Given: ΔABC where CB=a, AC=b, and AB=c and a2+b2=c2
Construct: Right ΔDEF such that AC¯¯¯¯¯¯¯¯≅FD¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯
Prove: ΔABC is a right triangle
Read the statements of proof. Then, answer the question.
Statements Reasons
1. ΔABC where CB=a, AC=b, and AB=c and a2+b2=c2 1. Given
2. Construct right ΔDEF such that AC¯¯¯¯¯¯¯¯≅FD¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯ 2. Construction
3. (AB)2=(AC)2+(CB)2 3. Substitution Property
4. 4. Pythagorean Theorem
5. AC=FD
CB=DE
5. Definition of congruence
6. (EF)2=(AC)2+(CB)2 6.
7. (EF)2=(AB)2 7. Transitive Property
8. 8. Calculations (square root)
9. EF¯¯¯¯¯¯¯¯≅AB¯¯¯¯¯¯¯¯ 9. Definition of congruence
10. ΔABC≅ΔFED 10. SSS Congruence Theorem
11. ∠C≅∠D 11. CPCTC Theorem
12. m∠C=m∠D 12. Definition of congruence
13. m∠D=90° 13.
14. m∠C=90° 14.
15. ΔABC is a right triangle 15. Definition of right triangles
What reason justifies statement 6?
(1 point)
Responses
Pythagorean Converse
Pythagorean Converse
Reflexive Property
Reflexive Property
Substitution Property
Substitution Property
Pythagorean Theorem
Pythagorean Theorem
1 answer
The correct reason that justifies statement 6, which asserts that \((EF)^2 = (AC)^2 + (CB)^2\), is:
Pythagorean Theorem
This is because statement 6 applies the Pythagorean Theorem to the right triangle DEF, where the sides \(FD\) and \(DE\) correspond to sides \(AC\) and \(CB\) of triangle ABC, respectively, and \(EF\) corresponds to the hypotenuse \(AB\).
