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The proof below may or may not be correct. If the proof is incorrect, determine the first step number that is not justified and the reason it is not justified.
Given: start overline, A, B, end overline, \cong, start overline, B, C, end overline
AB
≅
BC
and start overline, A, C, end overline, is perpendicular to, start overline, B, D, end overline, .
AC
⊥
BD
.
Prove: start overline, B, D, end overline
BD
bisects start overline, A, C, end overline
AC
.
.... The diagram associated with this problem is located below the proof table, and is available as a tactile printout. There may be a clarifying note below the diagram.
Step Statement Reason
1
start overline, A, B, end overline, \cong, start overline, B, C, end overline
AB
≅
BC
start overline, A, C, end overline, is perpendicular to, start overline, B, D, end overline
AC
⊥
BD
Given
2
angle, A, D, B∠ADB is a right angle
Perpendicular lines form right angles
3
triangle, A, B, D△ABD is a right triangle
The triangle contains a right angle
4
triangle, C, B, D△CBD is a right triangle
The triangle contains a right angle
5
start overline, B, D, end overline, \cong, start overline, B, D, end overline
BD
≅
BD
Reflexive Property
6
triangle, A, B, D, \cong, triangle, C, B, D△ABD≅△CBD
HL
7
start overline, A, D, end overline, \cong, start overline, D, C, end overline
AD
≅
DC
Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
8
start overline, B, D, end overline
BD
bisects start overline, A, C, end overline
AC
A segment bisector intersects the segment to form two congruent segments
A
B
C
D
An image is shown. To download tactile, press SHIFT-CONTROL-U for UEB or SHIFT-CONTROL-E for Nemeth.
Note: the segment A, CAC is a straight segment.
Answer
Attempt 1 out of 2
The proof is
incorrect
and step number
is the first unjustified step due to a
missing prior step
. Submit Answer
1 answer
The proof provided is indeed incorrect, and the first unjustified step is step number 6.
The reason step 6 is not justified is that, although it states the two triangles (△ABD and △CBD) are right triangles and asserts that they are congruent by using the Hypotenuse-Leg (HL) congruence theorem, it has not established that their hypotenuses and legs are indeed congruent based on the given information.
To use the HL theorem, you must show that both triangles have one leg congruent, and the hypotenuse must also be congruent; this is not clearly demonstrated from the previous steps. In particular, there has been no confirmation or evidence provided that \(AD ≅ BC\) or that the two hypotenuses \(AB\) and \(BC\) are equal, which is necessary to justify step 6.
Thus, the first unjustified step is step 6 due to missing prior evidence/steps to confirm the necessary congruent aspects of the triangles involved.
1 answer