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 (square^{circ}).

Explanation:

Step1: Define the smallest angle

Let the measure of the smallest angle be $x = 63^{\circ}$.

Step2: Find the measure of the first - mentioned angle

The measure of one angle is 10° greater than the smallest angle. So this angle $A_1=x + 10=63+10 = 73^{\circ}$.

Step3: Find the measure of the second - mentioned angle

The third angle is 11° greater than twice the smallest angle. Twice the smallest angle is $2x$, and the third angle $A_2=2x + 11=2\times63+11=126 + 11=137^{\circ}$.

Step4: Use the angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is 360°. Let the fourth angle be $A_3$. We know that $x+(x + 10)+(2x + 11)+A_3=360$. Substitute $x = 63$ into the equation: $63+(63 + 10)+(2\times63+11)+A_3=360$. First, simplify the left - hand side: $63+73+137+A_3=360$. Then, $63+73+137=273$. So, $273+A_3=360$. Solve for $A_3$: $A_3=360 - 273=87^{\circ}$.

Answer:

$73$