Calculate Angle BAD in an Obtuse Triangle with Perpendicular Height

Q: Why can't I use the 115° angle directly in my calculation?

The 115° angle is ∠ABC, but we need ∠ABD for triangle ADB. These are different angles! ∠ABD is supplementary to ∠ABC, so ∠ABD = 180° - 115° = 65°.

Q: How do I know which triangle to focus on?

Look for the right angle marker in the diagram. Triangle ADB has the right angle at D, so that's where you apply the triangle angle sum rule.

Q: What does 'supplementary angles' mean here?

Supplementary angles are two angles that add up to 180°. When you see angles on a straight line like ∠ABC and ∠ABD, they're supplementary!

Q: Why is angle ADB exactly 90°?

The small square symbol at point D indicates a right angle. This means AD is perpendicular to BC, making ∠ADB = 90°.

Q: Can I solve this without using supplementary angles?

No, you must use supplementary angles! The given 115° is not inside triangle ADB. You need to find its supplement (65°) to work with the right triangle.

Q: How do I remember the triangle angle sum rule?

Think: "All triangles have 180° total". So if you know two angles, subtract their sum from 180° to find the third: 180° - 90° - 65° = 25°.

Right Triangle Angles with Supplementary Relationships

ABC is an obtuse triangle.

Calculate the size of angle $∢\text{BAD}$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find the size of angle B A D.

00:11 Remember, adjacent angles add up to one hundred eighty degrees.

00:16 Substitute the given angle values to solve for angle D B A.

00:27 And that gives us the size of angle D B A.

00:32 We know A D is perpendicular, as given in the data.

00:36 Remember, the sum of all angles in a triangle is one hundred eighty degrees.

00:47 Now, let's isolate angle B A D.

01:00 And there we have our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABC is an obtuse triangle.

Calculate the size of angle $∢\text{BAD}$ .

Step-by-step solution

To calculate $\angle \text{BAD}$ , follow these steps:

Step 1: Understand from the problem and diagram that $\angle ABC = 115^\circ$ .
Step 2: Recognize that $\angle ABD$ is supplementary to $\angle ABC$ . Thus, $\angle ABD = 180^\circ - 115^\circ = 65^\circ$ .
Step 3: Use the right-triangle property in triangle $ADB$ (since $\angle ADB = 90^\circ$ ) to find $\angle BAD$ .

Now, calculate $\angle BAD$ using the sum of angles in a triangle:

In triangle $ADB$ , sum of angles is:

$\angle ADB + \angle BAD + \angle ABD = 180^\circ$

Substitute the known values:

$90^\circ + \angle BAD + 65^\circ = 180^\circ$

$\angle BAD = 180^\circ - 90^\circ - 65^\circ = 25^\circ$

Therefore, the size of angle $\angle \text{BAD}$ is $\boxed{25^\circ}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Supplementary Angles: Adjacent angles on straight line sum to 180°
Triangle Sum: All angles in triangle equal 180°, so 90° + 65° + angle = 180°
Verification: Check that $\angle BAD + \angle ABD + 90° = 180°$ gives 25° + 65° + 90° = 180° ✓

Common Mistakes

Avoid these frequent errors

Using the obtuse angle directly in calculations
Don't use ∠ABC = 115° directly in triangle ADB calculations = wrong answer! The 115° angle is outside the right triangle. Always find the supplementary angle ∠ABD = 180° - 115° = 65° first, then use triangle angle sum.

Practice Quiz

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Is DE side in one of the triangles?

FAQ

Everything you need to know about this question

Why can't I use the 115° angle directly in my calculation?

The 115° angle is ∠ABC, but we need ∠ABD for triangle ADB. These are different angles! ∠ABD is supplementary to ∠ABC, so ∠ABD = 180° - 115° = 65°.

How do I know which triangle to focus on?

Look for the right angle marker in the diagram. Triangle ADB has the right angle at D, so that's where you apply the triangle angle sum rule.

What does 'supplementary angles' mean here?

Supplementary angles are two angles that add up to 180°. When you see angles on a straight line like ∠ABC and ∠ABD, they're supplementary!

Why is angle ADB exactly 90°?

The small square symbol at point D indicates a right angle. This means AD is perpendicular to BC, making ∠ADB = 90°.

Can I solve this without using supplementary angles?

No, you must use supplementary angles! The given 115° is not inside triangle ADB. You need to find its supplement (65°) to work with the right triangle.

How do I remember the triangle angle sum rule?

Think: "All triangles have 180° total". So if you know two angles, subtract their sum from 180° to find the third: 180° - 90° - 65° = 25°.

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