📐 geometry

1. **State the problem:** We are given two adjacent angles formed by intersecting lines: one angle is $3x^\circ$ and the other is $(2x + 16)^\circ$. We need to find the value of $x

1. **State the problem:** We have two parallel lines $r$ and $s$ intersected by three transversals $t$, $u$, and $w$. Given angles $d = 42^\circ$, $g = 143^\circ$, and $h > d$, we

1. **State the problem:** We are given a triangle with base and height in the ratio $7:2$ and area $630$ cm$^2$. We need to find the base and height of the triangle. 2. **Formula f

1. **Problem Statement:** We are given a triangle $ABC$ with interior points $D, E, F, G$ where $G$ is the circumcenter of $\triangle ABC$. We need to find the length of segment $C

1. **Problem Statement:** We have $\triangle XYZ$ with points $U$, $V$, $W$ on sides $XY$, $YZ$, and $XZ$ respectively, and $H$ is the circumcenter of $\triangle XYZ$. We need to f

1. **State the problem:** We need to prove that $\triangle PQS \cong \triangle RQS$ given that $\angle QSP \cong \angle QSR$ are right angles, $S$ is the midpoint of $PR$, $PQ \con

1. **State the problem:** We need to prove that $\triangle JKN \cong \triangle MKL$ given that $K$ is the midpoint of $JM$ and $NL$, $\angle L \cong \angle N$, and $LM \cong JN$. 2

1. The user requests a circle, not a cylinder. 2. A circle is a 2D shape defined as the set of all points equidistant from a center point.

1. **State the problem:** We have two right triangles $ABC$ and $DCE$ sharing a base line $BC$ and $CE$. We want to rotate and reposition triangle $DCE$ so that it forms a polygon

1. **Problem 1:** Given $\angle 4$ and $\angle 7$ are supplementary, prove $j \parallel k$. 2. **Step 1:** Recall the definition of supplementary angles: two angles whose measures

1. **State the problem:** We need to find the total surface area of a cone with slant height $l = 9$ m and diameter $d = 16$ m. 2. **Formula for total surface area of a cone:**

1. **State the problem:** We need to find the surface area of a cylinder with height $h=5$ mm and radius $r=9$ mm. 2. **Formula for surface area of a cylinder:**

1. **State the problem:** We have a circle with two intersecting chords creating angles $a$ and $b$ inside the circle. Given arcs of $37^\circ$, $164^\circ$, and $45^\circ$, we wan

1. **State the problem:** We need to find the values of the unknown angles $q$ and $s$ inside a circle with given arcs of $23^\circ$, $127^\circ$, and $48^\circ$. 2. **Recall the r

1. **State the problem:** We are given a rhombus ABCD with points $B(2, 10)$ and $D(6, 2)$, and $A$ lies on the x-axis. We need to find the coordinates of the midpoint $M$ of $BD$,

1. **State the problem:** We need to find the surface area of the three-dimensional figure represented by the net. 2. **Analyze the net:** The net consists of:

1. **State the problem:** We have a triangle with angles $x^\circ$, $135^\circ$, and $2x^\circ$. We need to classify the triangle by its angles. 2. **Use the triangle angle sum pro

1. **State the problem:** We have a triangle with sides $4.2$, $r$, and $2r$, and one angle of $30^\circ$ opposite the side $4.2$. We want to classify the triangle by its angles an

1. **State the problem:** Calculate the volume of the given triangular prism with base edges and height labeled. 2. **Formula for volume of a triangular prism:**

1. **State the problem:** Katie wants to cover a triangular prism with glitter. Each square meter requires 60 g of glitter. We need to find the total amount of glitter required to

1. **State the problem:** Determine if triangles DEF and OPQ are congruent based on the given transformations and side/angle comparisons. 2. **Recall congruence criteria:** Triangl