📐 geometry

1. **Stating the problem:** We are given two intersecting lines at point O forming vertical angles. One angle, \(\angle POR\), is given as 146°. 2. **Understanding vertical angles:

1. **Problem Statement:** Determine the values of the missing variables and the measures of each unknown angle for the given angle relationships involving parallel lines, transvers

1. **Problem statement:** We have a cuboid with three visible faces having areas 12, 15, and 20 square meters. The edges of the cuboid are integers. We need to find the volume of t

1. **Stating the problem:** We have three squares arranged vertically between two parallel horizontal lines. The top square is rotated so that one corner touches the top line, form

1. Problem 3 involves two parallel lines cut by a transversal with angles labeled a, b, c, d and one angle given as 56°. 2. The key rule is that alternate interior angles are equal

1. **State the problem:** We have an isosceles triangle with two equal sides of length $n$ and the angle between these sides is $52^\circ$. The perpendicular height from the vertex

1. **State the problem:** We have an isosceles triangle with two equal sides of length $n$ and the angle between these sides is $52^\circ$. The perpendicular height from the vertex

1. **Problem Statement:** We need to identify the transformation that results in a horizontal reflection and a dilation by a scale factor of $\frac{1}{2}$ of an L shape. 2. **Under

1. **Problem statement:** Given a circular body with intersecting chords in a quadrilateral shape, with points A, B, C, D, and E inside the figure.

1. **State the problem:** We need to find the perimeter of a right-angled triangle with one angle of 43° and the side adjacent to this angle (base) measuring 13.8 m. 2. **Identify

1. **Problem statement:** Given a stadium-shaped figure with rectangle length $AB = CD = 100$ m, width $BC = AD = 40$ m, and diagonal $AC = 118.7$ m, find $\sin(\angle ACD)$, $\cos

1. **Problem statement:** We need to find the height $y$ of an isosceles triangle where the two equal sides each measure 11.5 m and the angle adjacent to the height is 57°. 2. **Un

1. **State the problem:** We need to find the area of a rectangular rug with width $8$ m and height $2 \frac{7}{12}$ m. 2. **Formula:** The area $A$ of a rectangle is given by:

1. The problem states that point Q is the midpoint of line PR, and we are given points P(-4, 6) and Q(k, -2). We need to find the value of $k$ given that the distance $PQ = 17$ uni

1. **Problem statement:** We have a right-angled triangle and an isosceles triangle joined together. The equal sides of the isosceles triangle are 25 cm each, and the base of the r

1. **Problem Statement:** Prove the projection law in any triangle $\triangle ABC$ using vector methods: $$a = b \cos C + c \cos B$$ where $a$, $b$, and $c$ are the sides opposite

1. **Problem statement:** In the figure, AB \parallel CE and DF \parallel AC. Given angles are \(\angle BAC = 71^\circ\), \(\angle EFC = 80^\circ\), and \(\angle BFE = 4x^\circ\).

1. **Problem statement:** Given that lines AB // CD and BP // AD, with angles \(\angle EBF = 68^\circ\) and \(\angle CDE = 58^\circ\), find (i) \(\angle AEB\) and (ii) \(\angle ABE

1. Let's clarify the step where we calculate $180 \times 5 = 900$.\n\n2. This step comes from the formula for the sum of the interior angles of a polygon: $$\text{Sum of interior a

1. **State the problem:** We have a 7-sided polygon (heptagon) with known angles 165, 150, 123, 100, 170, and two unknown angles labeled $x+11$ and $x$. We want to find the value o

1. **Stating the problem:** We have a 7-sided polygon (heptagon) with sides 165, 150, 123, 100, 170, and two unknown sides labeled $x+11$ and $x$. We want to find the value of $x$.