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Proving Angles Congruent Chapter 2: Reasoning and Proof1 Objectives 1 To prove and apply theorems about angles
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Proving Angles Congruent Chapter 2: Reasoning and Proof2 Key Concepts A statement that you prove true is a ____________. A paragraph proof is written as sentences in a paragraph. A __________ is a convincing argument that uses deductive reasoning. “Given”: lists what you know from the hypothesis of the theorem “Prove”: the conclusion of the theorem Diagram: records the given information visually
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Proving Angles Congruent Chapter 2: Reasoning and Proof3 Key Concepts Given: 1 and 2 are vertical angles Prove: 1 2 Proof: By the Angle Addition Postulate, m 1 + m 3 = 180 and m 2 + m 3 = 180. By substitution, m 1 + m 3 = m 2 + m 3. Subtract m 3 from each side. You get m 1 = m 2, or 1 2. Theorem: Vertical angles are congruent.
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Proving Angles Congruent Chapter 2: Reasoning and Proof4 Find the value of x.
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Proving Angles Congruent Chapter 2: Reasoning and Proof5 Key Concepts Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 3 Proof: By the definition of supplementary angles, m 1 + m 2 = 180 and m 3 + m 2 = 180. By substitution, m 1 + m 2 = m 3 + m 2. Subtract m 2 from each side. You get m 1 = m 3, or 1 3. Theorem: If two angles are supplements of the same angle, then the two angles are congruent.
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Proving Angles Congruent Chapter 2: Reasoning and Proof6 Key Concepts Given: 1 and 2 are supplementary 3 and 4 are supplementary 2 4 Prove: 1 3 Proof: By the definition of supplementary angles, m 1 + m 2 = 180 and m 3 + m 4 = 180. By substitution, m 1 + m 2 = m 3 + m 4. Since 2 4, by the definition of congruence m 2 = m 4. By substitution m 1 + m 4 = m 3 + m 4. Subtract m 4 from each side. You get m 1 = m 3, or 1 3. Theorem: If two angles are supplements of congruent angles, then the two angles are congruent.
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Proving Angles Congruent Chapter 2: Reasoning and Proof7 Key Concepts Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem: All right angles are congruent. Theorem: If two angles are congruent and supplementary, then each is a right angle.
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Proving Angles Congruent Chapter 2: Reasoning and Proof8 Write a paragraph proof using the given, what you are to prove, and the diagram. Given: WX = YZ Prove: WY = XZ ●● ●●
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