📐 geometry

1. **Problem Statement:** Given a circle with diameter \(\overline{EF}\) and tangent \(\overline{EH}\) at point \(E\), and the measure of arc \(\widehat{EFG} = 204^\circ\), find:

1. **Problem statement:** Given quadrilateral ABCD with AB = DC and \(\angle ABC = \angle BCD\) (both acute), prove that AD is parallel to BC. 2. **Key information:**

1. **State the problem:** We have a right triangle with legs $CA = \sqrt{5} - 2$ and $CB = \sqrt{5} + 2$, and we need to find the hypotenuse $AB$. 2. **Formula used:** In a right t

1. **Problem 1: Given polygons ABCH and EDCH are congruent, with C on BD, m(\angle AHC) = 110^\circ, BC = 5 cm. Find the requested values.\n\n2. Since ABCH \cong EDCH, correspondin

1. **Problem Statement:** We have polygon ABCD reflected about the y-axis to form polygon EFGD. Given angles at vertices A (120°) and G (60°), and right angles at B, C, F, and G, w

1. **Problem Statement:** Given two congruent polygons ABCD and AFED, with C as the midpoint of segment MD, and MC = 3 cm, find the length of AE.

1. **Problem statement:** We have two polygons ABCD and EFGH. Polygon EFGH is the image of ABCD by a translation.

1. **Problem Statement:** We have two congruent polygons ABCD and FECD sharing side CD. Given perimeter of ABCD is 24 cm and CD = 5 cm, find perimeter of polygon ABEFD. 2. **Unders

1. The problem is to draw a 10 degree angle and label it. 2. An angle is formed by two rays with a common endpoint called the vertex.

1. **Problem:** Given a circle with radius $8$ m and central angle $45^\circ$, find the arc length. 2. **Formula:** The arc length $s$ of a circle is given by

1. مسئله: دو ضلع مثلث به طول‌های $2$ و $5$ داده شده‌اند و زاویه بین آن‌ها بین $0$ و $90$ درجه است. مساحت مثلث برابر $3$ است. باید کسینوس زاویه بین دو ضلع را پیدا کنیم. 2. فرمول مسا

1. **Problem Statement:** If the diagonals of a parallelogram are equal, prove that the parallelogram is a rectangle.

1. **Problem Statement:** Given trapezium ABCD with AB || CD, E and F are midpoints of non-parallel sides AD and BC respectively. We need to prove that EF is parallel to AB and EF

1. **Problem statement:** Given trapezium ABCD with E and F as midpoints of non-parallel sides AD and BC respectively, prove that line segment EF is parallel to AB and that its len

1. **State the problem:** We have a triangle with side lengths in the ratio 2 : 6 : 7, and the longest side is 24.5 cm. We need to find the perimeter of the triangle. 2. **Understa

1. **Masalah Kapal Berlayar:** Diketahui kapal berlayar ke utara sejauh 80 km dan ke timur sejauh 192 km. Ditanya jarak terpendek (sisi miring) dari posisi awal ke posisi akhir. 2.

1. **Problem statement:** We have a rhombus with each side length $15$ cm and an area of $40$ cm$^2$. We need to find the angles of the rhombus. 2. **Formula used:** The area of a

1. **Problem statement:** Find the distance between the two lines \(l_1\) and \(l_2\) given by: \(l_1:\) \(x = t, y = t, z = t, t \in \mathbb{R}\)

1. **Problem Statement:** Prove that triangles $\triangle FGH$ and $\triangle IJH$ are congruent. 2. **Given Information:**

1. **Problem statement:** We have two congruent triangles, $\triangle ABC$ and $\triangle ABD$. Given: $AC=10$, $AB=6$, $\angle BAC=53^\circ$, $\angle ACB=37^\circ$, $BC=8$ (perpen

1. **Problem Statement:** We need to find the lengths of the sides labeled $x$ in the given figure. 2. **Formula and Rules:** Typically, to find unknown side lengths labeled $x$, w