To solve the problem, we need to analyze the angles formed by the lines in the diagram you described. Here are the relationships based on the provided information:
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Angle Relationships:
- Line C-F is perpendicular to line A-E at B, meaning that ∠C-B-E = 90° and ∠C-B-A = 90°.
- ∠D-B-E = 60° as given.
- Angles ∠C-B-D and ∠G-B-F are vertically opposite angles; thus, ∠C-B-D = ∠G-B-F.
- Angles ∠D-B-E and ∠A-B-G are also vertically opposite angles; thus, ∠D-B-E = ∠A-B-G.
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Calculating Angles:
- Since ∠D-B-E = 60° and angles D-B-E and A-B-G are vertically opposite, then ∠A-B-G = 60°.
- Now, because C-F is perpendicular to A-E, we know that ∠C-B-E + ∠A-B-G + ∠D-B-C = 180°. Since ∠C-B-E is 90° and ∠A-B-G is 60°, we can find ∠D-B-C: \[ 90° + 60° + m∠D-B-C = 180° \implies m∠D-B-C = 180° - 150° = 30°. \]
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Finding a:
- According to the problem, ∠D-B-C = 2a. We have just found that m∠D-B-C = 30°.
- Therefore, we can solve for a: \[ 2a = 30° \implies a = 15°. \]
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Find m∠A-B-G:
- As calculated, m∠A-B-G = 60°.
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Finding m∠C-B-D:
- From the angle relationships and as ∠C-B-D = ∠G-B-F, we also know: \[ ∠G-B-F + ∠A-B-G = 90° \rightarrow m∠G-B-F + 60° = 90° \implies m∠G-B-F = 30° \implies m∠C-B-D = 30°. \]
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Angle ∠E-B-F:
- Since ∠C-B-E + ∠E-B-F + ∠A-B-G = 180°, substituting known values gives: \[ 90° + m∠E-B-F + 60° = 180° \implies m∠E-B-F = 180° - 150° = 30°. \]
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Finding m∠D-B-E:
- From the given information, we know that m∠D-B-E = 60°, which is already provided.
Conclusion:
Based on the above analysis:
A. True: \( a = 15 \)
B. False: \( m∠A-B-G = 60^\circ \), not \( 70^\circ \).
C. False: \( a = 10 \).
D. False: \( m∠C-B-D = 30^\circ \), not \( 20^\circ \).
E. False: \( m∠E-B-F = 30^\circ \), not \( 70^\circ \).
F. False: \( m∠D-B-E = 60^\circ \), not \( 80^\circ \).
Therefore, the only correct statement is:
A: \( a = 15 \).