📐 geometry

1. **Problem statement:** Given rectangle UVWX with diagonals intersecting at point Y, and WY = 16, find VX. 2. **Important property:** In a rectangle, the diagonals are equal in l

1. **State the problem:** We need to find the values of $x$, $y$, and $z$ in a rhombus where the angles are given as $(-x+1)^\circ$, $(3z+9)^\circ$, $108^\circ$, and $(-2y+4)^\circ

1. **Problem Statement:** Given a figure with point Q and rays QS, QR, and QT, and two angles labeled 1 and 2 at vertex Q, we need to name the sides and vertices of the angles as r

1. The problem states that a slide forms a right triangle with a length (hypotenuse) of 9 feet and a height of 8 feet. We need to find the base length. 2. We use the Pythagorean th

1. **Problem Statement:** Determine if the given parallelogram is a rhombus, rectangle, or square based on the properties of its diagonals. 2. **Key Properties of Parallelograms:**

1. The problem asks if the given parallelogram can be concluded to be a rhombus, rectangle, or square based on the given information. 2. Important properties to recall:

1. **State the problem:** Find the volume of a cylinder with height $h = 17$ ft and radius $r = 6$ ft. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We are given a right triangle with angles $\angle A = (5x - 19)^\circ$, $\angle B = 90^\circ$ (right angle), and $\angle C = (2x + 32)^\circ$. We need to

1. **State the problem:** Find the volume of a cylinder with radius $r=2$ ft and height $h=15$ ft, rounding to the nearest tenth. 2. **Formula:** The volume $V$ of a cylinder is gi

1. **Problem Statement:** Find the volume of the solid composed of five 1-centimeter cubes arranged with four cubes in a 2 by 2 base layer and one cube stacked on top of the back r

1. **State the problem:** We need to find the volume of a solid composed of two right rectangular prisms joined together in an L-shape. 2. **Identify dimensions:** The height of bo

1. **State the problem:** We need to find the coordinates of the vertices P, Q, R, and S after reflecting them over the x-axis. 2. **Recall the reflection rule:** Reflecting a poin

1. **State the problem:** We have two triangles sharing a side. The left triangle is a right triangle with one angle of 65° and the right triangle has angles 50°, 80°, and an unkno

1. **Stating the problem:** We have two adjacent triangles sharing a vertex. One triangle has two angles marked as $x^\circ$, and the adjacent triangle has angles $43^\circ$ and $5

1. **Problem Statement:** Find the values of $x$, $y$, and $z$ in a right triangle with angles $45^\circ$ and $30^\circ$, and hypotenuse $14\sqrt{2}$. 2. **Relevant Formulas and Ru

1. **State the problem:** We have a rectangular cardboard 30 cm by 20 cm. Squares of side length $x$ cm are cut from each corner, and the sides are folded up to form an open box. T

1. The problem asks to name two points collinear to point D. Collinear points lie on the same straight line. Given the options AB and EF, if D lies on line AB, then points A and B

1. **Stating the problem:** We are given that the measure of angle 2 is $m\angle 2 = 5x + 4$ degrees. 2. **Understanding the figure and given information:** The figure shows two tr

1. **Problem statement:** Given that line $n$ bisects segment $CE$ at point $D$, and the lengths $CD = x + 6$ and $DE = 4x - 21$, find the length $CD$. 2. **Key concept:** When a p

1. **State the problem:** We are given that the measure of angle 2 is expressed as $m\angle 2 = 18 + 6x$ degrees, and another angle in the figure measures 132°. 2. **Identify the r

1. **State the problem:** We are given a triangle with two angles labeled as $2m - 3^\circ$ and $m + 6^\circ$. We need to find the value of $m$.