📐 geometry

1. **Problem Statement:** Fill in the blanks and calculate areas of given shapes including rectangles, polygons, parallelograms, and triangles.

1. **State the problem:** We are given a triangle $\triangle ABC$ with angles $A = 29^\circ$, $B = 36^\circ$, and side $b = 15.8$ cm. We need to find the length of side $a$ opposit

1. The problem is to find the supplementary angle of 135 degrees. 2. Supplementary angles are two angles whose measures add up to 180 degrees.

1. **Problem Statement:** Given a circle with center O and points P, Q, R, S on the circle such that PQ = SR = 10 cm, and a triangle ABC with \(\angle ACB = 90^\circ\) intersecting

1. **Problem statement:** Two angles are supplementary, and one angle measures 45 degrees. Find the measure of the other angle. 2. **Formula and rule:** Supplementary angles add up

1. **Problem Statement:** We are given a quadrilateral ABCD with diagonals intersecting at point E such that $AE = EC$ and $BE = ED$. We need to determine if ABCD is a parallelogra

1. **Problem Statement:** We need to find the coordinates of points D, E, and F after a 90° clockwise rotation around the origin. 2. **Rotation Formula:** For a point $(x,y)$, a 90

1. **Problem Statement:** We have two circles intersecting at point A. AB is tangent to the larger circle at A, AC is tangent to the smaller circle at A, and AD is a common chord.

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