Geometry and Measurement

Five identical angles meet at a point on one side of a straight line. What is the measure of each angle?

In an isosceles triangle, the vertex angle is $30^{\circ}$. Calculate the measure of each base angle.

Three angles at a point are in the ratio $1:2:3$. What is the size of the smallest angle?

The exterior angle of a triangle is $140^{\circ}$. If one of the opposite interior angles is $60^{\circ}$, what is the other opposite interior angle?

The interior angles of a triangle are $2x$, $4x$, and $6x$. What is the measure of the exterior angle to the smallest interior angle?

Two lines intersect. A pair of vertically opposite angles are $(2x + 10)^{\circ}$ and $(3x - 20)^{\circ}$. What is the sum of the other two vertically opposite angles?

In $\triangle ABC$, the exterior angle at vertex $B$ is $125^{\circ}$ and the exterior angle at vertex $C$ is $110^{\circ}$. What is the measure of the interior angle at vertex $A$?

The exterior angle at the vertex of an isosceles triangle is $110^{\circ}$. Find the measure of one of the base angles.

Three angles on a line are $x, y,$ and $z$. If $x = 2y$ and $y = 3z$, find the measure of angle $x$.

In $\triangle ABC$, $\angle A - \angle B = 15^{\circ}$ and $\angle B - \angle C = 30^{\circ}$. Find the measure of $\angle A$.

What is the supplement of a $100^\circ$ angle?

In the capital letter 'E', how many of the horizontal line segments are parallel to each other?

If two lines are drawn on a flat sheet of paper and they are not parallel, what must be true about them?

How many right angles are found inside a standard rectangle?

If $\angle 1$ and $\angle 2$ are alternate exterior angles formed by parallel lines and $\angle 1 = 135^\circ$, what is the measure of the angle supplementary to $\angle 2$?

A transversal cuts two parallel lines. If a pair of corresponding angles are $120^\circ$ and $3x^\circ$, what is the value of $x$?

Two lines are parallel. Co-interior angles are represented by $(4x + 10)^\circ$ and $(2x + 20)^\circ$. Solve for $x$.

A transversal cuts two parallel lines such that the difference between the two interior angles on the same side is $0^\circ$. What is the angle of the transversal?

Two parallel lines $m$ and $n$ are cut by transversal $t$. If the angle bisectors of all four interior angles are drawn, what shape do they form?

If two parallel lines are cut by a transversal, and the ratio of a pair of alternate interior angles is $(x+1):(2x-3)$, find the measure of the angles.

Which of these shapes has the fewest number of sides?

How many degrees are in a right angle?

Every rhombus is also a:

A triangle with sides of $6\text{ cm}, 6\text{ cm}$, and $6\text{ cm}$ is:

In a rhombus $ABCD$, the diagonal $AC$ forms an angle of $34^\circ$ with the side $AB$. What is the measure of $\angle ABC$?

Which property is shared by both a rhombus and a parallelogram?

The area of a square is $144\text{ cm}^2$. What is its perimeter?

In a regular $n$-gon, the ratio of an interior angle to an exterior angle is $7:2$. Find $n$.

A polygon has $n$ sides. If the number of sides is doubled, by how much does the sum of the interior angles increase?

In a rectangle $ABCD$, the angle between a diagonal and side $AB$ is $35^\circ$. What is the acute angle between the two diagonals?

What is the perimeter of a regular octagon with a side length of $4\text{ cm}$?

Find the area of a parallelogram with a base of $12\text{ cm}$ and a height of $2\text{ cm}$.

Calculate the perimeter of a rectangle with a length of $11\text{ cm}$ and a width of $9\text{ cm}$.

What is the area of a circle with a radius of $5\text{ m}$? (Use $\pi \approx 3.14$)

A window is designed in the shape of a rectangle with a semi-circle on top. The rectangle has a width of $80\text{ cm}$ and a height of $100\text{ cm}$. The diameter of the semi-circle is equal to the width of the rectangle. What is the total perimeter of the window's outer edge? (Use $\pi = 3.14$)

A square has an area of $169\text{ cm}^2$. What is its perimeter?

Find the side length of a regular octagon that has a perimeter of $64\text{ cm}$.

