Calculate Angle ABC in a Square with Diagonal: Advanced Geometry

Q: Why isn't angle ABC equal to 90° like the other square angles?

Great question! ABC refers to the angle formed by the diagonal AC, not the full corner angle of the square. The diagonal splits the 90° corner into two equal parts.

Q: How do I know the diagonal bisects the angle exactly in half?

In a square, the diagonal creates two congruent right triangles. Since these triangles are identical, the angles they create must be equal: $ 90° ÷ 2 = 45° $.

Q: Can I use this rule for any polygon with diagonals?

Only for squares and regular polygons. In irregular shapes, diagonals don't necessarily bisect angles equally. Always check the specific properties of each shape!

Square Angle Properties with Diagonal Bisection

ABCD is a square.

$∢\text{ABC}=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate ABC?

00:03 In a square all angles are right angles

00:10 The diagonal in a square bisects the angle

00:15 The angle equals half of the whole angle

00:25 Let's substitute appropriate values and solve for the angle

00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABCD is a square.

$∢\text{ABC}=\text{?}$

Step-by-step solution

Due to the fact that all angles in a square are equal to 90 degrees, and BC bisects an angle, we can calculate angle ABC accordingly:

$90:2=45$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Square Property: All interior angles in a square equal 90 degrees
Bisection Method: Diagonal AC divides angle ABC: $90° ÷ 2 = 45°$
Verification: Check that angle ABC + angle CBD = 90° total ✓

Common Mistakes

Avoid these frequent errors

Assuming the diagonal creates two 30° angles
Don't think diagonals create 30-60-90 triangles in squares = wrong 30° answer! Squares have 90° corners, not 120°. Always remember that square diagonals bisect the 90° corner angles into two equal 45° angles.

Practice Quiz

Test your knowledge with interactive questions

Look at the angles shown in the figure below.

What is their relationship?

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FAQ

Everything you need to know about this question

Why isn't angle ABC equal to 90° like the other square angles?

Great question! ABC refers to the angle formed by the diagonal AC, not the full corner angle of the square. The diagonal splits the 90° corner into two equal parts.

How do I know the diagonal bisects the angle exactly in half?

In a square, the diagonal creates two congruent right triangles. Since these triangles are identical, the angles they create must be equal: $90° ÷ 2 = 45°$ .

Would this be different in a rectangle?

Yes! In a rectangle that's not a square, the diagonal would create two different angles that still add up to 90°, but they wouldn't both be 45°.

Can I use this rule for any polygon with diagonals?

Only for squares and regular polygons. In irregular shapes, diagonals don't necessarily bisect angles equally. Always check the specific properties of each shape!

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