📐 geometry

1. **Problem statement:** We have an equilateral triangle with side length 1, and 10 points are selected inside it. We need to prove that there exist at least two points whose dist

1. **Problem Statement:** Find the measure of angle $z$ formed between a side of a regular dodecagon and a tangent line extended from a square attached to one side of the dodecagon

1. **Problem Statement:** Find the measure of angle $z$ formed between one side of a small square and an adjacent side of a regular 12-sided polygon (dodecagon). 2. **Key Informati

1. **Problem Statement:** We are given a regular polygon with an exterior angle measuring 60 degrees. We need to find the number of sides of this polygon. 2. **Formula Used:** The

1. **State the problem:** We need to find the measure of each interior angle of a regular octagon. 2. **Formula for the sum of interior angles:** The sum of interior angles of a po

1. The problem asks for the measure of one exterior angle of a regular octagon. 2. A regular octagon has 8 equal sides and 8 equal exterior angles.

1. The problem asks for the best description of a polygon that is a hexagon with six sides, where all sides are equal in length and all interior angles are equal. 2. A polygon with

1. **Problem statement:** We have a right triangle IJK with a right angle at J, angle I measuring 22°, side IJ = 16 units, and side JK = x units. We need to find the length of side

1. **State the problem:** We have a right triangle QRS with right angle at R, side QR = 80, angle RQS = 66°, and we need to find side RS = x. 2. **Identify the sides relative to an

1. **Problem 1: Total Surface Area of the Bird-bath** The bird-bath consists of a cylinder with a hemispherical depression at one end.

1. **Problem Statement:** Given a cyclic quadrilateral ABCD with diagonals AC and BD intersecting at P, and AB = DC, prove: a) $\frac{AP}{AB} = \frac{PD}{DC}$

1. **State the problem:** Given that $\overline{EB} \cong \overline{EC}$ and $\triangle AED$ is equilateral and equiangular, prove that $\triangle ACD \cong \triangle DBA$. 2. **Un

1. **State the problem:** We have triangle ABC with angles \(m\angle ACB = 3x - 14\) degrees, \(m\angle ABC = 3x + 9\) degrees, and an exterior angle \(m\angle CAD = 103\) degrees

1. **State the problem:** We are given a triangle ABC with angles labeled as follows:

1. **State the problem:** We need to find the measure of angle $\angle ORQ$ given two expressions for angles formed by intersecting lines: $(9x - 4)^\circ$ and $(3x + 40)^\circ$. 2

1. **State the problem:** We are given two angles formed by intersecting lines, with measures $(15x + 11)^\circ$ and $(5x + 69)^\circ$. We need to solve for $x$. 2. **Identify the

1. **Problem statement:** Given two parallel lines \(\overrightarrow{TV}\) and \(\overrightarrow{WY}\), and the angle \(m \angle VUS = 64^\circ\), find the measure of \(m \angle YX

1. **Problem Statement:** We are given two parallel lines \(\overleftrightarrow{JL}\) and \(\overleftrightarrow{MO}\) and several pairs of angles. We need to determine which pair o

1. **Problem Statement:** Given that $\overrightarrow{SU}$ and $\overrightarrow{VX}$ are parallel lines and $m \angle UTR = 139^\circ$, find $m \angle VWY$. 2. **Understanding the

1. The problem asks which pairs of angles are vertical angles given two parallel lines \(\overrightarrow{MO}\) and \(\overrightarrow{PR}\) intersected by a transversal \(\overright

1. **State the problem:** Given that $BA \cong BC$ and $AE \cong DC$, prove that $\triangle BDE$ is isosceles. 2. **Given information:**