📐 geometry

1. **Problem statement:** We have two parallel lines $L_1$ and $L_2$ intersected by two transversals, creating 12 angles numbered 1 through 12. Given $m\angle7=55^\circ$, and adjac

1. **Problem Statement:** Given two parallel lines L1 and L2 intersected by two transversals, find the measures of angles \(\angle 1\) through \(\angle 12\) given \(\angle 4 = 61^\

1. **Problem Statement:** Find the lateral area and surface area of each figure given. ---

1. **State the problem:** We need to find the total surface area of the pentagonal prism (pedestal) to determine how much tile is required to cover it. 2. **Identify the shape and

1. **State the problem:** We have a ladder leaning against a building forming a right triangle. The ladder is the hypotenuse of length 35 feet, the building height is one leg of le

1. **State the problem:** We need to find the value of $x$ in a square where the diagonal length is given as 8. 2. **Recall the formula:** For a square with side length $x$, the di

1. The problem involves finding unknown angles $\theta$ in various circle diagrams where points lie on the circumference and $O$ is the center. 2. Key formulas and rules:

1. **State the problem:** We have a figure with squares ABCD, DCEF, and FEPQ, and we want to find the sum of the lengths CAQ + EAQ + PAQ. 2. **Understand the problem:** Each of the

1. **State the problem:** We need to find the horizontal distance between two skyscrapers given the angle of inclination from the base of skyscraper A to the top of skyscraper B is

1. **State the problem:** Find the measure of NK in a circle where one leg is 9, the hypotenuse is 15, and the other leg is unknown (NK). 2. **Formula used:** Use the Pythagorean t

1. **Тодорхойлолт:** Өнцгийн хэмжээг транспортираар хэмжихэд өнцгийн хоёр талын хоорондох өнцгийн хэмжээг градусаар илэрхийлнэ. 2. **Формул ба дүрэм:** Өнцгийн хэмжээг градусаар хэ

1. **State the problem:** Find the surface area of a rectangular prism with dimensions height $h=3$ yd, width $w=5$ yd, and depth $d=1 \frac{1}{5}$ yd. 2. **Convert mixed number to

1. **State the problem:** We need to find the volume of a rectangular prism with dimensions height = 3 yards, length = 5 yards, and width = 1 \frac{1}{5} yards. 2. **Formula:** The

1. **State the problem:** We need to find the area of a right triangle with a base of 4 inches and a height of 3 inches. 2. **Formula:** The area $A$ of a triangle is given by the

1. **State the problem:** Find the area of a triangle with sides 7 mm, 6 mm, and 5 mm, where the 5 mm side is perpendicular to the base. 2. **Formula for the area of a triangle:**

1. **State the problem:** Given that line segment $\overline{AB}$ is parallel to line segment $\overline{DE}$, prove that triangle $\triangle ABC$ is similar to triangle $\triangle

1. The problem is to understand and use the formula for the volume of a sphere, which is given by $$V = \frac{4}{3} \pi r^3$$ where $V$ is the volume and $r$ is the radius of the s

1. **State the problem:** We have two triangles side by side with given side lengths. We need to find the length $x$ of the hypotenuse of the right triangle. 2. **Identify the tria

1. **Problem statement:** We have a circle with center $O$ and radius $OA = 7.5$ cm. The radius $OA$ is perpendicular to chord $AB$. Point $D$ lies on the circle such that $\angle

1. **Problem Statement:** Find the area of a regular hexagon with side length 10 inches. 2. **Formula:** The area $A$ of a regular polygon with $n$ sides of length $s$ can be found

1. The problem asks to find the area of a regular pentagon with a perimeter of 50 cm. 2. Recall the formula for the area of a regular polygon: $$\text{Area} = \frac{1}{2} \times \t