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Based on the characteristics of the diagram, the following relations hold true.
∠ 1 ≅ ∠ 3
∠ 2 ≅ ∠ 4
Analyzing the diagram, it can be seen that ∠ 1 and ∠ 2 form a straight angle, so these are supplementary angles. Similarly, ∠ 2 and ∠ 3 are also supplementary angles.
Therefore, by the Angle Addition Postulate, the sum of m∠ 1 and m∠ 2 is 180^(∘), and the sum of m∠ 2 and m∠ 3 is also 180^(∘). These facts can be used to express m∠ 2 in terms of m∠ 1 and in terms of m∠ 3.
Angle Addition Postulate | Isolate m∠ 2 |
---|---|
m∠ 1+m∠ 2 = 180^(∘) | m∠ 2 = 180^(∘)-m∠ 1 |
m∠ 2+m∠ 3 = 180^(∘) | m∠ 2 = 180^(∘)-m∠ 3 |
LHS-180^(∘)=RHS-180^(∘)
LHS * (- 1)=RHS* (- 1)
The previous proof can be summarized in the following two-column table.
Statements
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Reasons
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1. l_1 and l_2 lines
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1. Given
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2. ∠ 1 and ∠ 2 supplementary
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2. Definition of straight angle
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3. m∠ 1+m∠ 2=180^(∘)
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3. Definition of supplementary angles
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4. m∠ 2=180^(∘)-m∠ 1
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4. Subtraction Property of Equality
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5. ∠ 2 and ∠ 3 supplementary
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5. Definition of straight angle
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6. m∠ 2+m∠ 3=180^(∘)
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6. Definition of supplementary angles
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7. m∠ 2=180^(∘)-m∠ 3
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7. Subtraction Property of Equality
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8. 180^(∘)-m∠ 1=180^(∘)-m∠ 3
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8. Transitive Property of Equality
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9. m∠ 1=m∠ 3
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9. Subtraction and Multiplication Properties of Equality
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Consider the points A, B, C, and D on each ray that starts at the point of intersection E of the two lines.