Question

the measure of one angle of a quadrilateral is 10° greater than the smallest angle, the third angle is 11° greater than twice the smallest angle, and the fourth angle is 24° greater than the smallest angle. the measure of the smallest angle is 63°. the measure of the second angle is 73°. the measure of the third angle is 〇°.

Explanation:

Step1: Define the smallest angle

Let the measure of the smallest angle be $x$.

Step2: Express other angles in terms of $x$

The first - given angle: $x + 10$; the second - given angle: $73^{\circ}$; the third - given angle: $2x+11$.

Step3: Use the angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. So, $(x + 10)+73+(2x + 11)+(x+24)=360$.

Step4: Combine like terms

$x+10 + 73+2x + 11+x+24=360$, which simplifies to $(x + 2x+x)+(10 + 73+11+24)=360$, then $4x+118 = 360$.

Step5: Solve for $x$

Subtract 118 from both sides: $4x=360 - 118=242$. Then $x=\frac{242}{4}=60.5$.

Step6: Find the measure of the third angle

The third angle is $2x + 11$. Substitute $x = 60.5$ into it: $2\times60.5+11=121 + 11=132$.

Answer:

$132$