Question

An equilateral triangle CDE is constructed on a side \(\mathrm{CD}\) of square \(\mathrm{ABCD}\). The measure of \(\angle \mathrm{AEB}\) can be (1) \(150^{\circ}\) (2) \(45^{\circ}\) (3) \(30^{\circ}\) (4) \(20^{\circ}\)

Short answer

Answer: (3) \(30^{\circ}\)

Step-by-step solution

Analyze the square and equilateral triangle

Since ABCD is a square, all angles of the square are 90°, and all sides are equal. In the equilateral triangle CDE, all angles are 60°, and all sides are equal. The side CD is shared by both the square and the equilateral triangle.

Calculate the angles formed between square and triangle

Now, let's look at the angles in question. There are two important angles we need to find: angle ADC and angle CDE. Since ADC is an angle of the square, it is equal to 90°. Angle CDE is an angle of the equilateral triangle, so it is equal to 60°.

Find angle AEB using angles ADC and CDE

We know that angle AEB is supplementary to both angles ADC and CDE, that is, angle AEB = 180° - (angle ADC + angle CDE). By substituting the angle measures we found, we get angle AEB = 180° - (90° + 60°) = 180° - 150° = 30°.

Choose the correct answer

Based on our calculation, the measure of angle AEB is 30°. Therefore, the correct answer is: (3) \(30^{\circ}\)

Key concepts

Equilateral Triangle Properties

When it comes to understanding the nature of an equilateral triangle, certain fundamental properties set it apart from other triangle types. An equilateral triangle is, by definition, a polygon with three equal sides and, consequently, three equal angles. This uniformity in its construction leads to each angle measuring exactly 60 degrees, a fact derived from the knowledge that the sum of interior angles in any triangle amounts to 180 degrees.

Given its symmetry, an equilateral triangle also possesses other interesting characteristics. For instance, it is inherently also equiangular, meaning all angles are congruent. This symmetrical nature makes calculations involving equilateral triangles more straightforward. In addition, the triangle's altitude, which doubles as a median and angle bisector, creates two 30-60-90 special right triangles with well-known ratio properties. The altitude divides the base into two equal segments and intersects it at a 90-degree angle. This property often comes into play when solving various geometric problems involving equilateral triangles, such as finding the height or the area.

Angle Measurement in Shapes

The understanding of angle measurement is crucial when dealing with various geometric shapes, including triangles, squares, and other polygons. In essence, the measurement of an angle determines the degree of turn between two lines that intersect at a vertex. For any polygon, the sum of the interior angles can be calculated using the formula \( (n-2) \times 180^\text{{o}} \) where \( n \) is the number of sides.

Applying this knowledge to a shape like a square, we know that a square has four right angles, each measuring 90 degrees, leading to a total of 360 degrees for the sum of its interior angles. In triangle geometry, including equilateral, isosceles, and scalene, the sum is always 180 degrees. Knowing how to measure and calculate angles is intrinsic to solving complex problems, especially when shapes intersect or share sides, leading to supplemental (sums to 180 degrees) and complementary (sums to 90 degrees) angles, as well as the application of various angle theorems and properties in geometry.

Square Properties

Squares represent a special category of quadrilaterals with distinctive and quite useful properties. All four sides of a square are equal in length, and every interior angle is a right angle, that is, each measures 90 degrees. Due to this, the square is both a rectangle (opposite sides are equal and all angles are right angles) and a rhombus (all sides are equal in length), hence inheriting properties from both.

The diagonals of a square are equal in length and bisect each other at right angles, which is a defining characteristic not found in all quadrilaterals. These diagonals also bisect the square's angles, creating four congruent right triangles within the square. The diagonal length can be found using the Pythagorean theorem, as it forms a right triangle with two sides of the square. Additionally, the square has rotational symmetry of order 4, meaning it looks the same at quarter-turn rotations, and it has four lines of symmetry. These attributes make the square a highly regular shape and frequently advantageous in design and architecture.