📐 geometry

1. **Stating the problem:** We have a triangle inscribed in a circle with one angle inside the triangle labeled $45^\circ$, an exterior angle near the chord labeled $88^\circ$, and

1. **State the problem:** We have two intersecting chords inside a circle creating four angles. One angle is 137°, the angle opposite it is $x$, and another angle adjacent to $x$ i

1. **State the problem:** We have two intersecting lines forming four angles. One angle is 59° and we need to find the measures of angles $x$ and $y$. 2. **Recall important rules:*

1. **Problem statement:** Given an acute triangle $\Delta ABC$ with $AB < AC$ inscribed in circle $(O; R)$. Altitudes $BK$ and $CH$ are drawn. The bisector of angle $\widehat{BAC}$

1. **State the problem:** We need to determine if Olio will need more wire than Ellie to install around their painting and its border.

1. **State the problem:** We need to find the volume of a pentagonal prism where each side of the pentagonal base is 8 inches and the height of the prism is 5.5 inches. 2. **Formul

1. **State the problem:** We need to find the volume of a hexagonal prism with side length $4$ ft, height $3$ ft, and apothem $3.5$ ft. 2. **Formula for volume of a prism:**

1. **State the problem:** Find the volume of a cylinder with height $7$ km and radius $8$ km. 2. **Formula for volume of a cylinder:**

1. **Problem statement:** Find the angles $x$ and $y$ in the first figure where two parallel lines are intersected by a transversal, forming angles $72^\circ$, $y$, $x$, and $60^\c

1. **Problem statement:** Determine the perimeter of the right triangle with sides DE = 8.8 cm, hypotenuse DC = 10.6 cm, and angle ∠E = 34°. 2. **Formula and rules:** The perimeter

1. **State the problem:** We need to find the bearing from point O to point A. 2. **Understanding bearings:** Bearings are measured clockwise from the North direction.

1. **State the problem:** We need to find the bearing from point O to point A. 2. **Understanding bearings:** Bearings are measured clockwise from the north direction. The angle gi

1. **State the problem:** We have two similar right triangles. The smaller triangle has legs 3 m and 5 m, and the larger triangle has a base of 50 m and an unknown height (the heig

1. The problem asks to find the angles labeled $m$, $h$, and $l$. 2. To solve for angles in a geometric figure, we use the fact that the sum of angles around a point is $360^\circ$

1. **Stating the problem:** We have two parallel lines $\ell_1 \parallel \ell_2$ cut by a transversal, creating several angles including $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $k$

1. **Problem 1:** Given triangle ABC with right angle at B, BC = 10 (hypotenuse), AC = 10\sqrt{2}, find AB. 2. **Formula:** In a 45-45-90 triangle, the sides are in ratio $1:1:\sqr

1. **Classify angle pair 7 (angles 1 and 2):** - Given the top-left graph, angles 1 and 2 are vertical angles because they are opposite each other when two lines intersect.

1. **State the problem:** Find the total area of a composite figure consisting of three rectangles and one semicircle. 2. **Identify the shapes and their dimensions:**

1. **State the problem:** We need to find the area of an irregular polygon composed of multiple rectangles and squares with given side lengths. 2. **Analyze the figure:** The polyg

1. The problem asks to identify the rotation rule and the degree and direction of rotation for the transformation of points. 2. For point x: Original (9, 8) to x' (8, -9).

1. **State the problem:** Find the midpoint between the points $(2,4)$ and $(-3,9)$. 2. **Formula:** The midpoint $M$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: