📐 geometry

1. **State the problem:** We are given a triangle with angles measuring 81°, $x$°, and 40°. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** The sum of

1. **State the problem:** We are given a triangle with three angles: $(4x + 19)^\circ$, $111^\circ$, and $26^\circ$. We need to find the value of $x$. 2. **Recall the triangle angl

1. **Problem:** HIJK undergoes a dilation with a scale factor of 3 centered at the origin. Determine which statements are true. 2. **Understanding dilation:** A dilation centered a

1. **Problem statement:** Given triangle ABCD with point E on side AD such that BE is parallel to DC, and lengths AB = x, BC = 6, AC = 7, CD = 16, find the value of $x$. 2. **Key c

1. **Problem statement:** Find the perimeter of the shaded region in the first shape, which is a quarter circle inside a square with side length 20 cm. 2. **Understanding the shape

1. **State the problem:** We have two similar right triangles \(\triangle ABC \sim \triangle XYZ\). Given sides are \(AC=16\), \(CB=30\), \(AB=34\) in \(\triangle ABC\), and \(YZ=1

1. **Problem Statement:** We are given a circle with a tangent line \(\overrightarrow{GH}\) at point \(H\) and a chord \(\overline{FH}\). The angle between the tangent \(\overright

1. **Problem statement:** Find the measure of angle $\angle LNP$ formed by the intersection of chord $LM$ and secant $PN$ at point $N$ outside the circle. 2. **Relevant theorem:**

1. **Problem statement:** We are given a circle with a tangent line \( \overrightarrow{GH} \) touching the circle at point \( I \), and a chord \( IJ \). The angle between the tang

1. **Problem statement:** Given a circle with two tangent lines \( \overrightarrow{EF} \) and \( \overrightarrow{ED} \) touching the circle at points F and D respectively, and the

1. **Problem Statement:** Find the measure of the minor arc $JN$ given two secants intersecting outside the circle at point $L$ with an angle of $41^\circ$ between them.

1. **Problem statement:** We are given a circle with points R, T, S on its circumference and a tangent line PQ⃗ touching the circle at point R. 2. The angle between the tangent lin

1. **Problem statement:** We need to find the value of angle $x$ in a right triangle with a smaller isosceles triangle nested inside it. The smaller triangle has an angle of $48^\c

1. The problem is to understand and apply the Pythagorean theorem, which relates the lengths of the sides of a right triangle. 2. The Pythagorean theorem states that in a right tri

1. **Problem:** Find the height of a rectangular prism given the lateral area and the perimeter of the base. 2. **Formula used:** The lateral area $L$ of a rectangular prism is giv

1. **State the problem:** We are given three points $F(2,9)$, $G(-7,0)$, and $H(-1,1)$ that form a triangle. We need to find the perimeter of this triangle. 2. **Formula used:** Th

1. The problem asks for the equation Jasmine can use to find the area of a parallelogram cut from a 4 x 6 inch index card. 2. The formula for the area of a parallelogram is:

1. **Calculate the area of the triangle** The problem: Find the area of a triangle with base 12 cm and height 8 cm.

1. **State the problem:** We need to find the area of a sector of a circle with radius 5 cm and central angle 60°. 2. **Formula for the area of a sector:** The area $A$ of a sector

1. **State the problem:** We need to find the area of the yellow shaded region, which is the area of the square minus the area of the inscribed circle. 2. **Identify given values:*

1. **State the problem:** Given triangle ABC with BD as the bisector of segment AC, prove that triangle ABD is congruent to triangle CBD using a flowchart proof. 2. **Identify give