📐 geometry

1. Let's start by defining a right-angle triangle. A right-angle triangle is a triangle in which one of the angles is exactly $90^\circ$. 2. The side opposite the right angle is ca

1. The problem involves two intersecting lines forming vertical angles. We are given one angle as $102^\circ$ and the opposite angle as $(10x - 2)^\circ$. 2. Vertical angles are al

1. The problem involves understanding and applying theorems about tangent lines, secant segments, and their lengths with respect to circles, based on given points and segments in F

1. **Problem statement:** We have a right pyramid with vertex $V$ and square base $ABCD$ of side $16$ cm. Angle $AVC = 90^\circ$.

1. The set of all points in the plane that are the same distance away from a specific point, called the center, is a **circle**. 2. The distance from the center of the circle to it

1. Given a right triangle with legs $a=12$, $b=5$, and hypotenuse $c$, we use the Pythagorean theorem: $$c^2 = a^2 + b^2$$

1. The problem is to understand and prove why the radius of the Earth is calculated as it is. 2. One historical method, used by Eratosthenes, to estimate Earth's radius involves me

1. Problem 7 asks to find $l_\alpha$, $t_\beta$, and calculate $h_{\alpha\alpha'} - h_{\alpha\alpha}$. Given relationship: $$h_{\alpha\alpha'} - h_{\alpha\alpha}$$

1. We are given incomplete statements about angles formed by tangents, secants, and their intersections with circles. We will supply the missing words to make the theorems correct.

1. **Stating the problem:** We are given a parallelogram ABCD and need to find the values of $x$ and $y$ based on the angle expressions given: angle $A = y$, angle $E = 3x - 5$, an

1. **State the problem:** Calculate the area of spherical triangles given their angles and the radius of the sphere. 2. **Formula for the area of a spherical triangle:**

1. **Problem:** Given $m\angle GC = 149^\circ$ and $m\angle LSC = 39^\circ$, find $m\angle MC$. Step 1. Observe that points L, M, S are collinear. The angles around point C includi

1. Problem: Given $m\angle GC = 149^\circ$ and $m\angle LSC = 39^\circ$, find $m\angle MC$. Step 1: Identify the relationship between the angles (likely adjacent or related through

**I. Exterior Intersection Angles Problem:** Given angles and arcs around circle points A, B, C, D, E with secants/tangents intersecting outside.

1. Problem: Find the measures of angles formed by secants and tangents intersecting outside the circle. 2. Given angles and arcs: PA = 138°, AQ = 50°, DB = 125°, EA = 86°.

1. **State the problem:** We need to find the size of angle $x$ at point $A$ in a cyclic quadrilateral $ABCD$. The quadrilateral is inscribed in a circle, $BD$ passes through the c

1. The problem asks to find the measure of angle $B$ in a triangle where the angles at vertices $A$, $B$, and the third vertex are given as $3x$, $2x$, and an algebraic expression

1. We are given a triangle ABC with \(\angle B = 90^\circ\) and \(\angle A = 2\angle C\). We need to find the measures of \(\angle A\), \(\angle C\), and \(\angle ?\). 2. Since \(\

1. Given lines AB || CD, and angles around intersection point E as 11° and 17°, find $x$. Since AB || CD, alternate interior angles sum to $180^\circ$. The angles on a straight lin

1. ପ୍ରଥମ ସମସ୍ୟା: ଦିଆଯାଇଛି ଯେ କୋଣଗୁଡିକ (2y + 30)° ଏବଂ y° ହେଉଛି। ପଞ୍ଚମ ପରିମାଣର ଅନୁପାତ ଖୋଜାଯିବାକୁ ହେଉଛି। 2. ଇଠାରେ ଆମେ ଧରିବା ଯେ ପଞ୍ଚମ ପରିମାଣ ବୋଲି ଅର୍ଥ ହେଉଛି ଯେ ସେହି ଦୁଇ କୋଣର ଅନୁପାତ ହେଉ

1. **Theorem 1 (Example):** Given angles $\angle A$ and $\angle B$ such that $m\angle A = 30^\circ$ and $m\angle B = 45^\circ$. If $\angle A$ and $\angle B$ are complementary, show