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