📐 geometry

1. **State the problem:** We need to prove that two lines are parallel when they are cut by a transversal. 2. **Key concept:** When a transversal cuts two lines, certain angle pair

1. **Problem Statement:** We have two triangles, △ABC and △A'B'C', where △A'B'C' is a scaled copy of △ABC. We know the sides of △ABC: AB = 5, BC = 2, AC = 6, and the sides of △A'B'

1. **Problem Statement:** A solid cylinder has radius 18 cm and height 15 cm. A conical hole of radius $r$ cm and depth 12 cm is drilled on one end. The volume of the material remo

1. **Problem Statement:** We have a 30-inch by 16-inch piece of cardboard. Squares of side length $x$ are cut from each corner, and the sides are folded up to form an open box. We

1. **Problem Statement:** Given two parallel lines AB and CD, with angles \(\angle BNO = 128^\circ\) and \(\angle COM = 56^\circ\), find \(\angle MON\) and \(\angle DON\).\n\n2. **

1. **Problem Statement:** Determine the truth value of the following statements based on the given 3D figure with two parallelograms (planes) and points labeled as described. 2. **

1. **Problem Statement:** Determine the position of the given line relative to coordinate axes or other reference lines. 2. **Key Concepts:** The position of a line in 3D or 2D spa

1. **Homework 1: Determine if two lines are perpendicular, skew, or intersecting** - Problem: Given two lines in 3D or 2D space, determine if they are perpendicular, skew, or inter

1. **Stating the problem:** Given a circle with center P, angle $\angle BEC = 50^\circ$, angle $\angle EBC = 60^\circ$, and diameter $AD$ perpendicular to chord $EC$, find the meas

1. The problem is to create a graph with three vertical sections separated by two curved boundaries labeled "T" and "T-S" with sections labeled "1" and "T" as described. 2. To repr

1. **Problem:** Find the midpoint of the line segment joining the points $(-6, 2)$ and $(4, -8)$. 2. **Formula:** The midpoint $M$ of a segment joining points $(x_1, y_1)$ and $(x_

1. **Problem Statement:** Find the distance between each pair of points given. 2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the dis

1. Պատկերացնենք, որ ունենք եռանկյուն ABC, որտեղ <B=62°, <A=78°. 2. Եռանկյան անկյունների գումարը միշտ հավասար է 180°-ի, հետևաբար <C = 180° - <A - <B = 180° - 78° - 62° = 40°:

1. **Problem statement:** We have two containers: one cylindrical and one spherical. Both have the same volume. The cylinder has a height of 50 cm and a radius of 11 cm. We need to

1. **Problem statement:** We have two containers: one cylindrical and one spherical. Both have the same volume. The cylinder has a height of 50 cm and a radius of 11 cm. We need to

1. **Problem Statement:** Given a parallelogram ABFD with points D, A, B, and C on a circle, and lines BF and DF extended to meet DC and CB at points E and G respectively, with DC

1. **Problem Statement:** Given points W, X, Y, Z and lines WX, WY, WZ, XY, XZ, YZ, prove the theorem: "If W is a point, there exists at least one line not containing W." 2. **Unde

1. **Problem Statement:** Given points W, X, Y, Z and lines formed between pairs (W,X), (W,Y), (W,Z), (X,Y), (X,Z), and (Y,Z), prove the theorem: "If W is a point, there exists at

1. **Problem statement:** Given points A(-2, 5), B(4, 3), and O(0, 0), find the equation of line AB, length of AB, perpendicular distance from O to AB, area of parallelogram OABC,

1. **Problem Statement:** We have a children's park consisting of a quadrilateral ABCD and a semicircle AED. Given side lengths AB = 24 m, BC = 7 m, CD = 25 m, and the semicircle w

1. **Problem Statement:** We are given a circle with points A (top), B (bottom), P (left), Q (right), R (bottom-left), S (bottom-right), and center O. Points M and N lie on the ver