📐 geometry

1. **State the problem:** Given parallelogram HOME, prove that opposite sides are congruent: $\overline{HO} \cong \overline{ME}$ and $\overline{OM} \cong \overline{HE}$. 2. **Recal

1. **Stating the problem:** We have a circle with points P, Q, and S on the circumference and a point R outside the circle. The line passes through P, Q, and R. Given angles are $\

1. **Problem Statement:** Given a circle with points U, T, and V such that T lies on the circle, U and V are outside, and the angle at U between lines UV and UT is 50°. We need to

1. Problem 1: Find the scale of the building plan. Given:

1. **Problem Statement:** We have a circle with points L, M, N, and X on its circumference. Two triangles, LMN and XLN, are inscribed in the circle. The line segment LM is labeled

1. **State the problem:** Find the area of the quadrilateral with vertices $P(3,7)$, $Q(7,15)$, $R(17,10)$, and $S(13,2)$. 2. **Formula used:** We use the Shoelace formula (also ca

1. The problem shows two horizontal line segments, AB and CD, which appear parallel and have small arcs near points B and D respectively. 2. The conjecture involves comparing the l

1. The problem asks for the next step in constructing a line parallel to line CD using compass and straightedge. 2. In classical geometric constructions, to construct a parallel li

1. **Problem Statement:** We need to bisect angle $\angle ABC$ in the triangle with vertices $A$, $B$, and $C$. 2. **Understanding Angle Bisector:** An angle bisector divides an an

1. The problem asks for the first step in constructing a line perpendicular to vector $\overrightarrow{AB}$ passing through an external point $M$ using only a compass and straighte

1. **Problem Statement:** We are given a circle with points F, G, H, and A on its circumference. Triangle FGH is inscribed in the circle. The segment GH measures $21x - 2$ and segm

1. **Problem Statement:** We have a right triangle inscribed in a circle. The triangle has sides labeled 10.4, 8.4, and $x$, with a right angle between the sides 8.4 and $x$. We ne

1. **State the problem:** We have a triangle with angles 68°, 125°, and an unknown angle $x$. We need to find the measure of angle $x$. 2. **Formula used:** The sum of the interior

1. **Problem Statement:** We are given a triangle with angles and sides, and we need to find the measure of angle $x$. 2. **Given Information:**

1. **State the problem:** We are given a triangle with three angles labeled as $3x - 5$, $5x + 5$, and $3x - 15$. We need to find the measure of angle $x$. 2. **Recall the triangle

1. **Problem Statement:** Given $m\angle 2 = 55^\circ$ and $m\angle 3 = 80^\circ$, find $m\angle 4$.\n\n2. **Understanding the figure and relationships:** The angles are part of a

1. **Problem Statement:** Find the area of the shaded sector of a circle with radius $60$ units and arc length $150$ units.

1. **Problem Statement:** Find the area of a segment of a circle with radius $r=40$ and central angle $\theta=165^\circ$. 2. **Formula:** The area of a segment is given by the diff

1. **Problem Statement:** Find the area of the shaded sector of a circle with radius $45$ units and arc length $50$ units. 2. **Formula for arc length:** The arc length $s$ of a se

1. **Stating the problem:** We have a square ABCD with vertices A(3,7), B(3,3), and D(7,7). We need to find the coordinates of the fourth vertex C. 2. **Formula and rules:** In a s

1. **Problem statement:** We have a straight line JKLM with points K, L, M, and angles x and y. Given \( \cos x = \frac{1}{\sqrt{5}} \) and the ratio \( LM : KL = 1 : 2 \), we need