📐 geometry

1. **Problem 1:** Find the missing angle measures $\angle C$, $\angle A$, and side $a$ in triangle $ABC$ where $AB=20$ cm, $BC=19$ cm, and $\angle A=65^\circ$. Also find $\angle F$

1. **State the problem:** We are given a triangle ABC with sides AB = 10 cm, BC = 11 cm, and AC = 9 cm. We need to find the measures of angles \(\angle 4\), \(\angle B\), and \(\an

1. **Stating the problem:** We have triangle ABC with sides AC = 10 cm, BC = 11 cm, and the missing side AB = 2 cm. We need to find the measures of angles \(\angle 4\), \(\angle B\

1. **Problem:** Given triangle ABC with sides AB = 10 cm, BC = 11 cm, and side AC unknown, find the type of triangle and the measures of angles A, B, and C. 2. **Formula and rules:

1. **Problem Statement:** Find the missing side lengths and then use the Law of Cosines to find the indicated angles in triangles.

1. The problem is to find the formula for the volume of a cylinder. 2. The volume $V$ of a cylinder is given by the formula:

1. **Problem:** Given triangle with angles $m\angle B = 135^\circ$, $m\angle C = 30^\circ$, side $b = 35.2$ mm, find sides $a$, $c$ and triangle type. 2. **Formula:** Use Law of Si

1. **State the problem:** We need to find the volume of a cylinder with height $h=13$ feet and radius $r=11$ feet. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** Find the surface area of a sphere with radius $r = 10$ yards. 2. **Formula:** The surface area $A$ of a sphere is given by the formula:

1. **State the problem:** Find the total surface area of a sphere with radius $r=3$ ft. 2. **Formula:** The total surface area $A$ of a sphere is given by

1. **Problem statement:** We have a right triangle with legs of lengths 10 inches and 17 inches. We need to find the length of the hypotenuse. 2. **Formula used:** In a right trian

1. **State the problem:** We need to find the volume of a cone with height $h=12$ cm and radius $r=9$ cm. 2. **Formula:** The volume $V$ of a cone is given by the formula:

1. The problem asks which congruence rule explains why the two triangles GHJ and KMN are congruent. 2. The given information shows that in both triangles, two sides and the include

1. **State the problem:** We have two triangles, \(\triangle VWS\) and \(\triangle UST\), and we want to determine which congruence rule explains why these triangles are congruent.

1. The problem asks for the expression for the perimeter of a parallelogram with side lengths $s$ and $w$. 2. The perimeter $P$ of any parallelogram is the sum of the lengths of al

1. **State the problem:** We have two similar quadrilaterals KLMN and OPRS. We know the lengths of sides NK = 17, ML = 27, SO = 55, and we need to find the length of side PR. 2. **

1. **Problem Statement:** We have two similar triangles, JKL and MNO. We know sides LK = 5, KJ = 2 in triangle JKL, and side ON = 16 in triangle MNO. We need to find the length of

1. The problem asks us to find the new coordinates of the vertices of a square after a translation 3 units to the right and 8 units down. 2. The original vertices are given as:

1. **Problem Statement:** Find the azimuths of the lines AB, BC, CD, and DA given the angles at vertices A, B, C, and D. 2. **Understanding Azimuths:** Azimuth is the angle measure

1. **Problem Statement:** Find the azimuths of lines AB, BC, CD, and DA given angles between these lines and cardinal directions at points A, B, C, and D. 2. **Understanding Azimut

1. **State the problem:** We have a room with dimensions 5.8 m (length), 3.9 m (width), and 2.43 m (height). There are windows and doors with given dimensions. We need to find the