Find Angle ABC in a Rectangle with 30-Degree Reference Angle

Rectangle Angle Properties with Diagonal Relationships

ABCD is a rectangle.

ABC=? ∢\text{ABC}=?

AAABBBDDDCCC30

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Calculate angle ABC
00:04 The sum of angles in a triangle equals 180
00:20 Let's collect terms and isolate ABC
00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

ABCD is a rectangle.

ABC=? ∢\text{ABC}=?

AAABBBDDDCCC30

2

Step-by-step solution

Since we know that ABCD is a rectangle, we know that AC is parallel to BD.

Therefore, angles ACB and CBD are equal (30 degrees).

In a rectangle, we know that all angles are equal to 90 degrees, meaning angle ABD is equal to 90.

Now we can calculate angle ABC as follows:

9030=60 90-30=60

3

Final Answer

60

Key Points to Remember

Essential concepts to master this topic
  • Rectangle Property: All interior angles are 90 degrees by definition
  • Technique: Use parallel lines and alternate interior angles: ACB=CBD=30° ∠ACB = ∠CBD = 30°
  • Check: Verify that ABC=90°30°=60° ∠ABC = 90° - 30° = 60°

Common Mistakes

Avoid these frequent errors
  • Confusing which angle equals 30 degrees
    Don't assume ∠ABC = 30° directly from the diagram = wrong answer! The 30° shown is ∠ACB, not ∠ABC. Always identify which specific angle is marked, then use rectangle properties to find the requested angle.

Practice Quiz

Test your knowledge with interactive questions

Look at the angles shown in the figure below.

What is their relationship?

\( \)αβ

FAQ

Everything you need to know about this question

Why is angle ACB equal to 30 degrees if it's not directly labeled?

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The diagonal AC creates alternate interior angles with the parallel sides of the rectangle. Since ABCD is a rectangle, opposite sides are parallel, making ACB=CBD=30° ∠ACB = ∠CBD = 30° .

How do I know that angle ABD is 90 degrees?

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In any rectangle, all four corner angles are 90°. So angle ABD (which is the same as angle ABC plus angle CBD) must equal 90°.

What's the difference between angle ABC and angle ABD?

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Angle ABD is the full corner angle of the rectangle (90°). Angle ABC is just part of that corner, from side AB to the diagonal AC.

Can I solve this without using the parallel line property?

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Yes! You can also use the fact that in rectangle ABCD, ABC+ACB=90° ∠ABC + ∠ACB = 90° (complementary angles in a right triangle). So ABC=90°30°=60° ∠ABC = 90° - 30° = 60° .

Why isn't the answer 30 degrees like the marked angle?

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The marked 30° is angle ACB, not angle ABC! Always read the diagram carefully to identify which angle is actually being asked for in the question.

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