in an isosceles triangle ABC ,AB=AC.if AB and AC are produced to D and E respectively so that BD=CE.prove thatBE=CD
To prove that BE = CD in an isosceles triangle ABC, where AB = AC and BD = CE, we can use the fact that the opposite angles in an isosceles triangle are equal.
Here's how we can prove it step by step:
1. Draw triangle ABC and extend lines AB and AC to points D and E, respectively, such that BD = CE.
```plaintext
B
/ \
BD/ \EC
/ \
/_______\
A D C
```
2. Since AB = AC, triangle ABC is isosceles. This means that angle ABC = angle BAC.
```plaintext
B
/ \
BD/ \EC
/ \
/_ABC___\
A D C
```
3. Since BD = CE, we can conclude that angle BDC = angle CEB because they are corresponding angles formed by the transversal line DE.
```plaintext
B
/ \
BD/ \EC
/ θ \
/_ABC___\
A D C
```
Now we have triangle BDC and triangle BEC where angle BDC = angle BEC.
4. Since angle ABC = angle BAC (isosceles triangle property) and angle BDC = angle BEC (proven in step 3), the sum of angle ABC and angle BDC is equal to the sum of angle BAC and angle BEC.
So angle ABC + angle BDC = angle BAC + angle BEC.
5. The sum of angle ABC and angle BDC is equal to 180 degrees (straight angle), and the sum of angle BAC and angle BEC is also equal to 180 degrees (straight angle).
6. Therefore, angle ABC + angle BDC = angle BAC + angle BEC = 180 degrees.
```plaintext
B
/ \
BD/ \EC
/ \
/_ABC___\
A D C
```
7. In a triangle, the sum of all interior angles is always 180 degrees. Since angle ABC + angle BDC = angle BAC + angle BEC = 180 degrees, the remaining angle in triangle BDC must be equal to the remaining angle in triangle BEC.
8. Therefore, angle BCD (or angle CBD) = angle BCE (or angle CBE).
```plaintext
B
/ \
BD/ \EC
/ \
/_ABC___\
A D C
```
9. Using the fact that opposite angles in an isosceles triangle are equal, we can conclude that angle BCD = angle CBE.
10. Since angle BCD = angle CBE, and BD = CE, we can conclude that triangle BCD is congruent to triangle CBE by the Angle-Side-Angle (ASA) congruence criterion.
```plaintext
B
/ \
BD/ \EC
/ \
/_ABC___\
A D C
```
11. By proving the congruence of triangle BCD and triangle CBE, we can conclude that the corresponding sides are equal. Therefore, BE = CD.
```plaintext
B
/ \
BD/ \EC
/ \
/_ABC___\
A D C
```
Hence, we have proven that in an isosceles triangle ABC, where AB = AC and BD = CE, BE = CD.