📐 geometry

1. **State the problem:** We are asked to determine if two triangles are similar and to give reasons for their similarity. 2. **Recall the similarity criteria for triangles:**

1. **Problem statement:** Determine if 18-cm by 24-cm tiles can cover a floor measuring 6.48 m by 15.12 m without cutting tiles. 2. **Convert floor dimensions to centimeters:**

1. **State the problem:** We are given two intersecting lines forming vertical angles. One angle measures $81.7^\circ$ and the angle vertically opposite to it is labeled $h$. We ne

1. The problem asks to find the value of angle $r$ given that it forms a linear pair with an angle of $69.1^\circ$. 2. Recall that angles forming a linear pair are supplementary, m

1. **Stating the problem:** We need to find the volume of a three-dimensional stepped block shape with given dimensions. 2. **Understanding the shape:** The shape consists of two p

1. **State the problem:** Find the volume of a composite solid made of two rectangular prisms stacked along the length.

1. The problem asks which statement is true based on the diagram of parallelograms, rhombuses, squares, and rectangles. 2. From the description, squares are inside both rhombuses a

1. **State the problem:** We are given a parallelogram ABCD with angles expressed in terms of $x$: - $m\angle DAB = 85 - 2x$

1. **Problem Statement:** Given two parallel lines $p \parallel q$ cut by a transversal, find the value of angle $w$ adjacent to the $101^\circ$ angle on line $q$.

1. **Problem statement:** Find the measures of angles $\angle b$ and $\angle d$ given that lines $m \parallel n$ and the angles $a=60.7^\circ$, $c=119.3^\circ$, and $d=136.9^\circ$

1. **Problem statement:** Explain how the angle in a semicircle theorem enables construction of a right-angled triangle with a given hypotenuse AB. 2. **Theorem used:** The angle i

1. **State the problem:** Calculate the perimeter and area of an irregular L-shaped polygon with given side lengths: vertical left edge = 10.7 yd, top horizontal segment = 1.9 yd,

1. **State the problem:** We have a figure composed of a rectangle and a semicircle on top. The rectangle is 36 ft wide and 31 ft tall. The semicircle has its diameter equal to the

1. **State the problem:** We have a regular pentagon with side length 1 and a diagonal of length $x$. Using the symmetry and similar triangles inside the pentagon, we want to find

1. **Problem:** You are facing southeast and then turn 180° clockwise. What direction are you facing now? 2. **Formula and rules:** Turning 180° means facing the opposite direction

1. **State the problem:** We have a regular pentagon with side length 1 and a diagonal of length $x$. We want to use the symmetry and similar triangles to find a relationship betwe

1. **State the problem:** We have two identical rectangular plots of land. The first plot has vertices at approximately $(20,0)$, $(20,80)$, $(100,80)$, and $(100,0)$. The second p

1. **State the problem:** We have two identical rectangular plots of land. The first plot has vertices at (0,0), (120,0), (0,120), and (120,120). The second plot is shifted 100 yar

1. **State the problem:** We have a right triangle with three sides: the shorter leg, the longer leg, and the hypotenuse. - The shorter leg is 9 inches shorter than the longer leg.

1. **Problem Statement:** Find the area and perimeter of two trapezoids with given dimensions.

1. The problem asks to plot a map with a scale where 5 units on the map correspond to a length of 1 unit in reality. 2. This means the scale factor is $\frac{1}{5}$, so every 5 uni