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Based on the diagram, the following relation holds true.
If ∠ A ≅ ∠ D and ∠ C ≅ ∠ F, then ∠ B ≅ ∠ E.
To begin, apply the Triangle Angle Sum Theorem to both triangles. m∠ A + m∠ B + m∠ C = 180 ^(∘) m∠ D + m∠ E + m∠ F = 180 ^(∘) Now, since it is given that ∠ A ≅ ∠ D and ∠ C ≅ ∠ F, substitute m∠ A for m∠ D and m∠ C for m∠ F into the second equation. m∠ A + m∠ B + m∠ C = 180 ^(∘) m∠ A + m∠ E + m∠ C = 180 ^(∘) Next, subtract the second equation from first. m∠ A + m∠ B + m∠ C &= 180 ^(∘) ^- m∠ A + m∠ E + m∠ C &= 180 ^(∘) m∠ B - m∠ E &= 0 Finally, solve the equation above for m∠ B.
m∠ B - m∠ E = 0 ⇓ m∠ B = m∠ E
With this, ∠ B ≅ ∠ E by the definition of congruence. The above proof is organized in a two-column proof table below.
Statements
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Reasons
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1. ∠ A ≅ ∠ D and ∠ C ≅ ∠ F
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1. Given
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2. m∠ A = m∠ D and m∠ C = m∠ F
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2. Definition of congruence
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3. m∠ A + m∠ B + m∠ C =180 ^(∘) and m∠ D + m∠ E + m∠ F =180^(∘)
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3. Triangle Angle Sum Theorem
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4. m∠ A + m∠ B + m∠ C =180 ^(∘) and m∠ A + m∠ E + m∠ C =180^(∘)
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4. Substitution
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5. m∠ B - m∠ E = 0
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5. Subtracting both equations
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6. m∠ B = m∠ E
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6. Addition Property of Equality
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7. ∠ B ≅ ∠ E
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7. Definition of congruence
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