📐 geometry

1. **Problem Statement:** We have an acute triangle ABC with excenters G (opposite C) and H (opposite B). Points J and I lie on AB and AC respectively such that \(\angle BHJ = 90^\

1. **Stating the problem:** We need to find the area of the pentagon with given side lengths: two diagonal sides of 5 units each, two vertical sides of 4 units each, and a base of

1. **Stating the problem:** We have a pentagon with side lengths: two diagonal sides of 5 units each, two vertical sides of 4 units each, and a base of 8 units. 2. **Goal:** Calcul

1. The problem is to classify the given quadrilateral based on the markings and properties shown. 2. A quadrilateral is a polygon with four sides. Common types include parallelogra

1. **Problem statement:** Find the missing angles for problems 22 to 32. 2. **Formula and rules:**

1. **State the problem:** We have a right rectangular prism with two known dimensions: length $5$ cm and width $3$ cm. The volume is given as $75$ cubic centimeters. We need to fin

1. **State the problem:** We have a right rectangular prism with dimensions $?$ (unknown), 8 cm, and 4 cm. The volume is given as 96 cubic centimeters. We need to find the missing

1. **State the problem:** We need to find the area of a parallelogram with a base of 15 m and a height of 7 m. 2. **Formula:** The area $A$ of a parallelogram is given by the formu

1. **State the problem:** We have a rectangular window with width 38 inches and height 69 inches. We need to find the area and perimeter of the window, and understand which measure

1. **Stating the problem:** We are given two parallel lines AB and CD intersected by transversals creating angles of 2°, 100°, and 100°. We need to analyze the relationships betwee

1. **Stating the problem:** We have two parallel lines AB and CD with a transversal intersecting them, creating several angles. Given angles at points M and N on line CD are 98° an

1. **Stating the problem:** We have a person standing 1.5 m from a window of height 90 cm (0.9 m), and a tree 19.5 m from the window with height 7 m. We want to analyze the angles

1. **Problem statement:** Given two parallel lines AB and CD, and two intersecting lines creating angles 58° and 61° between them, find the measures of angles 1 and 2 formed at the

1. **Stating the problem:** We are given two parallel lines \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\) with transversal lines intersecting them, forming angles. We know \(

1. The problem shows four rectangles divided into segments with given lengths: 3, 9, 5 for the first; 4, 6, 8 for the second; 5, 2, 2, 7 for the third; and 7, 6, 3, 2 for the fourt

1. **State the problem:** We are given a triangle with one side length 4.2 and need to find the length of side UT. 2. **Identify the triangle type and given information:** Since th

1. **State the problem:** We need to find the measure of angle $m\angle DFE$ given a portion of a circle with chord $D$ and points $D$, $F$, and $E$ on or near the circle. 2. **Rec

1. **State the problem:** We need to find the angle $a$ at vertex $C$ in a scalene triangle $ABC$ drawn on a unit grid, where the side lengths are not aligned to the grid. 2. **Ide

1. **State the problem:** We have three similar solid shapes A, B, and C. - Surface area of A = 4 cm²

1. **Find the missing length $x$ in the right triangular prism with volume 432 cm³.** The volume formula for a triangular prism is:

1. **Problem Statement:** Find the missing height $H$ of a right triangular prism with volume $432$ cm³ and base sides $12$ cm and $8$ cm. 2. **Formula:** Volume of a prism is give