📐 geometry

1. **Problem 1:** Given triangle $ABC$ with point $H$ on $AB$ such that $AH \perp AB$ and $CH \perp AB$. We want to understand the properties of $\triangle ACB$ and the point $H$.

1. The problem involves finding the measure of angle $b$ in a triangle where the external angles at $a$ and $c$ are given as $65^\circ$ and $132^\circ$ respectively. 2. Recall that

1. **State the problem:** We have two intersecting diagonal lines crossing a horizontal line, forming angles labeled as follows: the upper left and right angles next to the horizon

1. The problem involves two intersecting lines forming vertical angles with labels 3x, x, x, and y. 2. Vertical angles are equal, so the angle labeled 3x is equal to the angle labe

1. The problem involves a right triangle with angles labeled $130^\circ$, $y$, and $x$, and a right angle of $90^\circ$. We need to find the values of $x$ and $y$. 2. Recall that t

1. **State the problem:** We have two intersecting lines forming vertical angles. The angles around the intersection are labeled as $x$, $x$, $3x$, and $y$. We need to find the rel

1. **State the problem:** We need to find the area of quadrilateral EFGH with given side lengths: EH = 8 cm (vertical), HG = 11 cm (horizontal), GF has a vertical height of 4 cm, a

1. **State the problem:** We need to find the bearing from point F to point G on a square grid map. 2. **Identify coordinates:** Assume each square represents 1 km. Let the coordin

1. Problem 1: Find the bearing from F to G on the square grid map. 2. To find the bearing, first determine the horizontal and vertical distances between points F and G using the sc

1. **Problem 1: Find cos \(\theta\) and hypotenuse \(d\) in the right triangle.** Given base segments 6.2 cm and 3.8 cm, total base = \(6.2 + 3.8 = 10\) cm.

1. **Problem 1:** The ratio of three angles in a triangle is 1:2:3. Find the measure of each angle in degrees. 2. Let the three angles be $x$, $2x$, and $3x$.

1. **Problem 1: Area of quadrilateral EFGH** The quadrilateral has sides EH = 10 cm (vertical), HG = 11 cm (horizontal), and GF = 5 cm (right vertical segment tilted). Right angles

1. The problem asks to find the triangle congruent to \(\triangle ABC\) with vertices in the correct order. 2. Given triangles:

1. **State the problem:** We are given a triangle ABC with angles at B and C measuring 70° and 30° respectively. We need to find the size of angle AC, which is the angle at vertex

1. **Problem 1: Prove every segment has a unique midpoint.** A midpoint $M$ of segment $AB$ is a point on $AB$ such that $AM = MB$.

1. The problem asks which congruence rule explains why the two triangles sharing side UW are congruent. 2. The triangles share side UW, so side UW is congruent to itself by the Ref

1. **State the problem:** We need to determine which congruence rule explains why triangles UVT and VWT are congruent. 2. **Analyze the given information:**

1. **State the problem:** We need to find the total area of a shape composed of two semicircles side-by-side, each with a diameter of 2 cm. 2. **Identify the radius:** The radius $

1. The problem asks to calculate the area of the shape formed by two shaded quarter circles, each with radius 2 cm. 2. Each quarter circle has an area of \( \frac{1}{4} \pi r^2 \).

1. The problem asks to find the area of a semicircle with a diameter of 21 inches. 2. Recall the formula for the area of a full circle: $$A = \pi r^2$$ where $r$ is the radius.

1. The problem asks to calculate the area of a shape formed by two quarter circles, each with radius 2 cm, combined to form a figure-eight or hourglass shape. 2. Each quarter circl