Square Geometry: Analyzing Perpendicular Diagonals in a Given Square

Q: How do I know if two lines are perpendicular?

Two lines are perpendicular when they meet at exactly $ 90° $. Look for the small square symbol at the intersection or calculate the angle mathematically.

Q: Why do the corner angles of a square matter for diagonals?

Each corner angle is $ 90° $, and diagonals bisect these corners, creating two $ 45° $ angles. This is key to finding the diagonal intersection angle!

Q: What's the triangle angle sum rule?

In any triangle, all three angles add up to exactly $ 180° $. So if you know two angles are $ 45° $ each, the third must be $ 180° - 45° - 45° = 90° $.

Q: Do all squares have perpendicular diagonals?

Yes, always! It's a fundamental property of squares. The diagonals not only intersect at $ 90° $, but they also bisect each other and have equal lengths.

Q: Can I use this method for other shapes?

This triangle angle analysis works for any quadrilateral! Just remember that only squares, rectangles, and rhombuses have special diagonal properties worth memorizing.

Square Diagonals with Perpendicular Angle Analysis

Are the diagonals of the given square perpendicular?

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Step-by-step video solution

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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 written solution

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Understand the problem

Are the diagonals of the given square perpendicular?

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.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Definition: Perpendicular lines intersect at exactly 90-degree angles
Technique: Find diagonal intersection angle using triangle angle sum = 180°
Check: All four angles at diagonal intersection equal 90° ✓

Common Mistakes

Avoid these frequent errors

Assuming diagonals are perpendicular without proving it
Don't just guess that square diagonals are perpendicular = wrong reasoning! Visual appearance can be deceiving without mathematical proof. Always calculate the actual angle using triangle properties and angle relationships.

Practice Quiz

Test your knowledge with interactive questions

Determine which lines are parallel to one another?

FAQ

Everything you need to know about this question

How do I know if two lines are perpendicular?

Two lines are perpendicular when they meet at exactly $90°$ . Look for the small square symbol at the intersection or calculate the angle mathematically.

Why do the corner angles of a square matter for diagonals?

Each corner angle is $90°$ , and diagonals bisect these corners, creating two $45°$ angles. This is key to finding the diagonal intersection angle!

What's the triangle angle sum rule?

In any triangle, all three angles add up to exactly $180°$ . So if you know two angles are $45°$ each, the third must be $180° - 45° - 45° = 90°$ .

Do all squares have perpendicular diagonals?

Yes, always! It's a fundamental property of squares. The diagonals not only intersect at $90°$ , but they also bisect each other and have equal lengths.

Can I use this method for other shapes?

This triangle angle analysis works for any quadrilateral! Just remember that only squares, rectangles, and rhombuses have special diagonal properties worth memorizing.

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