To determine the measure of the greatest angle in the given quadrilateral, we follow these steps:
1. Identify the expressions for the angles:
- The angles given in the problem are [tex]\( x^{\circ} \)[/tex], [tex]\( (x+10)^{\circ} \)[/tex], [tex]\( (x+20)^{\circ} \)[/tex], and [tex]\( (x+30)^{\circ} \)[/tex].
2. Recall the sum of the angles in a quadrilateral:
- The sum of the interior angles in any quadrilateral is [tex]\( 360^{\circ} \)[/tex].
3. Set up the equation:
[tex]\[
x + (x + 10) + (x + 20) + (x + 30) = 360
\][/tex]
4. Combine like terms:
[tex]\[
4x + 60 = 360
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Subtract 60 from both sides:
[tex]\[
4x = 300
\][/tex]
- Divide by 4:
[tex]\[
x = 75
\][/tex]
6. Calculate the measures of each angle:
- First angle: [tex]\( x = 75^{\circ} \)[/tex]
- Second angle: [tex]\( x + 10 = 75 + 10 = 85^{\circ} \)[/tex]
- Third angle: [tex]\( x + 20 = 75 + 20 = 95^{\circ} \)[/tex]
- Fourth angle: [tex]\( x + 30 = 75 + 30 = 105^{\circ} \)[/tex]
7. Identify the greatest angle:
- The angles are [tex]\( 75^{\circ} \)[/tex], [tex]\( 85^{\circ} \)[/tex], [tex]\( 95^{\circ} \)[/tex], and [tex]\( 105^{\circ} \)[/tex].
- The greatest angle among these is [tex]\( 105^{\circ} \)[/tex].
Thus, the measure of the greatest angle in the quadrilateral is [tex]\( \boxed{105} \)[/tex].
