📐 geometry

1. **Problem Statement:** We need to determine if certain pairs of lines are perpendicular or parallel by comparing their slopes. 2. **Formulas and Rules:**

1. **Problem Statement:** Find the area of the polygon with vertices A(-1, 4), B(2, 3), C(2, -4), D(-1, -3), E(-1, -2), F(-4, -3), G(-4, 2), H(-1, 3). 2. **Method:** Use the Shoela

1. **State the problem:** We are given two pairs of parallel lines: $h \parallel l$ and $j \parallel k$. We need to find the values of angles $z$ and $x$ based on the given angle m

1. The problem asks to find the area of a parallelogram with given dimensions. 2. The formula for the area of a parallelogram is:

1. **State the problem:** Emil rides a Ferris wheel with a radius of 28 meters, which completes 3 full rotations. We need to find the total distance Emil travels on the ride, round

1. **Problem:** Find missing angles $x$ and $y$ given angles $60^\circ$ and $72^\circ$. 2. **Formula:** Angles on a straight line sum to $180^\circ$.

1. **State the problem:** We need to find the area of triangle $\Delta ABC$ with vertices at $A(-2, 3)$, $B(4, 3)$, and $C(1, -1)$. 2. **Formula for area of a triangle given coordi

1. **State the problem:** We have a trapezoid-like quadrilateral with sides 5.9in (left vertical), 6.4in (right slanted), and 8in (bottom). A diagonal splits the figure into two re

1. **State the problem:** We need to find the length of the segment $UV$ on a number line where $U$ is at $38$ and $V$ is at $42$. 2. **Formula used:** The length between two point

1. **Problem Statement:** We need to find the length of the segment $QR$ on a number line where $Q$ is at 16 and $R$ is at 20. 2. **Formula Used:** The length between two points on

1. **State the problem:** We have a rectangle with two diagonals crossing, forming four angles at the intersection. One of these angles is given as $67^\circ$. We need to find the

1. **Problem Statement:** Determine if the given pairs of angles are congruent (equal in measure) based on the diagram described. 2. **Understanding Angles and Congruence:** Two an

1. **Problem Statement:** We need to determine which diagram shows $\triangle STU$ rotated $45^\circ$ clockwise about point $H$. 2. **Rotation Formula:** To rotate a point $(x,y)$

1. **State the problem:** We are given two supplementary angles ∠1 and ∠2 with measures m∠1 = $3x + 27$ degrees and m∠2 = $8x + 21$ degrees. We need to find the measure of ∠1. 2. *

1. **Problem Statement:** We need to determine which diagram shows $\triangle FGH$ rotated $115^\circ$ counterclockwise about point $G$. 2. **Understanding Rotation:** Rotation abo

1. Problem: Find each interior angle of a regular dodecagon. Formula: Each interior angle of a regular polygon with $n$ sides is given by $$\text{Interior angle} = \frac{(n-2) \tim

1. **State the problem:** We have an existing train track center line given by the equation $$y = -0.25x + 130$$. 2. **Understand the setup:**

1. **State the problem:** We need to find the slope of the line that forms the hypotenuse (longest side) of two right triangles sharing the same line. 2. **Recall the slope formula

1. **State the problem:** We have two proofs involving segment congruence and segment addition.

1. **State the problem:** We are given a circle with points A, B, C, E, and G on the circumference and a tangent line FED touching the circle at point E.

1. **State the problem:** We need to find the surface area of a rectangular prism with dimensions $2$ cm, $4$ cm, and $3$ cm. 2. **Formula for surface area of a rectangular prism:*