📐 geometry

1. **Problem statement:** Find the tangent ratios for angles $\angle Y$ and $\angle Z$ in a right triangle with vertices $X$, $Y$, and $Z$, where $X$ is the right angle. 2. **Given

1. **State the problem:** We have a right triangle NML with a right angle at M. Angle N is 46° and side NL (opposite angle M) is 4.1 units. We need to find side NM (adjacent to ang

1. **Problem statement:** We are given a right triangle PQR with right angle at P, and segment PQ is parallel to segment TS. We need to find which ratio is equivalent to $\tan Q$.

1. The problem states that triangles $\triangle QRP$ and $\triangle STP$ are similar, denoted as $\triangle QRP \sim \triangle STP$. 2. Similar triangles have corresponding angles

1. **Énoncé du problème :** Calculer les longueurs $AB$, $BC$ et $ME$ dans le triangle $ABC$ avec les données : $BE=5$, $EA=10$, $BM=2$, $MC=4$, $AF=16$, $AC=24$.

1. **Problem statement:** Given a triangle with sides JK = 7 cm, IJ = 8 cm, and IK = 5 cm, construct the perpendicular bisector of segment [K] and find point E, the reflection of K

1. **State the problem:** We need to find the circumference $C$ of a circle with diameter $d = 10$ m. 2. **Formula:** The circumference of a circle is given by $$C = \pi d = 2\pi r

1. The problem asks to find the circumference of a circle with radius $r = 8$ inches. 2. The formula for the circumference $C$ of a circle is:

1. **Problem statement:** Find the angles marked with letters given that CAB is always a straight line.

1. **State the problem:** Find the volume of a cylinder and the area of a circle with radius $r=21$ cm. 2. **Formulas:**

1. **State the problem:** Find the volume of a cube with side length 12 cm. 2. **Formula:** The volume $V$ of a cube with side length $s$ is given by:

1. **State the problem:** We are given two triangles, \(\triangle TUV\) and \(\triangle WXY\), with some angles known. We need to determine if these triangles are similar. 2. **Lis

1. **State the problem:** We are given two sides AB and BC of a regular polygon with $n$ sides, and two angles at vertex B: $148^\circ$ and $50^\circ$. We need to find the number o

1. **State the problem:** We have a triangle with angles labeled $y$, $x$, $50$, $70$, and $z$. We are asked to find the value of $x$. 2. **Recall the triangle angle sum rule:** Th

1. **State the problem:** We have a triangle with angles labeled 12°, 50°, and 70°, and we need to find the values of $x$, $y$, and $z$ which presumably correspond to these angles

1. **State the problem:** We have a trapezoid with height $h = 11$ cm, area $A = 550$ cm², and one base $b_1 = 10$ cm. We need to find the length of the other base $b_2$. 2. **Form

1. The problem asks to complete the table by measuring the side lengths of triangles △ABC and △DEF, then calculate the ratios $\frac{AB}{DE}$, $\frac{BC}{EF}$, and $\frac{AC}{DF}$.

1. **Stating the problem:** We are asked to find the height of a tree using indirect measurement. A 4-meter flagpole casts a 5-meter shadow, and a nearby tree casts a 20-meter shad

1. **Problem 4:** Given angles around a circle: 137°, $x^\circ$, 223°, and 111.5°. We need to find $x$. 2. The sum of angles around a point on a circle is 360°.

1. **State the problem:** A carpenter claims that in a 30°-60°-90° triangle with hypotenuse 12 m, the other two sides must be 6 m and 10.39 m. 2. **Recall the properties of a 30°-6

1. **State the problem:** We are given two triangles, \(\triangle ABC\) and \(\triangle DEF\), with \(\angle A \cong \angle D\). The sides including these angles are \(AB = 6\) cm,