4. Proving Angle Relationships
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Use the Supplement Theorem and the Congruent Supplements Theorem.
Statements
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Reasons
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1. ∠ 1 and ∠ 3, ∠ 2 and ∠ 4 are vertical angles
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1. Given
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2. m∠ 1+m∠ 2=180
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2. Supplement Theorem
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3. m∠ 2+m∠ 3=180
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3. Supplement Theorem
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4. ∠ 1≅ ∠ 3
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4. Congruent Supplements Theorem
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5. m∠ 2+m∠ 3=180
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5. Supplement Theorem
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6. m∠ 3+m∠ 4=180
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6. Supplement Theorem
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7. ∠ 2≅ ∠ 4
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7. Congruent Supplements Theorem
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Let's begin by remembering what Theorem 2.8 states.
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If two angles are vertical angles then they are congruent. |
We can make a graph that illustrates this situation.
In the graph above both ∠ 1 and ∠ 3, as well as ∠ 2 and ∠ 4, are vertical angles. Our job is to prove that ∠ 1≅ ∠ 3 and ∠ 2≅ ∠ 4.
Statements
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Reasons
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1. ∠ 1 and ∠ 3, ∠ 2 and ∠ 4 are vertical angles
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1. Given
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2. m∠ 1+ m∠ 2=180
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2. Supplement Theorem
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3. m∠ 2+ m∠ 3=180
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3. Supplement Theorem
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4. ∠ 1≅ ∠ 3
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4. Congruent Supplements Theorem
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5. m∠ 2+ m∠ 3=180
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5. Supplement Theorem
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6. m∠ 3+ m∠ 4=180
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6. Supplement Theorem
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7. ∠ 2≅ ∠ 4
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7. Congruent Supplements Theorem
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