Right Triangle Angle Calculation: Finding ∠CBD with Given 30° Angle

Altitude Properties with Angle Relationships

ABC Right triangle

If BAC=30 ∢\text{BAC}=30

Calculate the size of CBD ∢\text{CBD} AAABBBCCCDDD30α

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Calculate CBD
00:06 The sum of angles in a triangle equals 180
00:16 Let's group terms and isolate C
00:38 This is angle C
00:50 Now let's use exactly the same method in triangle DBC to find A
01:00 Let's group terms and isolate A
01:11 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

ABC Right triangle

If BAC=30 ∢\text{BAC}=30

Calculate the size of CBD ∢\text{CBD} AAABBBCCCDDD30α

2

Step-by-step solution

Note that angle BDA equals 90 degrees, therefore we can deduce that angle BDC also equals 90 degrees.

Let's look at triangle ABC and calculate angle C, since angles A and B are given to us:

1809030=60 180-90-30=60

Now let's focus on triangle BDC and calculate angle alpha, since we have calculated the other two angles.

1809060=30 180-90-60=30

3

Final Answer

30

Key Points to Remember

Essential concepts to master this topic
  • Right Triangle Sum: All angles in triangle always equal 180 degrees
  • Altitude Rule: BD perpendicular to AC creates two 90° angles at D
  • Verification: Check triangle BDC: 90°+60°+30°=180° 90° + 60° + 30° = 180°

Common Mistakes

Avoid these frequent errors
  • Forgetting that altitude creates right angles
    Don't assume angle BDC is just any angle = wrong calculations! The altitude BD is perpendicular to AC, so angle BDC must be 90°. Always recognize that altitudes create right angles at the point of intersection.

Practice Quiz

Test your knowledge with interactive questions

Indicates which angle is greater

FAQ

Everything you need to know about this question

Why is angle BDA equal to 90 degrees?

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Point D lies on the altitude from B to side AC. By definition, an altitude is perpendicular to the opposite side, which means it forms 90° angles with that side.

How do I know angle BDC is also 90 degrees?

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Since BD is perpendicular to AC at point D, it creates two right angles: angle BDA and angle BDC. Both angles are on a straight line AC, so each must be 90°.

Why do I need to find angle ACB first?

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To solve triangle BDC, you need all three angles. You know angle BDC = 90°, so finding angle ACB (which equals 60°) gives you two angles to find the third.

Can I solve this without using triangle ABC?

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No! You need angle ACB from triangle ABC to complete the calculation in triangle BDC. The triangles are connected - they share angle C.

What if I calculated angle ACB incorrectly?

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Double-check your work: In right triangle ABC, A+B+C=180° ∠A + ∠B + ∠C = 180° . So 30°+90°+C=180° 30° + 90° + ∠C = 180° , giving ∠C = 60°.

How can I verify my final answer of 30°?

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Check triangle BDC: CBD+BDC+BCD=30°+90°+60°=180° ∠CBD + ∠BDC + ∠BCD = 30° + 90° + 60° = 180° ✓. The angles sum correctly!

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