📐 geometry

1. **State the problem:** We have trapezoid ABCD with legs BC and AD, DE perpendicular to CB, and given angles $m\angle A = 86^\circ$ and $m\angle C = 74^\circ$. We want to find th

1. If two lines intersect, then the vertical angles are always **congruent**. This is called the **Vertical Angles Theorem**.

1. **Problem Statement:** We need to construct a pentagon with exactly one right angle (90 degrees) and exactly one acute angle (less than 90 degrees). 2. **Understanding Angles in

1. **State the problem:** We are given the original vertices of a triangle and the vertices of a new triangle after a transformation. We need to determine which rule was applied to

1. **State the problem:** We need to find the volume of a cylinder with radius $r=5$ cm and height $h=10$ cm. 2. **Formula for the volume of a cylinder:**

1. **Problem Statement:** Determine if the pairs of lines in each graph are parallel, perpendicular, or neither. 2. **Key Concepts:**

1. **Problem Statement:** Find the area of the shaded region in a trapezoid with an inscribed rectangle.

1. **Problem Statement:** We have a rectangular prism with dimensions width = 4 mm, depth = 8 mm, and height = 11 mm. The top and bottom faces are shaded. We need to find:

1. **Problem Statement:** Find the total surface area of a solid made from 1 meter cubes arranged in a stepped shape with a base layer of 4 by 4 cubes, a taller section 3 cubes hig

1. **State the problem:** Prove that triangles $\triangle UVX$ and $\triangle SYW$ are congruent given that $\angle UXV \cong \angle SWY$, $\angle U \cong \angle S$, and $XY \cong

1. Let's start by stating the problem: You want to understand what a scale factor is and how it works. 2. A scale factor is a number that scales, or multiplies, some quantity. In g

1. **State the problem:** We are given two points A(-4, -8) and B(-5, -10) and a scale factor of 5. We want to understand how the scale factor affects these points. 2. **Formula fo

1. **State the problem:** We have a quadrilateral with the top and bottom sides parallel. On the right side, two angles are labeled: the top angle is $x^\circ$ and the bottom angle

1. **State the problem:** We have an isosceles trapezoid with two equal non-parallel sides. The angles adjacent to these sides are given as $61^\circ$ and $(2x + 15)^\circ$. We nee

1. **State the problem:** We are given three geometric figures with unknown side lengths labeled as $x$. We need to find $x$ in each case.

1. **Problem Statement:** Rotate the Gas Station, a rectangle with vertices A, B, C, D, by 180 degrees about the origin. 2. **Formula for Rotation:** To rotate a point $(x,y)$ by 1

1. **Problem 7: Find the diagonal $x$ of a rectangle with length 24 and width 7.** 2. The diagonal of a rectangle can be found using the Pythagorean theorem: $$x = \sqrt{\text{leng

1. **State the problem:** We have a right triangle ABC with a right angle at C, angle $\angle A = 45^\circ$, and hypotenuse $AB = 12$. We need to find the length of side $BC$. 2. *

1. **State the problem:** We have a square pyramid with a base side length of 12 units and a vertical height of 15 units. We need to find the length $s$, which is the slant height

1. **State the problem:** We are given the radius $r=7$ cm and the volume $V=3846.5$ cm³ of a cylinder. We need to find the height $h$ of the cylinder. 2. **Formula for the volume

1. **State the problem:** We are given the height $h=3$ cm and the volume $V=8478$ cm$^3$ of a cylinder, and we need to find the radius $r$. 2. **Formula for the volume of a cylind