📐 geometry

1. **Stating the problem:** We have three intersecting lines forming angles labeled $a\angle 5$, $44\angle 5$, $66\angle 5$, $c\angle 5$, and $b\angle 5$. We need to find the measu

1. **Stating the problem:** We are given a figure with angles labeled involving the Greek letter \(\xi\) and numbers 44, 66, and angles \(a\), \(b\), and \(c\) to find.

1. **State the problem:** We need to find the size of angle $\angle CME$ in the triangular prism $ABCDEF$ where $M$ is the midpoint of $AC$, $AC=40$ cm, $\angle CAE=35^\circ$, and

1. **State the problem:** We have a square with area 100 cm² inscribed inside a circle. We need to find the area of the circle. 2. **Recall formulas:**

1. **State the problem:** We need to find the area of a right-angled triangle with perpendicular sides of lengths 3 m and 14 m. 2. **Formula:** The area $A$ of a right-angled trian

1. The problem asks for the coordinates of point M, which is the midpoint of the line segment CD. 2. The midpoint formula for two points $C(x_1, y_1)$ and $D(x_2, y_2)$ is:

1. The problem states that triangles $\triangle GHI$ and $\triangle JKL$ are congruent, i.e., $\triangle GHI \cong \triangle JKL$. 2. By the definition of congruent triangles, corr

1. **State the problem:** We need to find the surface area of a square-based pyramid with a square base side length of 30 mm and four identical triangular faces each having a heigh

1. **State the problem:** We need to find the surface area of a square-based pyramid where the base edges are 6 cm and each triangular face has a height of 19 cm. 2. **Formula for

1. **State the problem:** We have a regular 9-sided polygon (nonagon) with interior angles of 140 degrees and exterior angles of 40 degrees. We want to find the angle $x$ formed at

1. **Problem statement:** We have a circle with radius $5$ cm and a chord of length $6$ cm. We need to find the area of another circle concentric with the first one that is tangent

1. **Problem statement:** Given a circle $(O)$ and a point $A$ outside it, two tangents $AB$ and $AC$ touch the circle at points $B$ and $C$ respectively. $H$ is the intersection o

1. **State the problem:** We have two concentric circles. The outer circle has a radius of 10 cm. The area of the inner circle is half the area of the outer circle. We need to find

1. **Problem statement:** We need to find the area of a swimming pool formed by two identical circles of radius 9 m, where each circle passes through the center of the other. 2. **

1. Problem statement: Consider parallelogram QRST with diagonals intersecting at U. 2. Given: the angle at vertex R formed by RU and RS is $32^\circ$.

1. **State the problem:** We have parallelogram VWXY with diagonals WY and VX intersecting at Z. Given segment WZ = 9, angle X = 64°, angle Y = 39°, and segment ZY = 4x + 1, find $

1. **State the problem:** We have a triangle with side lengths 13, 16, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **State the problem:** We have a triangle with side lengths 10, 14, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **State the problem:** We have a triangle with side lengths 11, 12, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **Problem statement:** We have a circle with radius $2\sqrt{2}$ and a shaded right triangle formed by two radii and a chord subtending a $135^\circ$ angle at the center. 2. **Go

1. The problem states: In parallelogram ABCD, point E lies on CD, and we need to find angle \(\angle ABC\).\n\n2. Recall that in a parallelogram, opposite angles are equal and adja