📐 geometry

1. **State the problem:** We have a circle divided into five sectors with angles 31°, $f$, 73°, $g$, and one unknown angle. We need to find the size of angle $f$ and then use it to

1. The problem is to find which angles are complementary. 2. Complementary angles are two angles whose measures add up to 90 degrees.

1. The problem involves identifying relationships between pairs of angles formed by intersecting lines at point F, where a right angle is indicated. 2. Important rules:

1. The problem asks to determine which pairs of angles are adjacent among the given pairs: $\angle 5$ and $\angle 10$, $\angle 2$ and $\angle 9$, $\angle 2$ and $\angle 10$, $\angl

1. **State the problem:** We have a right triangle with hypotenuse 30m, base 14m, and we need to find the angle $x$ opposite the base. 2. **Formula used:** In a right triangle, the

1. **State the problem:** We have a right-angled triangle with two sides given: one side is 7 cm and the other is 5 cm. We need to find the length of the hypotenuse. 2. **Formula u

1. **Problem:** Find an equation of the sphere with center $(-1,4,5)$ that just touches the a. xy-plane,

1. **Stating the problem:** We are given the formula for the area of a rectangle or parallelogram, which is $A = b \times h$, where $A$ is the area, $b$ is the base, and $h$ is the

1. **State the problem:** We need to find the area of a triangle with all sides measuring 4 cm. 2. **Identify the type of triangle:** Since all sides are equal (4 cm), this is an e

1. **State the problem:** We need to find the central angle $x$ in a circle given an inscribed angle of $70^\circ$ that subtends the same arc. 2. **Recall the rule:** The central a

1. **Problem Statement:** Identify a pair of alternate exterior angles, a pair of corresponding angles, and a pair of alternate interior angles formed by two parallel lines $m$ and

1. **State the problem:** We have a cylindrical water bottle with height $15$ cm and inner radius $4$ cm. The water height is currently $8$ cm, and the bottle will be full (height

1. **Problem Statement:** Given a line segment $\overline{CD}$, construct its perpendicular bisector and label the intersection point as $E$.

1. **Problem Statement:** Describe the reflections and/or rotations that carry a square onto itself. 2. **Understanding the square's symmetry:** A square has several symmetries inc

1. **State the problem:** We have triangle EGD with point F on segment EG such that EF = $x$ and FG = $x + 10$. We know ED = 24 and GD = 54. We want to find the length EF. 2. **Ide

1. **Stating the problem:** We are given two triangles, one acute with one angle marked and one side marked with a single tick, and the other a right triangle with the vertical leg

1. **Problem statement:** We have an isosceles triangle $\triangle ABC$ with $AB = CB$, area $= 30$ cm$^2$, and base $AC = 12$ cm. We need to find the measure of $\angle ABC$ to th

1. **State the problem:** We are given two angles expressed in terms of $x$: $4x - 1$ degrees and $2x + 5$ degrees, and we need to find the measure of angle $m\angle MYW$. The diag

1. **State the problem:** We are given two angles at point Y, one measuring $(4x - 1)^\circ$ and the other $(2x + 5)^\circ$, and these angles are adjacent and form a right angle (9

1. **State the problem:** We have a right triangle with vertices K, L, and J, and a perpendicular segment KM from K to LJ. Given lengths are $KM=6$ and $MJ=8$. We need to find the

1. **Stating the problem:** We are given a geometry problem involving inequalities (inequations). We need to analyze the geometric conditions and solve the inequalities accordingly