📐 geometry

1. **State the problem:** We are given a parallelogram ABCD with angles expressed in terms of $x$: - $m\angle DAB = 85 - 2x$

1. **Problem Statement:** Given two parallel lines $p \parallel q$ cut by a transversal, find the value of angle $w$ adjacent to the $101^\circ$ angle on line $q$.

1. **Problem statement:** Find the measures of angles $\angle b$ and $\angle d$ given that lines $m \parallel n$ and the angles $a=60.7^\circ$, $c=119.3^\circ$, and $d=136.9^\circ$

1. **Problem statement:** Explain how the angle in a semicircle theorem enables construction of a right-angled triangle with a given hypotenuse AB. 2. **Theorem used:** The angle i

1. **State the problem:** Calculate the perimeter and area of an irregular L-shaped polygon with given side lengths: vertical left edge = 10.7 yd, top horizontal segment = 1.9 yd,

1. **State the problem:** We have a figure composed of a rectangle and a semicircle on top. The rectangle is 36 ft wide and 31 ft tall. The semicircle has its diameter equal to the

1. **State the problem:** We have a regular pentagon with side length 1 and a diagonal of length $x$. Using the symmetry and similar triangles inside the pentagon, we want to find

1. **Problem:** You are facing southeast and then turn 180° clockwise. What direction are you facing now? 2. **Formula and rules:** Turning 180° means facing the opposite direction

1. **State the problem:** We have a regular pentagon with side length 1 and a diagonal of length $x$. We want to use the symmetry and similar triangles to find a relationship betwe

1. **State the problem:** We have two identical rectangular plots of land. The first plot has vertices at approximately $(20,0)$, $(20,80)$, $(100,80)$, and $(100,0)$. The second p

1. **State the problem:** We have two identical rectangular plots of land. The first plot has vertices at (0,0), (120,0), (0,120), and (120,120). The second plot is shifted 100 yar

1. **State the problem:** We have a right triangle with three sides: the shorter leg, the longer leg, and the hypotenuse. - The shorter leg is 9 inches shorter than the longer leg.

1. **Problem Statement:** Find the area and perimeter of two trapezoids with given dimensions.

1. The problem asks to plot a map with a scale where 5 units on the map correspond to a length of 1 unit in reality. 2. This means the scale factor is $\frac{1}{5}$, so every 5 uni

1. **Problem 4:** Find the measure of each angle in the first diagram where lines AC and BD intersect at point E, and angle \( \angle AEC = 146^\circ \). 2. **Key fact:** When two

1. **State the problem:** We need to find the volume of a regulation men's basketball given its diameter is approximately 9.55 inches. 2. **Formula for the volume of a sphere:**

1. **State the problem:** We need to find the area of the base $B$, the height $h$ of the prism, and the volume $V$ of the prism. 2. **Given:** The base is a pentagon composed of a

1. **Problem Statement:** Find the value of $x$ in the first right triangle where the hypotenuse is $x$, one side is $7\sqrt{6}$, and the angles are $45^\circ$ and $60^\circ$.

1. **State the problem:** We have triangle ABC with vertices $A(0,2)$, $B(-2,2)$, and $C(-2,0)$ in quadrant two and three. We need to create triangle DEF in quadrant one such that

1. **State the problem:** We need to find the angle $x^\circ$ in a right triangle where side $BC = 7.4$ units and side $CD = 5.6$ units, with a right angle at $C$. 2. **Identify th

1. **State the problem:** We have points A(-5, -2), B(-3, -2), and D(-5, -4) in the third quadrant. We need to find point C to form a square ABCD. 2. **Find point C:** Since ABCD i