📐 geometry

1. **Stating the problem:** We are given a circle with center $O$ and points $A, B, C, D, E$ on the circumference. We want to find the value of angle $x$ at vertex $C$ of triangle

1. The problem involves analyzing angles and arcs in circles with inscribed polygons and chords. 2. Key circle theorems to use:

1. **Problem statement:** In the figure, ABC is a straight line. E lies on BD produced such that AD = AE. Prove that $a + b = 180^\circ$. 2. **Formula and rules:** The sum of angle

1. **Stating the problem:** We have a triangle ABC with angles at B and C given as 20° and 30° respectively. Points D and E lie on sides AC and AB, and angles at D and E are labele

1. The problem states that the area of triangle ABD is 20. 2. The formula for the area of a triangle is $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$.

1. **Stating the problem:** We have a triangle ABC with points D on AC and E on AB. Lines BD and CE intersect at F. Given angles at B (20°), C (30°), and angles $z$ near E and $y$

1. **State the problem:** We need to find the length of the hypotenuse $c$ of a right triangle with legs of lengths 11 and 5. 2. **Formula used:** The Pythagorean Theorem states th

1. **Problem Statement:** We need to find the length of the hypotenuse $c$ in a right triangle where the legs are $a=10$ and $b=11$. 2. **Formula:** The Pythagorean Theorem states

1. **Problem Statement:** We are given a right triangle with a hypotenuse of length 17 and one leg of length 13. We need to find the length of the other leg, $x$, and round it to t

1. **State the problem:** We need to find the value of $x$ in a right triangle where the hypotenuse is 18, one leg is 12, and the other leg is $x$. 2. **Formula used:** In a right

1. **State the problem:** We need to find the value of $x$, the hypotenuse of a right triangle with legs 10 and 12. 2. **Formula used:** For a right triangle, the Pythagorean theor

1. **State the problem:** We have a right triangle with legs of lengths 21 and 16, and we need to find the length of the hypotenuse $x$. 2. **Formula used:** In a right triangle, t

1. **State the problem:** We have a right triangle with legs 10 and $x$, and hypotenuse 13. We need to find the value of $x$ rounded to the nearest tenth. 2. **Formula used:** In a

1. **Problem Statement:** List all the lines in the given figure with points G, C, L, B, I, and the slanting line through points 13, 17, and k. 2. **Lines Definition:** A line exte

1. **Stating the problem:** We are given three triangles with side lengths: - Top triangle: 7, 7, 25

1. نبدأ ببيان المسألة: لدينا مثلث قائم الزاوية مع زوايا 30 و60 و90 درجات، وطول أحد الأضلاع المقابلة للزاوية 30 هو 3. 2. القاعدة المهمة في مثلث 30-60-90 هي أن أطوال الأضلاع تتناسب ك

1. **Problem statement:** Given a right triangle أ د ج with a right angle at د and side lengths ٤ and ٤, find the length of the hypotenuse أ ج and verify the length of segment ه ج

1. The problem is to understand the triangle congruence postulates SSS, SAS, and ASA and see examples for each. 2. Triangle congruence postulates are rules that allow us to determi

1. **Problem statement:** Given a circle with center $C$, points $A$, $B$, and $D$ on the circumference, and $\angle ABD = 38^\circ$, find the measure of $\angle CAD$ in triangle $

1. **Problem statement:** We have a circle with center $C$ and radius 8 cm. The angle $\angle FEG = 45^\circ$ is given.

1. **State the problem:** Prove that point E bisects both diagonals AC and BD in a parallelogram ABCD. 2. **Given:** AB || CD and AD || BC (definition of a parallelogram).