Isosceles Triangle: Calculate ∠BAD Using Height Properties

Q: Why does the height from A bisect angle BAC?

In an isosceles triangle, the height from the vertex to the base creates two congruent right triangles. This means the height must split the vertex angle exactly in half!

Q: How do I know which angle is the vertex angle?

The vertex angle is at the point where the two equal sides meet. Since AB = AC, the vertex angle is $ ∠BAC = 70° $.

Q: What if the triangle wasn't isosceles?

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

Q: How can I verify my answer is correct?

Check that $ ∠BAD + ∠CAD = 70° $. Since both should be 35°, we get $ 35° + 35° = 70° $ ✓

Angle Bisector with Height Properties

ABC isosceles triangle (AB=AC).

Given AD height.

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's determine angle B, A, D.

00:13 We have an isosceles triangle and a perpendicular line according to the given data.

00:20 In an isosceles triangle, the perpendicular line is also the median.

00:27 It's also an angle bisector. Nice, right?

00:34 This means angle B, A, D is half of angle A.

00:41 Now, substitute the value you have for angle A and solve for angle B, A, D.

00:46 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABC isosceles triangle (AB=AC).

Given AD height.

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

Step-by-step solution

In $\triangle ABC$ , since $AB = AC$ , the triangle is isosceles, and the base angles $\angle ABC$ and $\angle ACB$ are equal. Given that $\angle BAC = 70^{\circ}$ , the sum of the angles in a triangle is $180^{\circ}$ .
Therefore, the two base angles together are $180^{\circ} - 70^{\circ} = 110^{\circ}$ .
Since these angles are equal, each is $\frac{110^{\circ}}{2} = 55^{\circ}$ .

Since $AD$ is the height from $A$ to $BC$ , it bisects $\angle BAC$ .
Thus, $\angle BAD = \frac{\angle BAC}{2} = \frac{70^{\circ}}{2} = 35^{\circ}$ .

Therefore, the size of $\angle BAD$ is $\textbf{35 degrees}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Isosceles Property: Height from vertex to base bisects the vertex angle
Angle Bisection: $∠BAD = \frac{70°}{2} = 35°$
Check: Both $∠BAD$ and $∠CAD$ should equal 35° ✓

Common Mistakes

Avoid these frequent errors

Using base angles instead of vertex angle for bisection
Don't use the 55° base angles to find ∠BAD = gets 27.5°! The height bisects the vertex angle at A, not the base angles. Always divide the vertex angle (70°) by 2 to get 35°.

Practice Quiz

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Can a triangle have a right angle?

FAQ

Everything you need to know about this question

Why does the height from A bisect angle BAC?

In an isosceles triangle, the height from the vertex to the base creates two congruent right triangles. This means the height must split the vertex angle exactly in half!

How do I know which angle is the vertex angle?

The vertex angle is at the point where the two equal sides meet. Since AB = AC, the vertex angle is $∠BAC = 70°$ .

Do I need to find the base angles first?

No! You can go directly from the vertex angle to your answer. Since the height bisects $∠BAC$ , just divide: $∠BAD = \frac{70°}{2} = 35°$ .

What if the triangle wasn't isosceles?

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

How can I verify my answer is correct?

Check that $∠BAD + ∠CAD = 70°$ . Since both should be 35°, we get $35° + 35° = 70°$ ✓

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