Construct: Right △DEF such that AC¯¯¯¯¯¯¯¯≅FD¯¯¯¯¯¯¯¯ and CB¯¯¯¯¯¯¯¯ ≅ DE¯¯¯¯¯¯¯¯
Prove: △ABC is a right triangle
Read the statements of proof. Then, answer the question.
Statements Reasons
1. △ABC where CB =a, AC=b, and AB=c and a2 +b2 =c2 1. Given
2. Construct right △DEF such that AC¯¯¯¯¯¯¯¯≅FD¯¯¯¯¯¯¯¯ and CB¯¯¯¯¯¯¯¯ ≅ 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. △ABC ≅ △FED 10. SSS
11. ∠C ≅ ∠D 11. CPCTC
12. m∠C = m∠D 12. Definition of congruence
13. m∠D = 90° 13.
14. m∠C = 90° 14.
15. △ABC is a right triangle 15. Definition of right triangles
What reason justifies statement 13?
(1 point)
Responses
Transitive Property
Transitive Property
Definition of right triangles
Definition of right triangles
Definition of supplementary angles
1 answer
The reason that justifies statement 13, which states that \( m\angle D = 90^\circ \), is derived from the properties of the constructed triangle \( \triangle DEF \). Since \( \triangle DEF \) was constructed as a right triangle (as stated in the construction step), \( m\angle D \) is defined to be 90 degrees.
Thus, the correct reason for statement 13 is:
Definition of right triangles.
