📐 geometry

1. **Problem statement:** We are given a quadrilateral WUTV with diagonals WT and UV, where WT is parallel to UV.

1. **State the problem:** We are given the diameter $D=11.6$ m of a sphere and need to find its volume $V$. 2. **Formula:** The volume of a sphere is given by

1. **State the problem:** We need to find the unknown angle $x$ in three different triangles given the other angles. 2. **Recall the rule:** The sum of the interior angles in any t

1. **Stating the problem:** We have a square-based pyramid with base vertices C, D, E, F and apex B. Point A is the intersection of the diagonals CF and DE of the base.

1. The problem asks to write a congruency statement for triangles \(\triangle JKL\) and \(\triangle QRS\) given that \(\triangle JKL \cong \triangle QRS\). 2. A congruency statemen

1. The problem states that we have a triangle with three interior angles: $20^\circ$, $3x$, and $x$. We need to write an equation for the sum of these angles and then find the valu

1. **State the problem:** We are given a rectangle with height 5 cm and width $r - 4$ cm. The area of the rectangle is 10 cm$^2$. We need to find the value of $r$. 2. **Formula for

1. **State the problem:** We have a triangle with one angle measuring 36° and two sides labeled as 2t and 4t. We need to find the value of $t$. 2. **Understand the problem:** Since

1. **State the problem:** We have a triangle with three interior angles: 38°, 76°, and 6c°. We need to find the value of $c$. 2. **Recall the triangle angle sum rule:** The sum of

1. **Stating the problem:** We are asked to analyze pairs of vectors in a triangular prism: $\overrightarrow{DF}$ and $\overrightarrow{EH}$, $\overrightarrow{GI}$ and $\overrightar

1. **Problem Statement:** Identify which pairs of lines in the cube are skew lines. Skew lines are lines that are neither parallel nor intersecting and do not lie in the same plane

1. **State the problem:** Given that BD bisects \(\angle ABC\) and \(\angle ADC\), prove that \(\triangle ABD \cong \triangle CBD\). 2. **Recall the definition of angle bisector:**

1. **State the problem:** We have two similar triangles ABC and ADE. Given AB = 8 cm, AE = 12 cm, ED = 3 cm, and AC = x cm. We need to find the two possible values of $x$. 2. **Rec

1. **State the problem:** We are given two pentagons, ABCDE and A'B'C'D'E', where A'B'C'D'E' is a reflection of ABCDE. We need to determine the line of reflection. 2. **Identify co

1. **State the problem:** We have triangle ABC dilated about the origin to create triangle A'B'C'. We need to find the scale factor of the dilation. 2. **Recall the dilation formul

1. **Problem statement:** Given triangles \(\triangle ABC \sim \triangle ADE\), find: a) the scale factor from \(\triangle ABC\) to \(\triangle ADE\),

1. The problem asks to find the scale factor between triangles \(\triangle ABC\) and \(\triangle ADE\), then use it to find the perimeter of \(\triangle ABC\) and the area of \(\tr

1. **Problem statement:** Given a right triangle KLM with angles at K and L both 45°, and side KM = 20, find the length of side LM. 2. **Understanding the problem:** Since angles K

1. The problem involves a rectangle with a total length of 18 cm and two horizontal bars inside it, one with a 3 cm segment on the right and the other with a 2 cm segment on the le

1. **State the problem:** We need to find the length of the missing side $w$ in a quadrilateral where the sides and angles are given as follows: one side is 20 m, and the angles ad

1. **State the problem:** We need to find the missing angle $P$ in a triangle where one angle is $20^\circ$ and the sides adjacent to angle $P$ are 15 cm and 18 cm, with the side o