On the left constructed triangle FED with right angle D. On the right, triangle ABC with AC = b, CB = a, and AB = c.

Question

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.
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. 10. SSS Congruence Theorem
11. 11. CPCTC Theorem
12. m∠C=m∠D 12. Definition of congruence
13. m∠D=90° 13. Definition of right triangles
14. m∠C=90° 14. Transitive Property
15. ΔABC is a right triangle 15. Definition of right triangles
Which statement belongs on line 11?

(1 point)
Responses

∠C≅∠D
angle cap c is congruent to angle cap d

m∠C=90°
m angle cap c is equal to 90 degrees

EF=c
cap e cap f is equal to c

∠A≅∠F

Answer 1

The statement that belongs on line 11 should express the relationship of congruence between the angles in triangles ABC and DEF. Since you are trying to relate the angles of the two triangles based on the congruence of the triangles, the correct option is:

∠C≅∠D

This indicates that angle \(C\) in triangle \(ABC\) is congruent to angle \(D\) in triangle \(DEF\), which is necessary to complete the proof that triangle \(ABC\) is a right triangle.