Square Geometry: Analyzing Perpendicular Diagonals in a Given Square

Video Solution

Solution Steps

00:00 Are the diagonals in a square perpendicular?

00:04 The diagonals are angle bisectors in a square, therefore the angle is 45

00:08 A triangle with equal base angles is isosceles

00:13 The sum of angles in a triangle equals 180

00:17 Subtract the known angles from 180 to find the angle

00:24 Adjacent angles sum to 180, therefore they're all 90 as well

00:27 Because the angle is right, the lines are perpendicular

00:31 And this is the solution to the question

Step-by-Step Solution

Let's remember that perpendicular lines are lines that intersect at a 90-degree angle.

According to the properties of the square, all angles measure 90 degrees and the diagonals are bisectors.

We will focus on the upper triangle formed by the diagonals intersecting each other.

Since all angles measure 90 degrees, the diagonals form two 45-degree angles.

We will draw this as follows:

Calculate the missing third angle in the triangle, marked with a question mark, as follows.

The sum of the angles of a triangle equals 180 degrees, so the formula to find the third angle is:

$180-45-45=$

$180-45=135$

$135-45=90$

Since the third angle equals 90 degrees, its complementary angle also equals 90 degrees:

Since the diagonals form a 90-degree angle between them, they are indeed perpendicular and perpendicular to each other.