📐 geometry

1. **State the problem:** We are given two pairs of parallel diagonal lines crossing, with one angle marked 51° and the adjacent angle marked $a^\circ$. We need to find the value o

1. **Problem Statement:** Calculate the missing angles a, b, c, d, e, f, g, h, i, j, k, l in various configurations involving parallel lines and transversals. 2. **Key Concepts:**

1. **State the problem:** We have a segment $CE$ with two bisectors: $OR$ which is a perpendicular bisector, and $ST$ which is another bisector. We want to conclude something about

1. **Problem Statement:** We want to understand what perpendicular lines are and how to identify them. 2. **Definition:** Two lines are perpendicular if they intersect at a right a

1. **Stating the problem:** We want to understand the difference between parallel lines and intersecting lines. 2. **Definition of parallel lines:** Parallel lines are two lines in

1. Problem: Mamy ostrosłup o podstawie kwadratowej, krawędź boczna ma długość $\sqrt{13}$ m, a $\tg\angle SBO = \frac{2}{3}$. Należy obliczyć objętość ostrosłupa. 2. Zauważmy, że $

1. **Problem statement:** Given that triangle $ABC$ is congruent to triangle $BDE$, with $AB=15$ cm, $AC=13$ cm, and $DE=8$ cm, find: (i) length of $BC$

1. **State the problem:** We have a frustum ABQP formed by removing a smaller cone VAB from a larger cone VPQ.

1. **State the problem:** We have an isosceles triangle with two equal angles and a third angle that is 30° more than twice either of the equal angles. We need to find the measures

1. **State the problem:** We are given the interior angles of a heptagon (7-sided polygon) as $140^\circ$, $(5x -1)^\circ$, $(8x -61)^\circ$, $128^\circ$, $(10x -7)^\circ$, $(7x +

1. **State the problem:** Calculate the volume of a cylinder with radius $8$ and height $23$. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We are given that $HF \parallel JK$ and $HG \cong JG$. We need to prove that $\triangle FHG \cong \triangle KJG$ by ASA (Angle-Side-Angle) congruence. 2.

1. **Problem Statement:** We have four figures, each made up of six squares. Three of these figures can be folded to form a cube, and one cannot. We need to identify which figure c

1. The problem is to understand the Pythagoras theorem. 2. The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right

1. **State the problem:** We need to compare the areas of triangles $\triangle XDC$ and $\triangle YDC$ formed inside two identical rectangles. 2. **Understand the figure and given

1. **State the problem:** We have a right triangle $\triangle ABC$ with a right angle at $B$, angle $A = 60^\circ$, and side $AB = 13$. We need to find the lengths of $AC$ (the hyp

1. **State the problem:** We have a triangle with base angles $55^\circ$ and $38^\circ$, and an altitude (height) of $108$ units from the top vertex to the base. We need to find th

1. **State the problem:** We have an n-sided polygon with three known interior angles: 125\degree, 156\degree, and 175\degree. The remaining \(n-3\) interior angles are each 111\de

1. **Problem Statement:** We are given a trapezium ABDE and a parallelogram CDEF with \(BC = CF\) and \(\angle CDE = 123^\circ\). We need to find \(\angle CBF\) and \(\angle BFE\).

1. **Problem Statement:** We are given a 3D pipe diagram with points $A$, $B$, $C$, $D$, and $F$. The lengths $a = 6$ ft and $h = 1.5$ ft are provided. We need to find the angle $\

1. **State the problem:** We need to find the coordinates of points $A$, $B$, $C$, and $D$ after a $270^\circ$ anticlockwise rotation about the origin. 2. **Formula for rotation:**