \cdot 20^\circ = 60^\circ
\][/tex]
[tex]\[
4x = 4 \cdot 20^\circ = 80^\circ
\][/tex]
[tex]\[
5x = 5 \cdot 20^\circ = 100^\circ
\][/tex]
[tex]\[
6x = 6 \cdot 20^\circ = 120^\circ
\][/tex]
6. Identify the Smallest Angle:
From the calculated angles [tex]\(60^\circ, 80^\circ, 100^\circ,\)[/tex] and [tex]\(120^\circ\)[/tex], the smallest angle is [tex]\(60^\circ\)[/tex].
7. Compare with the Provided Choices:
The answer choices given are:
- [tex]\(45^\circ\)[/tex]
- [tex]\(60^\circ\)[/tex]
- [tex]\(36^\circ\)[/tex]
- [tex]\(48^\circ\)[/tex]
Identifying the smallest calculated angle as [tex]\(60^\circ\)[/tex], we confirm that it matches one of the provided options.
Thus, the smallest angle of the quadrilateral is [tex]\(\boxed{60^\circ}\)[/tex].
