📐 geometry

1. **State the problem:** We have a right triangle with legs of length $x$ and $x$, and the hypotenuse is $3\sqrt{2}$. We need to find the value of $x$ in simplest form. 2. **Formu

1. **State the problem:** We have a triangle with angles $x^\circ$, $20^\circ$, and $30^\circ$. We need to find the value of $x$. 2. **Recall the triangle angle sum property:** The

1. **Problem Statement:** Given that angles $\angle 1 \cong \angle 2$, determine which lines are parallel among $a$, $b$, $c$, and $d$. 2. **Relevant Property:** When two correspon

1. **State the problem:** We are given two parallel lines \(a\) and \(b\) cut by a transversal, with angles \( (2x - 5)^\circ \) on line \(a\) and \( (x + 20)^\circ \) on line \(b\

1. **State the problem:** We are given two parallel lines $a$ and $b$ cut by a transversal, with an angle of $155^\circ$ adjacent to angle $x$. We need to find the value of $x$. 2.

1. **State the problem:** We have two parallel lines $a$ and $b$ cut by a transversal. One angle formed is $55^\circ$ and we need to find the value of $x$ which is the angle vertic

1. **State the problem:** We have three vertical parallel lines cut by two transversals, creating segments labeled 12 and 8 on the top line, and segments 3y and y + 5 on the middle

1. **State the problem:** Given two parallel lines $a$ and $b$ cut by a transversal, determine the relationship between $m \angle 1$ and $m \angle 8$. 2. **Recall the angle relatio

1. **State the problem:** Given two parallel lines $a$ and $b$ cut by a transversal $t$, identify the relationship between $m \angle 2$ and $m \angle 6$. 2. **Recall the properties

1. Énonçons le problème : Calculer le volume d'une pyramide de hauteur $h$ dont la base est un triangle équilatéral de côté $a$ (tétrahèdre). 2. Formule du volume d'une pyramide :

1. **Problem:** Find the measure of angle $x$ in a triangle where the other two angles are $50^\circ$ and $60^\circ$. 2. **Formula:** The sum of angles in a triangle is always $180

1. **State the problem:** We need to determine which congruence test is used to prove that the purple triangle is congruent to the red triangle given the information about angles a

1. **Problem statement:** Identify the similar triangles and find the unknown side lengths $x$. 2. **Step 1: Identify similar triangles by comparing corresponding sides and angles.

1. **State the problem:** We have a circle with center O and points A, B, C, D on the circumference. DB is a diameter, and AC intersects DB at point P. Given lengths are AP = 12 cm

1. **State the problem:** We have a circle with center $O$ and points $A$, $B$, and $C$ on the circumference. Given angles:

1. **State the problem:** We need to find the value of $x$ in a triangle where the angles are $60^\circ$, $3x$, and $x + 90^\circ$. Note that the problem mentions an angle labeled

1. **State the problem:** We need to find the measure of angle $x$ at vertex $Q$ formed by rays $QR$ and $QS$. 2. **Analyze the given information:** Angle $PQR$ is given as $76^\ci

1. **State the problem:** We need to find the measure of angle $x$ given that angles $\angle YXZ = 97^\circ$ and $\angle WXY = x^\circ$ are adjacent angles on a straight line. 2. *

1. **Problem Statement:** Determine the type of triangle (scalene, isosceles, or equilateral) formed by the given vertices. 2. **Formula Used:** To classify triangles by side lengt

1. **Problem Statement:** Determine if the given pairs of triangles are congruent and by which congruence criterion (SSS, SAS, ASA, AAS, or not necessarily congruent). 2. **Importa

1. Сформулюємо задачу: на колі вибрано точки A, B і C так, що \(\angle ACB = 30^\circ\). Довжина меншої дуги AB дорівнює 25 см. Потрібно знайти довжину всього кола. 2. Відомо, що к