Question

A square \(A B C D\) has an equilateral triangle drawn on the side \(A B\) (interior of the square). The triangle has vertex at \(G\). What is the measure of the angle CGB? (a) \(60^{\circ}\) (b) \(80^{\circ}\) (c) \(75^{\circ}\) (d) \(90^{\circ}\)

Short answer

Answer: The measure of angle CGB is 90 degrees.

Step-by-step solution

Calculate Angle BGC

Since the equilateral triangle is drawn on side AB, it means that AB = BG = GA. In the square ABCD, all angles are equal to 90 degrees. Thus, angle ABC = 90 degrees. Since triangle ABG is equilateral, angle ABG = 60 degrees. Now, we can find angle BGC by subtracting angle ABG from angle ABC: Angle BGC = Angle ABC - Angle ABG = 90 - 60 = 30 degrees

Calculate Angle BCG

In the square ABCD, angle BCD = angle ABC = 90 degrees. Since the equilateral triangle is drawn on side AB, it means that AB = BG = GA. Therefore, triangles BCG and ABG are congruent which means angle ABG = angle BCG. Thus, angle BCG = 60 degrees.

Calculate Angle CGB

Now, we can find the angle CGB by using the angle sum property of a triangle. In triangle BCG, the sum of angles BGC, BCG, and CGB should be equal to 180 degrees. Angle CGB = 180 - (Angle BGC + Angle BCG) = 180 - (30 + 60) = 180 - 90 = 90 degrees So, the measure of angle CGB is 90°, which corresponds to option (d).

Key concepts

Properties of Squares

A square is a special type of quadrilateral where all four sides are equal in length, and all four angles are right angles, measuring exactly 90 degrees each. This means that in any square, each internal angle contributes to the square's distinct shape and properties. Since all sides and angles are congruent, squares are also considered regular quadrilaterals.
Understanding these properties is important because it allows us to deduce other information. For example, the diagonals of a square bisect each other at right angles (90 degrees) and are of equal length. This symmetry in geometry makes squares particularly interesting while solving problems that involve congruence and angle measurements. In our aforementioned problem, knowing that angle ABC equals 90 degrees in square ABCD helps determine the subsequent steps in finding the unknown angles.

Equilateral Triangles

An equilateral triangle is one in which all three sides are of equal length, which also means that all three internal angles are equal. Each angle in an equilateral triangle measures 60 degrees. This property of having equal sides and angles categorizes equilateral triangles as regular polygons.
When considering an equilateral triangle in a problem, like in our exercise, recognizing that every angle within such a triangle is 60 degrees simplifies the process of finding unknown angles. In our problem, triangle ABG is equilateral. This directly tells us that angle ABG measures 60 degrees and assists us in calculating angles related to it, like angles BGC and BCG.

Angle Properties

Angles are fundamental in shaping how we understand geometry. In the context of a triangle, the angle sum property is a vital concept. The sum of the internal angles in any triangle is always 180 degrees. This principle is essential for calculating unknown angles when some angles are already known or assumed, as in our problem.
Additionally, the recognition of right angles (90 degrees) is crucial, particularly in squares. The problem-solving exercise mentioned earlier combines the angle properties of both squares and triangles. By subtracting known angles using properties like the angle sum property and understanding equilateral triangle angles, we can solve for unknown ones, such as finding angle CGB in triangle BCG.

Triangle Congruence

Triangle congruence occurs when two triangles are identical in terms of side lengths and angle measures. Several criteria can establish triangle congruence, such as ASA (Angle-Side-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side).
In our problem, the triangles BCG and ABG are congruent by the SSS criterion, since AB = BG = GA. Congruence implies that corresponding angles in congruent triangles are equal. That's why angle BCG equals angle ABG, which allows us to solve for unknowns by leveraging their equalities. Understanding triangle congruence is crucial in geometry, permitting deductions of unknown lengths and angles through the relationships of congruent figures.