Given:
Segment AB is congruent to Segment DE
Prove:
Segment AD is congruent to BE
Note: (for the illustration)
C is the midpoint of Segment AE, B is the midpoint of Segment AC and D is the midpoint of Segment CE.
To prove that Segment AD is congruent to Segment BE, we can use the concept of congruent triangles. By showing that triangles ABD and BEC are congruent, we can conclude that their corresponding sides are also congruent.
Let's break down the proof step-by-step:
1. Given: Segment AB is congruent to Segment DE
This tells us that AB = DE.
2. B is the midpoint of Segment AC
This means that AB is congruent to BC, and we can also conclude that AC = 2 * AB.
3. D is the midpoint of Segment CE
Similarly, we can deduce that DE is congruent to CE, and EC = 2 * DE.
4. C is the midpoint of Segment AE
Since C is the midpoint of AE, AC is congruent to EC (since C splits AE into two equal halves).
Now, let's prove that triangles ABD and BEC are congruent:
5. Angle BAD is congruent to angle BEC
This can be proven by noting that BAC is congruent to CEB (as alternate interior angles).
6. AB is congruent to BE (Given)
As stated, AB = DE.
7. Angle ABD is congruent to angle BEC
By combining steps 5 and 6, we can conclude that ABD is congruent to BEC (as vertical angles).
8. By using the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangles ABD and BEC are congruent.
Once we have proven the congruence of triangles ABD and BEC, we can use the corresponding sides of congruent triangles to determine that AD is congruent to BE. Specifically, we can say that:
In triangle ABD, AD is congruent to BD (since it is an isosceles triangle).
In triangle BEC, BE is congruent to EC (since it is an isosceles triangle).
Since BD is congruent to EC (due to the congruence of BEC and ABD), we can conclude that AD is congruent to BE.
Thus, we have proven that Segment AD is congruent to Segment BE using the given information and the concept of congruent triangles.
To prove that segment AD is congruent to BE, we will use the property of midpoints.
Given:
Segment AB is congruent to segment DE.
C is the midpoint of segment AE.
B is the midpoint of segment AC.
D is the midpoint of segment CE.
We want to prove:
Segment AD is congruent to segment BE.
Proof:
Step 1: Draw a diagram
Draw line AE and label point C as the midpoint of AE.
Draw line segment AC, with B as the midpoint of AC.
Draw line segment CE, with D as the midpoint of CE.
Step 2: Mark the congruent segments
Label the congruent segments as AB ≅ DE.
Step 3: Use the midpoint property
The midpoint property states that if a line segment has a midpoint, then it can be divided into two congruent segments.
Using this property, we can say that:
AC ≅ CB (as B is the midpoint of AC)
CE ≅ ED (as D is the midpoint of CE)
Step 4: Apply Transitive property
Since AB ≅ DE and AC ≅ CB, we can use the transitive property to say that AB ≅ CB.
Step 5: Use the midpoint property again
Using the midpoint property and the fact that AB ≅ CB, we can say that AD ≅ DB. This is because D is the midpoint of segment CE, and DB is one of the resulting segments when CE is divided at D.
Step 6: Apply the transitive property
Since AD ≅ DB and AD ≅ DE (given), we can apply the transitive property to say that DB ≅ DE.
Step 7: Use the midpoint property again
Using the midpoint property and the fact that DB ≅ DE, we can say that BE ≅ DE. This is because E is the midpoint of segment DB, and BE is one of the resulting segments when DB is divided at E.
Step 8: Apply the transitive property
Since BE ≅ DE and DB ≅ DE, we can apply the transitive property to say that BE ≅ DB.
Step 9: Apply the symmetric property
Since BE ≅ DB, we can use the symmetric property to say that DB ≅ BE.
Step 10: Apply the transitive property
Since DB ≅ BE and AD ≅ DB, we can apply the transitive property to say that AD ≅ BE.
Therefore, segment AD is congruent to BE, as required.
AD = AE-DE
BE = AE-AB
AB=DE, so
AD = AE-AB = BE
You can reword that using all the "congruents" instead of = signs.