Given: ΔABC where CB=a , AC=b , and AB=c and a2+b2=c2
Construct: Right ΔDEF such that AC¯¯¯¯¯¯¯¯≅FD¯¯¯¯¯¯¯¯ and CB¯¯¯¯¯¯¯¯≅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 CB¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯ 2. Construction
3. (AB)2=(AC)2+(CB)2 3. Substitution Property
4. (EF)2=(FD)2+(DE)2 4.
5. AC=FD
CB=DE
5. Definition of congruence
6. (EF)2=(AC)2+(CB)2 6. Substitution Property
7. (EF)2=(AB)2 7. Transitive Property
8. EF=AB 8. Calculations (square root)
9. EF¯¯¯¯¯¯¯¯≅AB¯¯¯¯¯¯¯¯ 9. Definition of congruence
10. ΔABC≅ΔFED 10.
11. ∠C≅∠D 11.
12. m∠C=m∠D 12. Definition of congruence
13. m∠D=90° 13. Definition of right triangles
14. m∠C=90° 14.
15. ΔABC is a right triangle 15. Definition of right triangles
What reason justifies statement 14?
(1 point)
Responses
Reflexive Property
Reflexive Property
Definition of right triangles
Definition of right triangles
Definition of right angles
Definition of right angles
Transitive Property
1 answer
The reason that justifies statement 14 (m∠C = 90°) is:
Transitive Property
This is because statement 13 states that m∠D = 90°, and statement 12 states that m∠C = m∠D (due to congruence of angles from the previous parts of the proof). By the transitive property, if m∠D = 90° and m∠C = m∠D, it follows that m∠C = 90°.
