Take three types of triangles (Acute, Obtuse and Right angled triangles) now mark mid point of any two sides.
Join these midpoint and prove that the line so formed is parallel to third side.| Hint: See Pair of corresponding angles are equal)

To prove that the line joining the midpoints of two sides of a triangle is parallel to the third side, we will apply the **Midpoint Theorem**. The Midpoint Theorem states that:

> "The line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length."

### Given:
- A triangle with three sides.
- We mark the midpoints of any two sides.
- The line joining these midpoints needs to be proven parallel to the third side.

Let’s break it down into the three types of triangles: **Acute**, **Obtuse**, and **Right-angled** triangles.

### 1. **For an Acute Triangle:**

#### Step 1: Setup
- Consider triangle \(ABC\).
- Let \(D\) be the midpoint of side \(AB\), and \(E\) be the midpoint of side \(AC\).
- We need to prove that line \(DE\) is parallel to side \(BC\).

#### Step 2: Apply the Midpoint Theorem
- According to the Midpoint Theorem, the line segment joining two midpoints of any two sides of a triangle is parallel to the third side.
- Thus, the line \(DE\) is parallel to side \(BC\) and \(DE = \frac{1}{2} BC\).

#### Step 3: Corresponding Angles
- The corresponding angles formed by the transversal \(AB\) cutting the parallel lines \(DE\) and \(BC\) are equal.

Thus, in an acute triangle, the line joining the midpoints of two sides is parallel to the third side.

### 2. **For an Obtuse Triangle:**

#### Step 1: Setup
- Consider an obtuse triangle \(PQR\), where angle \(QPR\) is obtuse.
- Let \(M\) be the midpoint of side \(PQ\), and \(N\) be the midpoint of side \(PR\).
- We need to prove that the line \(MN\) is parallel to side \(QR\).

#### Step 2: Apply the Midpoint Theorem
- By the Midpoint Theorem, the line joining the midpoints \(M\) and \(N\) is parallel to the third side \(QR\).
- Thus, \(MN \parallel QR\) and \(MN = \frac{1}{2} QR\).

#### Step 3: Corresponding Angles
- The transversal \(PQ\) cuts the parallel lines \(MN\) and \(QR\), and the corresponding angles formed are equal.

Thus, in an obtuse triangle, the line joining the midpoints of two sides is parallel to the third side.

### 3. **For a Right-angled Triangle:**

#### Step 1: Setup
- Consider a right-angled triangle \(XYZ\) with the right angle at \(X\).
- Let \(P\) be the midpoint of side \(XY\), and \(Q\) be the midpoint of side \(XZ\).
- We need to prove that line \(PQ\) is parallel to side \(YZ\).

#### Step 2: Apply the Midpoint Theorem
- By the Midpoint Theorem, the line joining the midpoints \(P\) and \(Q\) is parallel to the third side \(YZ\).
- Thus, \(PQ \parallel YZ\) and \(PQ = \frac{1}{2} YZ\).

#### Step 3: Corresponding Angles
- The transversal \(XY\) cuts the parallel lines \(PQ\) and \(YZ\), and the corresponding angles formed are equal.

Thus, in a right-angled triangle, the line joining the midpoints of two sides is parallel to the third side.

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### Conclusion:
For all three types of triangles (Acute, Obtuse, and Right-angled), the line joining the midpoints of two sides is parallel to the third side. This is a direct application of the **Midpoint Theorem**, and

Take three types of triangles (Acute, Obtuse and Right angled triangles) now mark midpoint of any two sides. Join these midpoints and prove that the line so formed is parallel to third side.| Hint: See Pair of corresponding angles are equal.

Explore how to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side. This solution covers the Midpoint Theorem applied to acute, obtuse, and right-angled triangles, with step-by-step explanations for students in Grades 8-10.

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