📐 geometry

1. **Problem:** Find the values of $x$ and $y$ where $O$ is the center of the circle and angles satisfy $AO0 + 108 + COD + 4AO8 = 360^6$. **Step 1:** Understand that the sum of an

1. مسئله: در شکل مقابل، O مرکز دایره است و BC و AD بر دایره مماس هستند. نشان دهید که BC برابر AD است. 2. فرمول و قوانین مهم: اگر دو خط مماس به دایره از یک نقطه خارج از دایره رسم شو

1. **Nêu bài toán:** Cho tam giác $ABC$ với $KJ \parallel BC$, $KI$ là phân giác của góc $AKJ$. Góc $ABC = 60^\circ$. Tính góc $IKJ$. 2. **Phân tích và công thức:** Vì $KJ \paralle

1. **State the problem:** We need to find the size of angle $x$ in a polygon with seven interior angles given as $108^\circ$, $121^\circ$, $62^\circ$, $274^\circ$ (reflex angle), $

1. **State the problem:** We need to find the size of angle $n$ in a pentagon where the other four interior angles are given as $58^\circ$, $29^\circ$, $73^\circ$, $71^\circ$, and

1. **State the problem:** We need to prove that points A(1,-3), B(-3,0), and C(4,1) form an isosceles right-angled triangle and then find its area. 2. **Formula and rules:**

1. **Problem statement:** Given a circle with points $P$, $Q$, and $R$ on it, and tangents $ST$, $TU$, and $SU$ at these points respectively. Lines $RQ$ and $ST$ extended meet at $

1. **Problem Statement:** Prove that parallelograms ABCD and PQRD have equal areas.

1. **Problem:** Find the unknown angle $x$ in a right triangle with angles $58^\circ$, $90^\circ$, and $x$. 2. **Formula:** The sum of angles in a triangle is always $180^\circ$.

1. **State the problem:** We have a composite figure made of two triangles sharing a side of 15 cm. The left triangle has sides 15 cm and 25 cm with an included angle of 50°. The r

1. **Problem:** Find the unknown angle \(\angle U\) in kite \(WVUT\) given \(\angle V = 74^\circ\) and \(\angle T = 96^\circ\). 2. **Properties of kites:** The sum of interior angl

1. **Problem:** Given the surface area of a cube is 54 cm², find its volume. 2. **Formula for surface area of a cube:**

1. **Problem:** The surface area of a cube is 54 cm². Find the volume of the cube. 2. **Formula and rules:**

1. The problem asks which congruence rule explains why the two triangles inside the parallelogram are congruent. 2. The triangles share a common side and have pairs of congruent an

1. The problem asks for a compound inequality describing the possible lengths of the third side $x$ of a triangle with two sides 0.8 and 1.4. 2. According to the triangle inequalit

1. **Problem statement:** We have a triangle with two sides of lengths 4.3 and 0.6, and we want to find the possible lengths of the third side $x$. 2. **Formula and rule:** The tri

1. **State the problem:** We have a triangle with two sides of lengths 12 and 16. We want to find the largest possible whole-number length for the third side. 2. **Recall the trian

1. **Problem Statement:** We have a triangle with two sides of lengths 17.9 and 13.9, and we want to find the possible lengths of the third side $x$. 2. **Triangle Inequality Theor

1. **Problem statement:** We have a triangle with two sides of lengths 9 and 19. We want to find the largest possible whole-number length for the third side. 2. **Formula and rule:

1. **Problem statement:** We have a triangle with two sides of lengths 4 and 7. We want to find the largest possible whole-number length for the third side. 2. **Triangle inequalit

1. **State the problem:** We need to determine if a triangle can have sides of lengths 11.4, 7.8, and 16.2. 2. **Recall the triangle inequality theorem:** For any triangle with sid