In the issoseles tri. ABC AB = BC The bisectors of angle B and angle C meet the opposite sides at D and E respectively. Prove that DE||BC.Solve using 9th grade maths 

In the given isosceles triangle ABC, where \( AB = BC \), the bisectors of angles \( B \) and \( C \) meet the opposite sides at points \( D \) and \( E \), respectively. We are tasked with proving that \( DE \parallel BC \).

### Step-by-Step Solution:

1. **Given:**
   - \( \triangle ABC \) is isosceles, with \( AB = BC \).
   - The angle bisectors of \( \angle B \) and \( \angle C \) meet the opposite sides \( AC \) and \( AB \) at points \( D \) and \( E \), respectively.
   - We need to prove that \( DE \parallel BC \).

2. **Draw the figure**:
   Start by drawing triangle \( ABC \) such that \( AB = BC \), and then draw the bisectors of \( \angle B \) and \( \angle C \), meeting sides \( AC \) and \( AB \) at points \( D \) and \( E \), respectively.

3. **Use the Angle Bisector Theorem**:
   According to the angle bisector theorem:
   - \( \frac{AD}{DC} = \frac{AB}{BC} \)
   - Since \( AB = BC \), it follows that \( AD = DC \). Therefore, point \( D \) is the midpoint of \( AC \).
   
   Similarly, applying the angle bisector theorem at \( C \):
   - \( \frac{AE}{EB} = \frac{AC}{BC} \)
   - Since \( AB = BC \), \( AE = EB \). Thus, point \( E \) is the midpoint of \( AB \).

4. **Properties of Midsegments in a Triangle**:
   - In any triangle, the line segment joining the midpoints of two sides is parallel to the third side and is half of its length.
   - From the earlier steps, we know that \( D \) is the midpoint of \( AC \) and \( E \) is the midpoint of \( AB \). Therefore, by the **midpoint theorem**, \( DE \parallel BC \) and \( DE = \frac{1}{2} BC \).

### Conclusion:
We have proven that \( DE \parallel BC \) by using the midpoint theorem, based on the properties of angle bisectors and the fact that \( AB = BC \).

Would you like further clarification on any part of the solution?

Here are 5 related questions:
1. What is the angle bisector theorem, and how is it applied in various triangles?
2. How do you prove the midpoint theorem in a triangle?
3. Can the midline property be applied to non-isosceles triangles? Why or why not?
4. How do congruent triangles help in proving parallel lines in geometrical problems?
5. What are the differences between angle bisectors and perpendicular bisectors in a triangle?

**Tip:** Understanding the properties of special triangles, such as isosceles and equilateral triangles, simplifies solving many geometry problems!

In the isosceles triangle ABC, AB = BC. The bisectors of angle B and angle C meet the opposite sides at D and E respectively. Prove that DE || BC. Solve using 9th grade maths.

This problem focuses on proving that DE is parallel to BC in an isosceles triangle where AB = BC, by using the Angle Bisector Theorem and Midline Theorem. The solution is ideal for Grade 9 students and provides a detailed step-by-step breakdown.

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