Calculate Angle BAD in an Isosceles Triangle with Median AD

Q: Why does the median also bisect the angle in isosceles triangles?

In isosceles triangles with AB = AC, the median from vertex A to base BC creates perfect symmetry. This single line acts as median, altitude, perpendicular bisector, AND angle bisector all at once!

Q: How do I know which angle to divide by 2?

Always divide the vertex angle - that's the angle between the two equal sides. In this problem, $ \angle BAC = 70° $ is at vertex A between equal sides AB and AC.

Q: What if the triangle wasn't isosceles?

If the triangle wasn't isosceles, the median would not bisect the angle! This special property only works when two sides are equal.

Q: Can I use this property for any isosceles triangle?

Yes! This works for any isosceles triangle. The median from the vertex angle to the base will always bisect that vertex angle, creating two equal angles.

Q: How can I verify my answer is correct?

Check that $ \angle BAD + \angle CAD = \angle BAC $. In this case: 35° + 35° = 70° ✓ Also verify both angles are equal since AD bisects the vertex angle.

Angle Bisector Property with Isosceles Triangles

ABC is an isosceles triangle.

AB = AC

AD is the median.

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

00:07 Let's find the angle B A D.

00:10 This triangle is isosceles, as the information states.

00:15 We know A D is a median from the given details.

00:19 In an isosceles triangle, the median also splits the angle exactly in half.

00:25 So, angle B A D is half of angle B A C.

00:32 And that's how we solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABC is an isosceles triangle.

AB = AC

AD is the median.

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

Step-by-step solution

To determine $\angle \text{BAD}$ in triangle $ABC$ , follow these steps:

The problem states that $ABC$ is an isosceles triangle where $AB = AC$ .
$AD$ is given as the median, which also acts as the angle bisector because $ABC$ is isosceles.
Since $\angle BAC = 70^\circ$ , and $AD$ bisects $\angle BAC$ , $\angle BAD = \angle CAD = \frac{\angle BAC}{2}$ .
Therefore, $\angle BAD = \frac{70^\circ}{2} = 35^\circ$ .

Thus, the measure of $\angle \text{BAD}$ is $\mathbf{35^\circ}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Isosceles Property: In isosceles triangles, the median to base is angle bisector
Technique: Divide vertex angle by 2: $\angle BAD = \frac{70°}{2} = 35°$
Check: Both halves equal: $\angle BAD = \angle CAD = 35°$ and sum to 70° ✓

Common Mistakes

Avoid these frequent errors

Using the median property without recognizing angle bisector
Don't just think AD is a median that only divides the base = missing the angle property! In isosceles triangles, this special line has multiple roles. Always remember that median to the base also bisects the vertex angle, so divide by 2.

Practice Quiz

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Is DE side in one of the triangles?

FAQ

Everything you need to know about this question

Why does the median also bisect the angle in isosceles triangles?

In isosceles triangles with AB = AC, the median from vertex A to base BC creates perfect symmetry. This single line acts as median, altitude, perpendicular bisector, AND angle bisector all at once!

How do I know which angle to divide by 2?

Always divide the vertex angle - that's the angle between the two equal sides. In this problem, $\angle BAC = 70°$ is at vertex A between equal sides AB and AC.

What if the triangle wasn't isosceles?

If the triangle wasn't isosceles, the median would not bisect the angle! This special property only works when two sides are equal.

Can I use this property for any isosceles triangle?

Yes! This works for any isosceles triangle. The median from the vertex angle to the base will always bisect that vertex angle, creating two equal angles.

How can I verify my answer is correct?

Check that $\angle BAD + \angle CAD = \angle BAC$ . In this case: 35° + 35° = 70° ✓ Also verify both angles are equal since AD bisects the vertex angle.

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