📐 geometry

1. **State the problem:** We have a triangle with angles labeled as $3w$, $2w$, and $35^\circ$. We need to find the value of $w$. 2. **Recall the triangle angle sum rule:** The sum

1. **State the problem:** We have a triangle with angles labeled as $45^\circ$, $11w$, and $16w$. We need to find the value of $w$. 2. **Recall the triangle angle sum rule:** The s

1. **State the problem:** We have a triangle with three interior angles: 71°, 73°, and 18a°. We need to find the value of $a$. 2. **Recall the triangle angle sum rule:** The sum of

1. The problem states that the triangle has three angles labeled as $7w$, $7w$, and $6w$. 2. We know the sum of the interior angles of any triangle is always $180^\circ$.

1. **State the problem:** We are given two angles formed by a transversal crossing two parallel lines. The angles are \( (15x + 8)^\circ \) and \( (9x + 26)^\circ \). We need to fi

1. **Problem statement:** Given parallelogram ABCD with $DA=4$, $DC=5$, and $\angle ADO=60^\circ$. (a) Prove that $AB + AO + BC = 2AC$.

1. The problem states that two triangles are similar, meaning their corresponding sides are proportional. 2. Given the large triangle sides: 3.6, 2.8, and 5.2, and the smaller tria

1. **State the problem:** We have points A(10,10) and B(-2,14).

1. The problem describes a circle with center $O$, a tangent $P_i$ touching the circle at point $i$, and a chord $iQ$ inside the circle. 2. The chord $iQ$ subtends an angle $3x$ at

1. **Problem Statement:** We have a circle with center $O$, a tangent line $Pi$ touching the circle at point $i$, and a chord $\overline{iQ}$.

1. Bài toán yêu cầu cho đường tròn tâm $O$ bán kính $R$ với đường kính $AB$. 2. Điểm $C$ là trung điểm của đoạn $OB$, tức là $C$ nằm trên đoạn thẳng nối $O$ và $B$ sao cho $OC = CB

1. **Problem Statement:** Construct quadrilateral ABCD with given sides $AB=4.5$ cm, $AC=CD=5$ cm, $AD=6$ cm, and angle $\angle BAC=60^\circ$. Then construct triangle PBC such that

1. **Problem statement:** Find the volume of the tetrahedron bounded by the coordinate planes $x=0$, $y=0$, $z=0$ and the plane $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1.$$\n\n

1. **Stating the problem:** We have a triangle ABC with a point D on BC such that AD is perpendicular to BC. The angle at A adjacent to this perpendicular is 40°. At vertex B, ther

1. **State the problem:** We need to find the measure of angle $C$ in a triangle where sides $a=18$, $b=22$, and $c=20$ are given. 2. **Formula used:** The Law of Cosines states:

1. **Problem statement:** An aquarium is in the form of a cuboid with external dimensions 80 cm x 80 cm x 40 cm. We need to find the area of paper required to cover the bottom, sid

1. **Problem:** The capacity of a closed cylindrical vessel of height 1 m is 15.4 L. Find the surface area of metal sheet needed to make it. 2. **Given:**

1. **State the problem:** We are given a rectangle where the length is 20% more than the breadth, and the breadth is 20 cm. We need to find the area of the rectangle. 2. **Identify

1. **Problem Statement:** We have two parallelograms ABCD and AFGD. AF and CD intersect at point E. Given that $AD = AE$ and $\angle DGF = 36^\circ$, we need to find $\angle BCD$.

1. **Problem Statement:** Find the volume of water left in a right circular cylinder of radius 60 cm and height 180 cm after placing a solid consisting of a right circular cone of

1. **State the problem:** Given a circle with radius 7 cm and a sector with central angle 150°, we want to find the length of the chord (the other side) opposite the 150° angle. 2.