Geometric Angle Problem: Verify 65° and 40° Triangle Relationships

Triangle Similarity with Angle-Angle Theorem

AAADDDFFFBBBCCC65°40°40°Which of the following are true?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Choose the correct answer
00:03 The triangles share the same vertex angle (a)
00:15 Equal angles according to given data (a)
00:23 The triangles are similar by A.A
00:37 Equal angles as a result of similar triangles
00:41 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

AAADDDFFFBBBCCC65°40°40°Which of the following are true?

2

Step-by-step solution

If the sum of the angles in a triangle equals 180, then angle F equals 75 and therefore angle C also equals 75.

The triangles are similar according to the angle-angle theorem

Answer D is correct

3

Final Answer

Answers (a) and (b) are correct.

Key Points to Remember

Essential concepts to master this topic
  • Angle Sum: All triangle angles sum to exactly 180 degrees
  • Technique: Find missing angles: 180° - 65° - 40° = 75°
  • Check: Verify AA similarity: two pairs of equal angles confirm triangles are similar ✓

Common Mistakes

Avoid these frequent errors
  • Assuming triangles are similar without checking all angles
    Don't just look at two given angles and assume similarity = wrong conclusions! You must calculate ALL angles first to verify which pairs are actually equal. Always find the third angle using 180° - (sum of known angles) before determining similarity.

Practice Quiz

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FAQ

Everything you need to know about this question

How do I know if triangles are similar from just angles?

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Use the AA (Angle-Angle) Theorem: if two angles in one triangle equal two angles in another triangle, the triangles are similar. You only need two pairs of equal angles!

Why does angle C equal angle F in this problem?

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Both triangles have angles of 65° and 40°. Since triangle angles sum to 180°, the third angle in each triangle must be: 180°65°40°=75° 180° - 65° - 40° = 75°

What does the ∼ symbol mean?

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The symbol means "similar to". Similar triangles have the same shape but different sizes - all corresponding angles are equal and sides are proportional.

Do I need to check all three angles for similarity?

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No! The AA Theorem says you only need two pairs of equal angles. The third pair will automatically be equal since all triangle angles sum to 180°.

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

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Check your arithmetic! If angles don't sum to 180°, you made a calculation error. All triangles must have angles totaling exactly 180° - this is a fundamental rule.

How do I write triangle similarity correctly?

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Write corresponding vertices in the same order: if angle A corresponds to angle A, B to D, and C to F, then write ABCADF \triangle ABC \sim \triangle ADF

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