Question

find the value of x and y if abcd ~ qsrp.
a
p
82°
97°
d
(x + 34)°
b
r
(3y - 13)°
q
83°
98°
c
s
x=
y=

Explanation:

Step1: Use property of similar polygons

For similar polygons, corresponding angles are equal. Angle $D$ in $ABCD$ corresponds to angle $R$ in $QSRP$. So, $x + 34=97$.

Step2: Solve for $x$

Subtract 34 from both sides of the equation $x + 34=97$. We get $x=97 - 34$.
$x = 63$

Step3: Use another pair of corresponding angles

Angle $C$ in $ABCD$ corresponds to angle $S$ in $QSRP$. This is just for verification as we already have enough to solve for $y$. Angle $A$ in $ABCD$ corresponds to angle $P$ in $QSRP$ which is correct as given ($82^{\circ}$ and $97^{\circ}$ respectively). Now, since the sum of angles in a quadrilateral is $360^{\circ}$, we can also use the fact that corresponding - angle equality holds. Let's use the equality of non - given corresponding angles. We know that the sum of angles in a quadrilateral is $360^{\circ}$. But we can also use the fact that since the polygons are similar, we can set up an equation for the other unknown angle. Angle $D$ in $ABCD$ corresponds to angle $R$ in $QSRP$. Now for the other unknown, we set up the equation for the remaining pair of corresponding angles. Let's assume we use the fact that the polygons are similar and equate the remaining non - used corresponding angles. We have $3y-13 = 83$.

Step4: Solve for $y$

Add 13 to both sides of the equation $3y-13 = 83$: $3y=83 + 13$.
$3y=96$. Then divide both sides by 3: $y=\frac{96}{3}$.
$y = 32$

Answer:

$x = 63$
$y = 32$