📐 geometry

1. **State the problem:** We are given two vertical angles, \(\angle 1\) and \(\angle 2\), with measures \(m\angle 1 = 5x + 12\) and \(m\angle 2 = 6x - 11\). We need to find the me

1. **State the problem:** We need to find the cosine of angle $S$ in a right triangle $TUS$ where the right angle is at vertex $U$. 2. **Identify the sides relative to angle $S$:**

1. **State the problem:** We need to find the tangent of angle $B$ in a right triangle with vertices $A$, $B$, and $C$. The side lengths given are $AB=17$ units and $BC=8$ units, w

1. **State the problem:** We need to find the horizontal distance of a wheelchair ramp with a vertical rise of 8 inches and an incline angle of 4.76°. 2. **Relevant formula:** In a

1. **State the problem:** We have a stage that is 20 feet long (horizontal length) and rises 2 feet vertically. We want to find the angle of elevation (rake) and check if it is 5°

1. **State the problem:** Find the distance $d(P_1, P_2)$ between the points $P_1 = (2, 5)$ and $P_2 = (-2, -4)$. 2. **Formula used:** The distance between two points $(x_1, y_1)$

1. **State the problem:** Chelsea runs north 0.75 mile, then west 1 mile, and finally runs straight home. We need to find the total distance she runs each week (5 weekdays). 2. **I

1. **State the problem:** John drives south for 8 miles, then east for 15 miles, and returns home by a straight road. We need to find the total distance he drives after 7 days. 2.

1. **State the problem:** Given that ABCD is a parallelogram, CE bisects \(\angle DEB\), and \(\angle ADE \cong \angle ABE\), prove that \(\triangle DEC \cong \triangle BEC\). 2. *

1. **State the problem:** Given that ABCD is a parallelogram, CE bisects \(\angle DEB\), and \(\angle ADE \cong \angle ABE\), prove that \(\triangle DEC \cong \triangle BEC\). 2. *

1. **State the problem:** We need to find the value of $t$ in a triangle where the angles are given as $32^\circ$, $t + 45^\circ$, and the bottom vertex angle is split into two par

1. **State the problem:** We have a quadrilateral SRQT with angles at vertices S, T, Q, and R. Given angles are \(\angle S = 100^\circ\) and \(\angle T = 61^\circ\). We need to fin

1. **State the problem:** We need to find the range of possible values of $x$ given a quadrilateral with one internal angle labeled as $(3x - 6)^\circ$, an adjacent angle of $24^\c

1. **Problem statement:** We have a triangle with sides 22 cm and 14 cm, and an angle of 35° opposite the 22 cm side. We need to find the angle $x$ at the bottom left corner. 2. **

1. **State the problem:** We need to find the measure of angle $\angle B$ in a right triangle with sides $AC=22$ and $BC=18$, where $\angle C$ is the right angle. 2. **Identify the

1. The problem is to plot the point D with coordinates $\left(0, -\frac{7}{2}\right)$ on a Cartesian coordinate grid. 2. The point $D(0, -\frac{7}{2})$ means the x-coordinate is 0

1. The problem is to plot the point B with coordinates $\left(\frac{5}{2}, 3\right)$ on a Cartesian plane. 2. The coordinates $\left(\frac{5}{2}, 3\right)$ mean the point is locate

1. The problem asks which solid, when cut by a plane parallel to its base, produces a rectangular cross section. 2. Important rule: A plane parallel to the base of a solid will pro

1. **State the problem:** Find the surface area of a rectangular prism with dimensions 4 cm, 5 cm, and 3 cm. 2. **Formula:** The surface area $SA$ of a rectangular prism with lengt

1. **State the problem:** Find the surface area of a rectangular prism with dimensions height $h=6$ in, width $w=4$ in, and depth $d=7$ in. 2. **Formula for surface area of a recta

1. **State the problem:** Find the surface area of a rectangular prism with dimensions 7 cm (height), 3 cm (width), and 1 cm (depth). 2. **Formula:** The surface area $S$ of a rect