📐 geometry

1. **Problem 5(a):** Find the ratio of PN to MN in simplest form given triangles ABC and MNP are similar. 2. Since triangles ABC and MNP are similar, corresponding sides are propor

1. Problem 13: Find the measure of the angle at point V formed by a tangent and a chord intercepting an arc of 160°. 2. Formula: The angle formed by a tangent and a chord is half t

1. **State the problem:** We are given a circle with points W, V, and X on the circumference. The arc WX measures $17x - 10$ degrees, and the inscribed angle at V intercepting arc

1. **Problem Statement:** We are given two parallel lines $l$ and $m$ cut by a transversal, and we need to find the measure of angle $\angle ABC$. 2. **Key Concept:** When two para

1. **Problem statement:** You are asked to find and cut out geometric bodies from various brochures, sort them, paste them into your notebook, and name them.

1. **Stating the problem:** Calculate the area of the irregular polygon described, which resembles a trapezoid with a rectangular notch cut out from the bottom-left interior.

1. **Problem:** A 20-ft ladder is leaning against a wall making a 62° angle with the floor. Find the distance from the base of the ladder to the wall. 2. **Formula:** Use the cosin

1. **State the problem:** We need to find various lengths and areas related to two circles: one centered at A with radius 24, and one centered at J with diameter 30. Given lengths

1. **Stating the problem:** We have a large triangle DFE with a right angle at H where FH is perpendicular to DE. We are given some angles and segments, and we need to find the val

1. **State the problem:** Find the area of the irregular L-shaped polygon with given side lengths: top side = 8 ft, left side = 3 ft, right vertical side = 5 ft, and bottom short h

1. **Stating the problem:** We have a quadrilateral inscribed in a circle with points T, U, S, V, and R inside the circle. We want to find expressions for the measures of angles \(

1. **Problem Statement:** Given two parallel lines L1 and L2 intersected by two transversals t1 and t2, find the measures of angles 1, 2, 3, and 4 using the given angles 109° and 8

1. **State the problem:** We need to find the volume of the box shown in the graph. 2. **Identify the dimensions:** The box is composed of unit cubes arranged in a rectangular pris

1. **State the problem:** We need to find the volume of a rectangular prism with length $3 \frac{5}{8}$ inches, width $2 \frac{1}{2}$ inches, and height 4 inches. 2. **Formula for

1. **Problem Statement:** Nathan is making a kite with two adjacent sides each 12 inches and a right angle between them. We need to find the length of the support rod, which is the

1. **State the problem:** Find the volume of a cylinder with height $16$ feet and diameter $10$ feet. 2. **Formula for the volume of a cylinder:**

1. The problem asks to identify which sides are legs and which are the hypotenuse in triangle $\triangle ABC$ with vertices $A$ at the left bottom, $B$ at the top, and $C$ at the r

1. The problem asks to identify which sides of triangle $\triangle ABC$ are legs and which are the hypotenuse. 2. In a right triangle, the hypotenuse is the side opposite the right

1. **State the problem:** We need to find the values of $x$ and $y$ given the angles in the triangle-like figure with angles $70^\circ$, $(x+y)$, $(3x+y)$, and $60^\circ$. 2. **Use

1. Given that $\ell_1 \parallel \ell_2$ and $\ell_3 \parallel \ell_4$, and an angle of 108° is marked. 2. (a) To find $\angle 1$, note that $\angle 1$ and 108° are alternate interi

1. **State the problem:** Find the surface area and volume of a rectangular prism with dimensions 10 cm (length), 6 cm (width), and 4 cm (height). 2. **Surface Area Formula:**