📐 geometry

1. **State the problem:** Given parallelogram ABCD with $m\angle DAC = 28^\circ$, $m\angle DCA = 25^\circ$, $AD = 18$, $AB = 10$, and $BE = 8$ on diagonal $BD$, find $m\angle ABC$

1. **State the problem:** We are given two similar triangles ABE and CDE with corresponding sides AB = 93 ft, BC = 98 ft, CE = 68.6 ft, and we need to find the length $d = ED$. 2.

1. **State the problem:** We have two similar right triangles \(\triangle NML \sim \triangle KMO\). Given sides are \(NL = 15\) m, \(ML = 12\) m, \(OM = 16\) m, and we need to find

1. **State the problem:** Calculate the surface area of a cylinder with height $h=10$ cm and diameter $d=6$ cm. 2. **Formula for surface area of a cylinder:**

1. **Problem:** Find the lateral surface area (LSA), surface area (SA), and volume (V) of a cube with side length $7$ meters. 2. **Formulas for a cube:**

1. **Problem statement:** Given a quadrilateral PQRS with parallel lines LM, SR, and PQ, and side lengths LM = 9 cm, PQ = 3 cm, and SR = 6 cm, find the ratio $\frac{LS}{PL}$. 2. **

1. **Problem statement:** Given trapezoid PQRS with sides $PQ=3$, $SR=9$, legs $PS=4$, $QR=6$, and a smaller trapezoid LMQP inside it such that $LM \parallel SR \parallel PQ$. The

1. **State the problem:** We have two parallel lines PQ and ST cut by a transversal. Given angles are $35^\circ$ at PQ and $140^\circ$ at ST. We need to find the value of $x$, the

1. **Problem 1: Find the area of the figure made up of 4 semi-circles and a square with radius 4 mm.** 2. The figure consists of a square and 4 semi-circles attached to each side.

1. **Stating the problem:** We have a rectangle with length $x$ and width $y$. We need to determine which statements about its perimeter and area are true. 2. **Formulas:**

1. The problem is to explain the properties of a kite as shown in the picture. 2. A kite is a quadrilateral with two pairs of adjacent sides that are congruent (equal in length). I

1. The picture shows a square, which is a special type of parallelogram. 2. A square has four equal sides and four equal angles, each measuring 90°.

1. The problem is to explain the properties of a square as shown in the picture and provide examples of its angles, sides, and diagonals. 2. A square is a special type of parallelo

1. The picture shows a rhombus, which is a special type of parallelogram. 2. Key properties of a rhombus include:

1. The problem is to understand the properties and characteristics of a rectangle, which is a special type of parallelogram. 2. A rectangle is defined as a parallelogram with four

1. **State the problem:** We are given triangle $\triangle KNO$ with $m\angle NKO = 30^\circ$ and angles $\angle NKO \cong \angle KNM \cong \angle MNT$. We need to find $m\angle NM

1. **Problem statement:** Adam has built a garden shed composed of a rectangular prism base and a triangular prism roof. The base measures 4 m by 4 m by 2 m, and the triangular pri

1. **State the problem:** We need to find the surface area of a right triangular prism with legs 4 cm and 3 cm, and a prism length of 10 cm along the hypotenuse direction at an ang

1. **State the problem:** We have a circle with radius $7$ cm divided into two sectors in the ratio $3:7$. We need to find the length of the minor arc. 2. **Understand the ratio:**

1. **State the problem:** We need to find the volume of the wall of a hollow steel pipe shaped like a right circular cylinder.

1. **Stating the problem:** We have a right triangle with angles 45°, 45°, and 90°. We want to understand the properties of this triangle. 2. **Formula and important rules:** In a