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) With diagram

Let's consider three types of triangles (acute, obtuse, and right-angled) and use the midpoint theorem to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side. The Midpoint Theorem states that a line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.

### Step-by-step proof with diagrams:

#### 1. **Acute Triangle:**

Consider an acute triangle \( \triangle ABC \), where all interior angles are less than 90 degrees.

- Let \( D \) be the midpoint of side \( AB \), and \( E \) be the midpoint of side \( AC \).
- We will prove that the line segment \( DE \) is parallel to the side \( BC \) and that \( DE = \frac{1}{2} BC \).

**Diagram:**
```
      A
     / \
    /   \
   D-----E
  /       \
 B---------C
```

- Since \( D \) and \( E \) are midpoints, by the **Midpoint Theorem**, \( DE \parallel BC \) and \( DE = \frac{1}{2} BC \).

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

#### 2. **Obtuse Triangle:**

Now, consider an obtuse triangle \( \triangle PQR \), where one of the angles is greater than 90 degrees (say, \( \angle PQR > 90^\circ \)).

- Let \( M \) be the midpoint of side \( PQ \), and \( N \) be the midpoint of side \( PR \).
- We will prove that the line segment \( MN \) is parallel to the side \( QR \).

**Diagram:**
```
  P
 / \
M-----N
/       \
Q---------R
```

- Since \( M \) and \( N \) are midpoints, by the **Midpoint Theorem**, \( MN \parallel QR \) and \( MN = \frac{1}{2} QR \).

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

#### 3. **Right-Angled Triangle:**

Finally, consider a right-angled triangle \( \triangle XYZ \), where \( \angle XYZ = 90^\circ \).

- Let \( P \) be the midpoint of side \( XY \), and \( Q \

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.

Explore the proof of the Midpoint Theorem with acute, obtuse, and right-angled triangles. This guide demonstrates how to prove that the line joining two midpoints is parallel to the third side using corresponding angles. Suitable for students in Grades 9-10.

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