📐 geometry

1. **Problem statement:** Find the angle between the line $MA$ and the plane $ABCD$ in a cube $ABCDEFGH$, where $M$ is the midpoint of edge $GH$.

1. **State the problem:** We need to find the value of $x$ that makes lines $A$ and $B$ parallel by using the property of same side exterior angles. 2. **Recall the rule:** When tw

1. **State the problem:** We need to find the value of $x$ that makes lines $A$ and $B$ parallel by using the property of alternate exterior angles. 2. **Recall the property:** Alt

1. **State the problem:** We need to find the area of an irregular L-shaped playground with given side lengths. 2. **Understand the shape:** The playground can be divided into two

1. **Problem statement:** Given two parallel lines M and N cut by a transversal, find the measure of angle $\angle a$ on line M, knowing the corresponding angle on line N is 55°.

1. **State the problem:** We have two parallel lines M and N cut by a transversal. We know the measure of an angle on line N is 75° and need to find the measure of angle $a$ on lin

1. **State the problem:** We need to find the area of an isosceles triangle with two equal sides of length 5 cm and an included angle of 65° between them. 2. **Formula used:** The

1. **State the problem:** We have two similar pentagons, A and B. Pentagon A has a base length of 6.4 cm and an area of 97.6 cm².

1. **State the problem:** We have a cuboid ABCDEFGH with given lengths AD = 9 cm, FD = 13 cm, and angle GHF = 49°. We need to find the size of angle FAH, rounded to the nearest deg

1. **State the problem:** We have a cuboid ABCDEFGH with length $17$ cm, width $5$ cm, and height $8$ cm. We need to find the angle between the line segment $AH$ and the plane $EFG

1. **State the problem:** We need to find the length of the diagonal connecting opposite vertices of a cuboid with dimensions 1 cm, 4 cm, and 3 cm. 2. **Formula:** The diagonal $d$

1. **State the problem:** We need to find the area of a shape composed of a right-angled triangle ABC (right angle at B) and a semicircle with diameter BC. 2. **Given:**

1. **State the problem:** We have two similar quadrilaterals RSTU and VWXY. We know some side lengths of RSTU and one side length of VWXY, and we need to find the length of side XY

1. **State the problem:** We have two similar quadrilaterals FGHI and JKLM. We know sides FG = 45, HI = 40.2, and JK = 15. We need to find the length of side KL. 2. **Recall the pr

1. **State the problem:** We are given two similar triangles, BCD and EFG. We know sides BC = 1.8, BD = 2 in triangle BCD, and side EG = 10 in triangle EFG. We need to find the len

1. **State the problem:** We have two similar triangles, EFG and HIJ. We know sides GE = 8, EF = 3.6 in triangle EFG, and side JH = 24 in triangle HIJ. We need to find the length o

1. **State the problem:** We have two similar triangles TUV and WXY. We know side lengths VU = 2, VT = 3, and YX = 12.6. We need to find the length of side YW. 2. **Recall the prop

1. **State the problem:** We have two similar triangles, LMN and L'M'N', where triangle L'M'N' is a dilation of triangle LMN by a scale factor of 4.

1. **State the problem:** We have a quadrilateral OPQR with side length 16 units.

1. **State the problem:** Quadrilateral PQRS is dilated by a scale factor of $\frac{3}{4}$ to form quadrilateral P'Q'R'S'. We are given the length of side S'R' as 18 and need to fi

1. The problem involves right triangles with a 45° angle and finding relationships between sides. 2. In a right triangle with a 45° angle, the sides opposite and adjacent to the an