Isosceles Triangle ABC: Calculate and Classify the 45° Vertex Angle

Q: How do I know which angles are equal in an isosceles triangle?

In an isosceles triangle, the base angles (angles opposite the equal sides) are always equal. Since AB = BC, angles BAC and BCA are the base angles, so they're both 45°.

Q: Why isn't angle ABC also 45°?

Angle ABC is the vertex angle between the two equal sides. It's different from the base angles and must be calculated using the triangle angle sum: $ 180° - 45° - 45° = 90° $.

Q: What makes this triangle special?

This is both an isosceles triangle (two equal sides) and a right triangle (90° angle). It's actually a 45-45-90 triangle, which is very common in geometry!

Q: How can I remember angle classifications?

Acute: Less than 90°Right: Exactly 90°Obtuse: Greater than 90° but less than 180°Since our angle ABC = 90°, it's a right angle.

Isosceles Triangles with Given Vertex Angles

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find angle ABC

00:04 The triangle is isosceles according to the given information

00:12 In an isosceles triangle, the base angles are equal

00:29 The sum of angles in a triangle equals 180

00:32 Therefore, we subtract the known angles from 180 to find ABC

00:41 This is angle ABC, an angle equal to 90 is a right angle

00:46 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

Step-by-step solution

Given that it is an isosceles triangle: $AB=BC$

It is possible to argue that: $BAC=ACB=45$

Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:

$180-45-45=90$

Since the angle ABC measures 90 degrees, it is a right triangle.

Final Answer

90°, right angle.

Key Points to Remember

Essential concepts to master this topic

Isosceles Rule: Base angles are always equal in isosceles triangles
Technique: Use angle sum 180° - 45° - 45° = 90°
Check: Verify all three angles sum to exactly 180° ✓

Common Mistakes

Avoid these frequent errors

Assuming the marked angle is the vertex angle at B
Don't assume angle ABC is 45° just because you see 45° marked! The diagram shows angle BAC = 45°, not angle ABC. This leads to wrong calculations and missing the right angle. Always identify which specific angle is given before applying isosceles properties.

Practice Quiz

Test your knowledge with interactive questions

Look at the angles shown in the figure below.

What is their relationship?

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FAQ

Everything you need to know about this question

How do I know which angles are equal in an isosceles triangle?

In an isosceles triangle, the base angles (angles opposite the equal sides) are always equal. Since AB = BC, angles BAC and BCA are the base angles, so they're both 45°.

Why isn't angle ABC also 45°?

Angle ABC is the vertex angle between the two equal sides. It's different from the base angles and must be calculated using the triangle angle sum: $180° - 45° - 45° = 90°$ .

What makes this triangle special?

This is both an isosceles triangle (two equal sides) and a right triangle (90° angle). It's actually a 45-45-90 triangle, which is very common in geometry!

How can I remember angle classifications?

Acute: Less than 90°
Right: Exactly 90°
Obtuse: Greater than 90° but less than 180°

Since our angle ABC = 90°, it's a right angle.

What if I calculated angle ABC as 45°?

Check your work! If ABC were 45°, then all three angles would be 45° + 45° + 45° = 135°, which is not 180°. Remember: triangle angles must always sum to exactly 180°.

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