Square Diagonal Properties: Classifying Triangles ABC and ACD

Q: Why are triangles ABC and ACD both right triangles?

Because ABCD is a square, angles B and D are both 90°. When you draw diagonal AC, it creates triangle ABC with a right angle at B, and triangle ACD with a right angle at D.

Q: How can I tell these triangles are isosceles?

In a square, all sides are equal! So $ AB = BC $ and $ AD = DC $. When a triangle has two equal sides, it's isosceles.

Q: Could these triangles ever be equilateral?

No! Equilateral triangles have all angles equal to 60°. But our triangles have one 90° angle, so they can't be equilateral. They're right isosceles triangles.

Q: What are the angles in these triangles?

Each triangle has angles of 90°, 45°, and 45°. The right angle comes from the square's corner, and the other two angles are equal because the triangle is isosceles.

Q: Do both triangles have the same properties?

Yes! Both ABC and ACD are congruent - they're exactly the same size and shape. They're both right isosceles triangles with the same angles and side lengths.

Triangle Classification with Square Diagonal

ABCD is a square with AC as its diagonal.

What kind of triangles are ABC and ACD?

(There may be more than one correct answer!)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 How can we define the triangles formed by the diagonal of a square?

00:03 In a square all angles are right angles

00:07 A triangle with a right angle is a right triangle

00:10 The diagonal in a square bisects the angle

00:14 A triangle with equal base angles is isosceles

00:19 The same thing happens in the second triangle,

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABCD is a square with AC as its diagonal.

What kind of triangles are ABC and ACD?

(There may be more than one correct answer!)

Step-by-step solution

Since ABCD is a square, all its angles measure 90 degrees.

Therefore, angles D and B are equal to 90°, that is, they are right angles,

Therefore, the two triangles ABC and ADC are right triangles.

In a square all sides are equal, therefore:

$AB=BC=CD=DA$

But the diagonal AC is not equal to them.

Therefore, the two previous triangles are isosceles:

$AD=DC$

$AB=BC$

Final Answer

Right triangles

Key Points to Remember

Essential concepts to master this topic

Square Properties: All angles are 90°, all sides are equal length
Diagonal Rule: Square diagonal creates two congruent right isosceles triangles
Check Types: Look for right angles (90°) and equal sides ✓

Common Mistakes

Avoid these frequent errors

Assuming all triangles in a square are equilateral
Don't think that equal sides means equilateral triangle = all angles 60°! In a square, the diagonal creates triangles with one 90° angle and two 45° angles. Always check if any angle is 90° first, then look at side lengths.

Practice Quiz

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In a right triangle, the side opposite the right angle is called....?

FAQ

Everything you need to know about this question

Why are triangles ABC and ACD both right triangles?

Because ABCD is a square, angles B and D are both 90°. When you draw diagonal AC, it creates triangle ABC with a right angle at B, and triangle ACD with a right angle at D.

How can I tell these triangles are isosceles?

In a square, all sides are equal! So $AB = BC$ and $AD = DC$ . When a triangle has two equal sides, it's isosceles.

Could these triangles ever be equilateral?

No! Equilateral triangles have all angles equal to 60°. But our triangles have one 90° angle, so they can't be equilateral. They're right isosceles triangles.

What are the angles in these triangles?

Each triangle has angles of 90°, 45°, and 45°. The right angle comes from the square's corner, and the other two angles are equal because the triangle is isosceles.

Do both triangles have the same properties?

Yes! Both ABC and ACD are congruent - they're exactly the same size and shape. They're both right isosceles triangles with the same angles and side lengths.

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