📐 geometry

1. Given: $\angle WVX \cong \angle VXY$ and $\angle XVY \cong \angle VXW$. 2. Note that side $VX$ is common to both triangles $\triangle VXY$ and $\triangle XVW$, so $VX \cong VX$

1. **State the problem:** We need to prove that triangles $\triangle FHI$ and $\triangle FHG$ are congruent given that $FH$ bisects angles $\angle GHI$ and $\angle GFI$. 2. **Given

1. **State the problem:** We need to prove that triangles $\triangle FHI$ and $\triangle FHG$ are congruent given that $FH$ bisects $\angle GHI$ and $\angle GFI$. 2. **Given:**

1. **Problem Statement:** Determine the relationship between given line segments and planes in a cube, and identify angle pairs using the provided word bank. 2. **Line Relationship

1. **Problem statement:** We have three atoms A, B, and C with atomic radii 5.1, 4.0, and 2.0 respectively. They form a triangle by connecting their centers. We need to find the me

1. **Stating the problem:** We need to find the area and perimeter of three triangles: ABC, PQR, and DEF. Each triangle has two side lengths given, but the third side is missing. 2

1. **Problem Statement:** Given parallelogram MATH with vertices M, A, T, H and diagonals intersecting at point S, find the following congruences and equalities. 2. **Recall Proper

1. **Problem Statement:** We are given a triangle ABC with an altitude from A to BC intersecting at F, and points E and D on segments AE and AD respectively, forming right angles.

1. **Problem Statement:** We are given a triangle ABC with various points and perpendicular segments, and we need to find the length of segment PA. 2. **Understanding the Problem:*

1. **State the problem:** We need to find the area of a parallelogram drawn on a grid where each square side is 1 yard. 2. **Formula for area of a parallelogram:**

1. **State the problem:** We need to find the surface area $A$ of a square prism with base edge length $s=14$ inches and height $h=18$ inches using the formula $$A = 2s^2 + 4sh.$$

1. **State the problem:** We need to find the total area of a mural shaped like a trapezoid with bases of lengths 22 yards and 32 yards, and a height of 24 yards. 2. **Formula used

1. The problem is to find the area of a circle given the radius and using the value of $\pi = 3.14$.\n\n2. The formula for the area of a circle is: $$A = \pi r^2$$ where $r$ is the

1. **Problem Statement:** Identify the shape of the cross section formed when a plane slices through each given solid.

1. **State the problem:** We need to find the surface area of a gas tank shaped as a cylinder with two hemispheres attached at each end. 2. **Given:**

1. **State the problem:** We need to find the total surface area of a shape made by joining a hemisphere on top of a cylinder. The hemisphere has radius $r=9$ cm and the cylinder h

1. **State the problem:** We need to find the total surface area of a shape made by joining a hemisphere on top of a cylinder.

1. **State the problem:** We need to prove that triangle ABC is a right triangle given that line segment BE is perpendicular to DE and that the measure of angle ABE equals the meas

1. **State the problem:** We are given two parallel lines \(\overleftrightarrow{PQ} \parallel \overleftrightarrow{RS}\) and a transversal \(\overleftrightarrow{TU}\) intersecting t

1. **State the problem:** Given that ABCD is a rhombus, prove that triangles $\triangle AEB$ and $\triangle CEB$ are congruent. 2. **Step 1:** Statement: ABCD is a rhombus. Reason:

1. **State the problem:** Given that \(\overline{BE} \cong \overline{FD}\), \(\overline{AE} \cong \overline{FC}\), and both \(\angle AEB\) and \(\angle CFD\) are right angles, prov