ACT 1 Geo: Apply theorems, definitions, and axioms

$Q_{1}:$ In the following figure, triangles $ABC$ and $DEF$ are similar. What is the value of $x$?

$Q_{2}:$ Which of the following represents $\frac{1}{2}$ of $\frac{1}{20}$?

$Q_{3}:$ In the following figure, rectangles $ABCD$ and $EFGD$ are similar. What is the perimeter of $EFGD$?

$Q_{4}:$ If $x$ is inversely proportional to $y$ and $x = 12$ when $y = 48$, then what is the value of $x$ when $y = 12$?

$Q_{5}:$ If $80%$ of a number is $122$, what is $40%$ of the number?

$Q_{6}:$ The front of a house, shown in the unscaled diagram below, was damaged in a hailstorm. New wood trim is needed around the perimeter of the house as well as around the door and window, but trim is not needed along the base of the house and door. The wall and the trim also need $2$ fresh coats of paint. The window is a square with $4$-foot sides, and the door is $6$ feet tall and $3$ feet wide. House trim costs $\$ 0.50$ per foot. How much money, to the nearest dollar must be budgeted to replace the trim?

$Q_{7}:$ In previous question, one gallon of paint covers about 400 square feet. How many gallons of paint are needed for the project detailed above?

$Q_{8}:$ A rectangle has a width of $5$ meters and a length of $14$ meters. If a similar rectangle has a width of $15$ meters, what is its perimeter, in meters?

$Q_{9}:$ Lines $AB$ and $l$ intersect at point $C$ in the following figure. What is the value of $x$ in degrees?

$Q_{10}:$ Line $l$ is a transversal of the parallel lines $m$ and $n$ as shown in the figure. The measures of two angles are given in terms of $x$ and $y$. What is the value of $x$ in degrees?

$Q_{11}:$ In the following figure, the area of square $ABCD$ is $16$ square units. What is the circumference of the circle centered at point $B $?

$Q_{12}:$ The area of a circle is $8\pi $ square units. What is the radius of the circle?

$Q_{13}:$ The circles in the following figure are centered at point $O$. In square centimeters, what is the area of the shaded region?

$Q_{14}:$ Given the trapezoid in the following figure, what is the value of $x$?

$Q_{15}:$ In the following figure, the points $A, C,$ and $D$ lie along the same line, and $ABC$ is a triangle with sides of the indicated length. If the measure of angle $CAB$ is $85$, what is the measure of angle $BCD$ in degrees?

$Q_{16}:$ Given the right triangle in the following figure, what is the value of $x$?

$Q_{17}:$ In the following figure, $AB = BC = AC$, and $BD$ bisects the angle $ABC$. What is the value of $x $?

$Q_{18}:$ The following figure represents the dimensions of three rooms of a house that are to be completely carpeted. In total, how many square feet of carpet will be needed?

$Q_{19}:$ If one of the interior angles of a triangle measures $72$, each of the following pairs could represent the measures of the remaining angles EXCEPT?

$Q_{20}:$ The lengths of the sides of a triangle are $3, 5,$ and $x$. How many possible values of x are there, if $x$ must be an integer?

$Q_{21}:$ What is the area of trapezoid $PMNR$?

$Q_{22}:$ The length of a rectangle is $5$ more than the side of a square, and the width of the rectangle is $5$ less than the side of the square. If the area of the square is $45$, what is the area of the rectangle?

$Q_{23}:$ In the figure below, the two diagonals divide square $WXYZ$ into four small triangles. What is the sum of the perimeters of those four triangles?

$Q_{24}:$ How many sides does a polygon have if the measure of each interior angle is $9$ times the measure of each exterior angle?

$Q_{25}:$ What is the area of trapezoid $ABCD$ in the figure below?