Calculate the Missing Angle B in Triangle ABC: Given 105° and 45°

Triangle Angle Sum with External Angle Measures

According to the data in the diagram, calculate a size of angle ABC.AAABBBCCC10545?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find angle ABC
00:05 Vertical angles are equal
00:15 The sum of angles in a triangle equals 180
00:21 Let's group terms and isolate ABC
00:48 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

According to the data in the diagram, calculate a size of angle ABC.AAABBBCCC10545?

2

Step-by-step solution

Note that from the diagram we are able to determine the value of angle BAC, since its corresponding angle is equal to 105.

Now in the triangle we are given two angles:

BAC=105 BAC=105

BCA=45 BCA=45

Since the sum of angles in a triangle is equal to 180, we can calculate angle ABC:

18010545=30 180-105-45=30

3

Final Answer

30

Key Points to Remember

Essential concepts to master this topic
  • Triangle Angle Sum: All three interior angles always equal 180 degrees
  • Technique: Subtract known angles: 180° - 105° - 45° = 30°
  • Check: Add all angles together: 105° + 45° + 30° = 180° ✓

Common Mistakes

Avoid these frequent errors
  • Confusing exterior angles with interior angles
    Don't use exterior angle measures directly as interior angles = wrong calculations! The 105° shown is actually an exterior angle, so the interior angle at A is 180° - 105° = 75°. Always identify whether angles are interior or exterior first.

Practice Quiz

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Indicates which angle is greater

FAQ

Everything you need to know about this question

How can I tell if an angle is interior or exterior?

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An interior angle is inside the triangle between two sides. An exterior angle is outside the triangle, formed when you extend one side. Look at the diagram carefully to see where the angle is positioned!

What if the given angles don't add up to 180°?

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Check if some angles might be exterior angles! Remember: exterior angle = 180° - interior angle. Convert all angles to interior angles first, then apply the triangle angle sum rule.

Why does the triangle angle sum always equal 180°?

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This is a fundamental property of triangles in flat (Euclidean) geometry. No matter what shape or size triangle you have, the three interior angles will always add up to exactly 180°.

Can I solve this without knowing the triangle angle sum rule?

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The triangle angle sum rule is essential for this type of problem. It's one of the most important geometric facts you'll use! Once you know two angles, you can always find the third.

What should I do if I get a negative angle?

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A negative result means you made an error! Check that you're using interior angles correctly. Interior angles in triangles are always between 0° and 180°, so negative values indicate a mistake.

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