Triangle Similarity Proof: Comparing 40° and 60° Angles in Two Triangles

Triangle Similarity with Angle-Angle Theorem

Angle B is equal to 40°

Angle C is equal to 60°

Angle E is equal to 40°

Angle F is equal to 60°

Are the triangles similar?

AAABBBCCCDDDEEEFFF

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Are the triangles similar?
00:03 Angles are equal according to given (A)
00:09 Angles are equal according to given (A2)
00:14 The triangles are similar according to A.A
00:26 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

Angle B is equal to 40°

Angle C is equal to 60°

Angle E is equal to 40°

Angle F is equal to 60°

Are the triangles similar?

AAABBBCCCDDDEEEFFF

2

Step-by-step solution

Given that the data shows that there are two pairs with equal angles:

B=E=40 B=E=40

C=F=60 C=F=60

The triangles are similar according to the angle-angle theorem, therefore triangle ABC is similar to triangle DEF.

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic
  • AA Theorem: Two triangles are similar if two angles match
  • Technique: Compare corresponding angles: B=E=40°, C=F=60°
  • Check: Third angles also match: A=D=80° (since 180°-40°-60°=80°) ✓

Common Mistakes

Avoid these frequent errors
  • Checking only one pair of equal angles
    Don't conclude similarity from just one angle pair like B=E=40° = triangles are similar! One matching angle isn't enough proof. Always find at least two pairs of equal corresponding angles to use the AA Theorem.

Practice Quiz

Test your knowledge with interactive questions

Angle B is equal to 60°

Angle C is equal to 55°

Angle E is equal to 60°

Angle F is equal to 50°

Are these triangles similar?

AAABBBCCCDDDEEEFFF

FAQ

Everything you need to know about this question

Why don't I need to check all three angles?

+

The AA (Angle-Angle) Theorem says that if two angles match, the third angle must also match! Since angles in a triangle always add up to 180°, once you know two angles are equal, the third is automatically equal too.

How do I know which angles correspond to each other?

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Look at the position and labeling of the triangles. In this problem, angle B corresponds to angle E (both are 40°), and angle C corresponds to angle F (both are 60°). The remaining angles A and D must correspond.

What if the triangles are oriented differently?

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Triangle orientation doesn't matter! Focus on the angle measurements, not how the triangles are drawn. Similar triangles can be flipped, rotated, or resized while maintaining the same angles.

Do I need to know the side lengths?

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No! For the AA Theorem, you only need angle measurements. Side lengths would be needed for other similarity tests like SSS or SAS, but not for AA.

What does 'similar' actually mean?

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Similar triangles have the same shape but different sizes. All corresponding angles are equal, and all corresponding sides are proportional (same ratio).

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