Question
Quadrilateral with vertices upper A, upper B, upper C, and upper D with diagonals modifying above upper B upper D with hash marks and modifying above upper A upper C with hash marks intersecting at point upper E.
Statements Reasons
1. ΔAEB≅ΔCEB
Quadrilateral ABCD
ΔAEB
and ΔCEB
are right triangles at vertex E
1. Given
2. ∠AEB
, ∠CEB
are right angles 2. Definition of a right triangle
3. BD¯¯¯¯¯¯¯¯⊥AC¯¯¯¯¯¯¯¯
3. Definition of right angles
4. ? 4. ?
5. BD¯¯¯¯¯¯¯¯
is the perpendicular bisector of AC¯¯¯¯¯¯¯¯
5. Definition of perpendicular bisector
6. ABCD
is a kite 6. If one diagonal is the perpendicular bisector of the other diagonal in a quadrilateral, then it is a kite.
Which is the best statement and reason to complete step 4 of the proof?
(1 point)
Responses
AE¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯
, CPCTC (Congruent Parts of Congruent Triangles are Congruent)
line segment cap A cap e is congruent to line segment cap c cap e , CPCTC (Congruent Parts of Congruent Triangles are Congruent)
EB¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯
, Definition of a Quadrilateral
line segment cap e cap b is congruent to line segment cap d cap e , Definition of a Quadrilateral
AD¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯
, Side-Side-Side Triangle Congruence Theorem
line segment cap A cap d is congruent to line segment cap c cap d , Side-Side-Side Triangle Congruence Theorem
AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
, CPCTC (Congruent Parts of Congruent Triangles are Congruent)
Statements Reasons
1. ΔAEB≅ΔCEB
Quadrilateral ABCD
ΔAEB
and ΔCEB
are right triangles at vertex E
1. Given
2. ∠AEB
, ∠CEB
are right angles 2. Definition of a right triangle
3. BD¯¯¯¯¯¯¯¯⊥AC¯¯¯¯¯¯¯¯
3. Definition of right angles
4. ? 4. ?
5. BD¯¯¯¯¯¯¯¯
is the perpendicular bisector of AC¯¯¯¯¯¯¯¯
5. Definition of perpendicular bisector
6. ABCD
is a kite 6. If one diagonal is the perpendicular bisector of the other diagonal in a quadrilateral, then it is a kite.
Which is the best statement and reason to complete step 4 of the proof?
(1 point)
Responses
AE¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯
, CPCTC (Congruent Parts of Congruent Triangles are Congruent)
line segment cap A cap e is congruent to line segment cap c cap e , CPCTC (Congruent Parts of Congruent Triangles are Congruent)
EB¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯
, Definition of a Quadrilateral
line segment cap e cap b is congruent to line segment cap d cap e , Definition of a Quadrilateral
AD¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯
, Side-Side-Side Triangle Congruence Theorem
line segment cap A cap d is congruent to line segment cap c cap d , Side-Side-Side Triangle Congruence Theorem
AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
, CPCTC (Congruent Parts of Congruent Triangles are Congruent)
Answers
GPT-4o mini
To complete step 4 of the proof, the best statement and reason is:
**AE̅̅̅̅̅̅ ≅ CE̅̅̅̅̅̅, CPCTC (Congruent Parts of Congruent Triangles are Congruent)**
This choice is appropriate because it follows from the initial congruence of triangles ΔAEB and ΔCEB, which you established in step 1. Since these triangles are congruent, their corresponding parts are also congruent by the principle of CPCTC, allowing you to conclude that AE is congruent to CE.
**AE̅̅̅̅̅̅ ≅ CE̅̅̅̅̅̅, CPCTC (Congruent Parts of Congruent Triangles are Congruent)**
This choice is appropriate because it follows from the initial congruence of triangles ΔAEB and ΔCEB, which you established in step 1. Since these triangles are congruent, their corresponding parts are also congruent by the principle of CPCTC, allowing you to conclude that AE is congruent to CE.