A circle is inscribed in a square. The ratio of the area of the circle to the area of the square is:

An L-shaped figure is made of two rectangles. One is $10 \times 4$ and the other is $6 \times 4$. If they are joined at a $4 \times 4$ corner, what is the total perimeter of the resulting shape?

The difference between the circumference and the radius of a circle is $37\text{ cm}$. What is the area of the circle? (Use $\pi = \frac{22}{7}$)

Find the volume of a rectangular prism with length $8$ cm, width $0.5$ cm, and height $4$ cm.

A triangular prism has a base area of $8\text{ cm}^2$ and a volume of $56\text{ cm}^3$. What is its length?

A cylinder has a radius of $1$ cm and a height of $5$ cm. What is its volume in terms of $\pi$?

Calculate the total surface area of a cube with a side length of $7$ cm.

A cylinder has a total surface area of $132\pi$ $cm^2$. If the radius is $6$ cm, find the height.

A rectangular prism measures $9$ cm by $2$ cm by $1$ cm. What is its volume?

A triangular prism has a volume of $180$ $cm^3$. The triangle base has a base of $6$ cm and a height of $10$ cm. What is the length of the prism?

An open cylindrical tank (no lid) has a radius of $1.5$ m and a height of $4$ m. If it costs $\$20$ per square meter to paint the entire outside and inside surface (excluding the base of the tank on the ground), what is the total cost? (Use $\pi \approx 3.14$)

A rectangular prism has a volume of $V$. If its length is doubled, its width is halved, and its height is tripled, what is the new volume?

A closed cylindrical can has a height of $14$ cm and a radius of $5$ cm. What is the total area of the metal sheet required to make $10$ such cans? (Use $\pi \approx 3.14$)

Is a triangle with side lengths $15$, $36$, and $39$ a right triangle?

What is the hypotenuse of a triangle with legs of $14\text{ cm}$ and $48\text{ cm}$?

A right triangle has a hypotenuse of $25$ and a leg of $20$. Find the other leg.

In a right triangle with sides $a$, $b$, and $c$, where $c$ is the hypotenuse, which of the following is always true?

A square has a diagonal of length $8\text{ cm}$. What is the area of the square?

Find the length of the hypotenuse for a right-angled triangle with leg lengths of $3\text{ cm}$ and $7\text{ cm}$.

A $25\text{ ft}$ ladder is leaned against a house. If the base of the ladder is $7\text{ ft}$ from the house, how high does the ladder reach?

Two cyclists start from the same point. One travels North at $15\text{ km/h}$ and the other travels East at $20\text{ km/h}$. How far apart are they after $2$ hours?

A ship leaves port and travels $35\text{ km}$ due East and then $12\text{ km}$ due North. It then travels $13\text{ km}$ further North. How far is the ship from the port?

Find the distance between the points $P(2, -3)$ and $Q(-4, 5)$.

Rotating $(3, -2)$ by $180^\circ$ about the origin gives:

Reflecting a point twice across the same line results in:

If the translation vector is $\begin{pmatrix} h \\ k \end{pmatrix}$, and the image of $(2, 2)$ is $(5, 1)$, what are $h$ and $k$?

A point at $(5, 0)$ is rotated $90^\circ$ counter-clockwise about the origin. Its image is:

Point $I(5, 5)$ is translated by $\begin{pmatrix} -2 \\ -2 \end{pmatrix}$ and then rotated $90^\circ$ counter-clockwise about the origin. What are the final coordinates?

Reflect the point $(0, -2)$ in the line $y = 1$. What are the coordinates of the image?

Point $H(3, 3)$ is rotated $90^\circ$ counter-clockwise about the origin. In which quadrant is $H'$?

A triangle is reflected across $x = 1$, then $x = 2$, then $x = 3$. This is equivalent to a reflection across which line?

A point $P$ is rotated $90^\circ$ CCW about the origin to $P'$. $P'$ is then reflected across $y = x$ to $P''$. What single transformation maps $P$ to $P''$?

Point $E(-3, -2)$ is reflected over the $x$-axis, then the $y$-axis, then the line $y = x$. What are the final coordinates?