Give the definition, property, definition, theorem or postulate that can be used toreach the “prove” from the given using deductive reasoning.5. Given: 2 + 𝑥 = 9Prove: 𝑥 = 76. Given: ∠1 and ∠2 are complementary.Prove: ∠1 and ∠2 are both acute.7. Given: 𝐶 is a midpoint of 𝑋𝑌.Prove: 𝐶𝑋 ≅ 𝑋𝑌.8. Given: ∠𝐴 and ∠𝐵 are vertical angles.Prove: ∠𝐴 and ∠𝐵 are congruent.
Question
Give the definition, property, definition, theorem or postulate that can be used toreach the “prove” from the given using deductive reasoning.5. Given: 2 + 𝑥 = 9Prove: 𝑥 = 76. Given: ∠1 and ∠2 are complementary.Prove: ∠1 and ∠2 are both acute.7. Given: 𝐶 is a midpoint of 𝑋𝑌.Prove: 𝐶𝑋 ≅ 𝑋𝑌.8. Given: ∠𝐴 and ∠𝐵 are vertical angles.Prove: ∠𝐴 and ∠𝐵 are congruent.
Solution
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Given: 2 + 𝑥 = 9 Prove: 𝑥 = 7
Definition/Property: The property of equality states that if two quantities are equal, then they can be replaced by each other in any equation or expression.
Steps:
- Subtract 2 from both sides of the equation 2 + 𝑥 = 9.
- This gives 𝑥 = 9 - 2.
- Simplifying the right side gives 𝑥 = 7.
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Given: ∠1 and ∠2 are complementary. Prove: ∠1 and ∠2 are both acute.
Definition/Property: Complementary angles are two angles whose measures add up to 90 degrees. An acute angle is an angle whose measure is less than 90 degrees.
Steps:
- Since ∠1 and ∠2 are complementary, the sum of their measures is 90 degrees.
- Therefore, each angle must be less than 90 degrees.
- Hence, ∠1 and ∠2 are both acute.
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Given: 𝐶 is a midpoint of 𝑋𝑌. Prove: 𝐶𝑋 ≅ 𝑋𝑌.
Definition/Property: The definition of a midpoint is a point that divides a segment into two congruent segments.
Steps:
- Since C is the midpoint of XY, by definition, CX ≅ CY.
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Given: ∠𝐴 and ∠𝐵 are vertical angles. Prove: ∠𝐴 and ∠𝐵 are congruent.
Definition/Property: The Vertical Angles Theorem states that vertical angles are always congruent.
Steps:
- Since ∠A and ∠B are vertical angles, by the Vertical Angles Theorem, ∠A ≅ ∠B.
Similar Questions
Determine whether each conclusion is true or false. If false, provide acounterexample.1. Given: ∠𝑋 and ∠𝑌 are complementary.Conclusion: ∠𝑋 and ∠𝑌 are both acute.2. Given: 𝑇 is a midpoint of 𝐴𝐵.Conclusion: 𝐴𝑇 ≅ 𝑇𝐵.3. Given: 𝑊, 𝑋, and 𝑌 are points on a plane.Conclusion: 𝑊, 𝑋, and 𝑌 are collinear.4. Given: ∠1 and ∠2 are supplementary.Conclusion: ∠1 and ∠2 form a linear pair.
Introductory Activity for congruency of trianglesDear Students, This vacation you will discover all about congruency by yourself!Two triangles are said to be congruent when the 3 sides and 3 angles of one triangle isexactly equal to the 3 sides and 3 angles of another triangle.If you observe the above triangles, the 3 sides and 3 angles of △ABC is exactly equal to 3sides and 3 angles of △DFE. This means △ABC is congruent to △DFE.We write it as △ABC ≅ △DFEACTIVITYAim:It is sufficient if I can show that the 3 sides of one triangle is equal 3 sides of anothertriangle.The 3 angles of both the triangles become automatically equal.The two triangles are congruent.Procedure:Have students cut straws into the following lengths:2 straws 8 centimeters in length2 straws 11 centimeters in length2 straws 5 centimeters in length
(b)If ∠1 and ∠2 are complementary angles, then one of them must have a measure greater than 45°.=Counterexample:m∠1=, °m∠2°
1 △TUW is equilateral Given2 TX≅UVGiven3 ∠VUW≅∠WTX Given4 TW≅UWAdditive Property of LengthAll right angles are congruentDefinition of angle bisectorDefinition of equilateral triangleDefinition of midpointVertical Angle Theorem5 △UVW≅△TXW SASSubmit
Which of the following are accepted without proof in a logical system? Check all that apply.A.TheoremsB.DefinitionsC.PostulatesD.Axioms
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