📐 geometry

1. **Problem statement:** We have a triangle with base $AB = 8m$ and a segment $DE$ inside the triangle parallel to $AB$ with length $DE = 5m$. The sides from the top vertex to $B$

1. **State the problem:** We have a parallelogram ABCD with sides labeled as follows: AB = 12x, BC = 10y + 10, CD is parallel to AB, and DA = 6x. We need to find the value of $y$.

1. **State the problem:** We need to find the exterior angles $a$, $b$, and $c$ of a triangle where the interior angles adjacent to $c$ and $b$ are $70^\circ$ and $67^\circ$ respec

1. **Problem statement:** Given that lines AB and CD are parallel, explain why angles $a$ and $d$ are equal, angles $b$ and $e$ are equal, and use these facts to prove that the sum

1. **Problem Statement:** We have a rectangle with top-left corner at $(20, 20)$, width $80$ units (x-direction), and height $60$ units (y-direction downward).

1. The problem asks to determine the coordinates of the other corners of a shape, but the original coordinates or shape details are missing. 2. To find other corners, we typically

1. **Problem statement:** We need to draw a rectangle in a Cartesian coordinate system with width 80 units and height 60 units. The top left corner of the rectangle is at coordinat

1. **Problem Statement:** You want to create a graph paper layout of the city streets and attractions based on the given angle and intersection relationships.

1. **Problem Statement:** You are tasked with designing a city layout with four streets: two parallel streets, and two transversal streets with specific angle relationships and att

1. The problem asks to name a chord for circle D. 2. A chord is a line segment with both endpoints on the circle.

1. The problem asks to name a central angle for circle D. 2. A central angle in a circle is an angle whose vertex is at the center of the circle and whose sides are radii extending

1. **State the problem:** We are given four triangles with incenter N and three segments from N to the sides forming right angles. For each triangle, the sum of the three segments

1. **Problem statement:** Find the length of side $a$ in a right triangle where the hypotenuse is 6 and the angle opposite side $a$ is 60°. 2. **Formula used:** In a right triangle

1. **Problem statement:** Find the measure of angle $\angle V$ in the parallelogram where $m\angle V = 71^\circ$, and given expressions $36x - 1$, $29x + 1$, and $50^\circ$ with $x

1. **Problem:** Find $m\angle V$ in the parallelogram where angles are given as $36x - 1$ and $29x + 1$, with $x=2$ and another angle $50^\circ$. 2. **Formula and rules:** In a par

1. **State the problem:** We need to determine if a triangle with side lengths 40 yards, 42 yards, and 58 yards is a right triangle. 2. **Recall the converse of the Pythagorean the

1. **State the problem:** We are given a polygon with points A, B, C, D, E, F, G and certain lengths and angle measures: $EG=3$, $EB=9$, $AF=8$, $m\angle EBG=19$, $m\angle EGF=29$,

1. **State the problem:** We are given lengths and angle measures in a polygonal figure with points A, B, C, D, E, F, G. We need to find the measure of angle $\angle CAF$ given $EG

1. **State the problem:** Given that $m\angle HLK = 56^\circ$, find $m\angle CLB$. 2. **Identify the relationship:** Angles $HLK$ and $CLB$ are vertical angles because they are for

1. **State the problem:** We are given angles $m\angle QOS = 53$, $m\angle POR = 56$, and $m\angle POQ = 22$. We need to find $m\angle ROS$. 2. **Understand the setup:** The rays $

1. **State the problem:** We need to find the measure of angle $x$ given that it and a $59^\circ$ angle are adjacent and together form a right angle of $90^\circ$. 2. **Use the Ang