Given: start overline, B, D, end overline

Question

BD
and start overline, A, C, end overline
AC
bisect each other.

Prove: start overline, A, B, end overline, \parallel, start overline, C, D, end overline
AB
∥
CD
.
In the proof shown in the table below, the statements and the first reason are filled in for you. Use the dropdown menu in the Reason column to select the reason for the next statement. Once you have selected a reason, a new statement will appear in a new row below. This will continue until you have selected the reason for the final statement, and then the answer area will appear with a submit button. The diagram associated with this problem is located at the bottom of the page, below the proof, and is available as a tactile printout. There may be a clarifying note below the diagram.
Step Statement Reason
1
start overline, B, D, end overline
BD
and start overline, A, C, end overline
AC
bisect each other
Given
2
start overline, A, E, end overline, \cong, start overline, E, C, end overline
AE
≅
EC


Transitive Property
3
start overline, D, E, end overline, \cong, start overline, E, B, end overline
DE
≅
EB


Substitution
4
angle, A, E, B, \cong, angle, C, E, D∠AEB≅∠CED

Reflexive Property
5
triangle, A, B, E, \cong, triangle, C, D, E△ABE≅△CDE

SSA
6
angle, A, \cong, angle, C∠A≅∠C

If two angles are supplements of the same angle (or congruent angles), then they are congruent
7
start overline, A, B, end overline, \parallel, start overline, C, D, end overline
AB
∥
CD


If two lines cut by a transversal form congruent alternate interior angles, then the two lines are parallel
The proof is complete. You may now press submit. You can still change reasons.
A
B
C
D
E
An image is shown. To download tactile, press SHIFT-CONTROL-U for UEB or SHIFT-CONTROL-E for Nemeth.
Note: start overline, D, B, end overline
DB
and start overline, A, C, end overline
AC
are segments.
Answer
Attempt 2 out of 2

answer are reflexive property transitive porperty substitution corresponding parts of congruent triangles are congruent (CPCTC) in a triangle, sides are opposite of the congruent angles are congruent a segment bisector divides a segment into two congruent segments a midpoint divides a segment into two congruent segments opposit e sides of a parallelogram are congruent the diagonal of a rectangle/square are congruent all sides of a rhombus/square are congruent congruent segments added to congruent segments form congreunt segments congreunt segments subtracted from congruent segments form congruent segments segments that are half the lengths of congruent segments are congruent segments that are twice the lengths of congruent segments are congruent sss sas asa aas ssa aa hl

Answer 1

Let's fill in the reasons for each step in the proof based on the information you've provided.

The proof steps you've outlined are systematic and hinge on the initial information that segments \( BD \) and \( AC \) bisect each other. As we choose the reasons, we will refer to the appropriate geometric concepts already provided.

Here's how we can proceed with the proof:

1. Given: \( \overline{BD} \) and \( \overline{AC} \) bisect each other.
2. \( \overline{AE} \cong \overline{EC} \)
   Reason: A segment bisector divides a segment into two congruent segments.
3. \( \overline{DE} \cong \overline{EB} \)
   Reason: A segment bisector divides a segment into two congruent segments.
4. \( \angle AEB \cong \angle CED \)
   Reason: Vertical angles are congruent.
5. \( \triangle ABE \cong \triangle CDE \)
   Reason: ASA (Angle-Side-Angle).
6. \( \angle A \cong \angle C \)
   Reason: Corresponding parts of congruent triangles are congruent (CPCTC).
7. \( \overline{AB} \parallel \overline{CD} \)
   Reason: If two lines cut by a transversal form congruent alternate interior angles, then the two lines are parallel.

The proof hence concludes that since \( \overline{AB} \) is parallel to \( \overline{CD} \) based on the congruence of alternate interior angles, the conclusion is reached successfully.

You may submit this proof based on the completion provided above, and all your statements align with proper geometric reasoning.

Answer 2

same answers Step Statement Reason
1
angle, A, C, E, \cong, angle, A, C, F∠ACE≅∠ACF
angle, E, C, B, \cong, angle, F, C, D∠ECB≅∠FCD
start overline, B, C, end overline, \cong, start overline, D, C, end overline
BC
≅
DC

Given
2
angle, A, C, B, \cong, angle, A, C, D∠ACB≅∠ACD

Transitive Property
3
start overline, A, C, end overline, \cong, start overline, A, C, end overline
AC
≅
AC


Transitive Property
4
triangle, A, B, C, \cong, triangle, A, D, C△ABC≅△ADC

SSS
5
start overline, A, B, end overline, \cong, start overline, A, D, end overline
AB
≅
AD


Transitive Property
The proof is complete. You may now press submit. You can still change reasons.
A
B
C
D
E
F
An image is shown. To download tactile, press SHIFT-CONTROL-U for UEB or SHIFT-CONTROL-E for Nemeth.
Note: start overline, A, B, end overline
AB
and start overline, A, D, end overline
AD
are segments.

Answer 3

Let's fill in the steps and reasons provided for your proof. Based on the information given, it looks like we are comparing angles and segments, and we are trying to prove the congruence of triangles.

Here’s how we can approach the proof step by step:

1. Step 1:

   • Statement: \( \angle ACE \cong \angle ACF \)

   • Reason: Given (this should include information indicating that \( \angle ACF \) and \( \angle ACE \) are congruent by a common measure, perhaps identified as vertical angles or equal angles).

   • Next Statement: \( \angle ECB \cong \angle FCD \)

   • Reason: If angles are congruent by the same angle or vertically, select "Given" or an equivalent if this detail is included in what was given.

   • Next Statement: \( \overline{BC} \cong \overline{DC} \)

   • Reason: Given (This states that segments are congruent via the information provided).

2. Step 2:

   • Statement: \( \angle ACB \cong \angle ACD \)
   • Reason: Transitive Property (as angles are compared through a series of equal connections).

3. Step 3:

   • Statement: \( \overline{AC} \cong \overline{AC} \)
   • Reason: Reflexive Property (This shows that a segment is congruent to itself).

4. Step 4:

   • Statement: \( \triangle ABC \cong \triangle ADC \)
   • Reason: SSS (Side-Side-Side) criterion (Since all three sides of one triangle correspond to three sides of another triangle by congruence).

5. Step 5:

   • Statement: \( \overline{AB} \cong \overline{AD} \)
   • Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

The proof demonstrates the relationship between the angles and segments leading to the conclusion that specific segments \( \overline{AB} \) and \( \overline{AD} \) are equal because the triangles they belong to are congruent.

You can now submit this proof format you have set up. Just ensure the reasons align directly with the statements you've made.