The angles of a cyclic quadrilateral are in the ratio 1:3:4:4. What is the measure of the largest angle?

The correct option is 120°.

Step 1: Understand the properties of a cyclic quadrilateral.
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. A key property of any cyclic quadrilateral is that the sum of its opposite angles is equal to 180°.

Also, the sum of all interior angles of any quadrilateral is always 360°.

Step 2: Set up the angles using the given ratio.
Let the angles of the cyclic quadrilateral taken in order be $A_{}$, $B_{}$, $C_{}$, and $D_{}$.
Given the ratio of the angles is 1:3:4:4, we can express the angles in terms of a common ratio variable, $x$:

$A=1x$

$B=3x$

$C=4x$

$D=4x$

Notice that opposite angles are ($A$, $C$) and ($B$, $D$).
Let's check the sum of opposite angles:
$A+C=x+4x=5x$
$B+D=3x+4x=7x$

Since the total sum of all four angles in any quadrilateral is 360°:

$x+3x+4x+4x=360°$

$12x=360°$

$x=\frac{360°}{12}=30°$

Step 3: Calculate the measures of each angle.

First angle = $1\times 30°=30°$

Second angle = $3\times 30°=90°$

Third angle = $4\times 30°=120°$

Fourth angle = $4\times 30°=120°$

Step 4: Identify the largest angle.
Comparing the calculated angle measures (30°, 90°, 120°, and 120°), the measure of the largest angle is 120°.