Question
In a triangle $\angle\text{ABC},$ $\angle\text{A}=50^\circ,$ $\angle\text{B}=60^\circ$ andFind the measures of the angles of the triangle formed by joining the mid-points of the sides of this triangle.
✓
Answer
In $\triangle\text{ABC},$ 
D and E are mid points of AB and BC. By Mid point theorem,$\text{DE}||\text{AC},\ \text{DE}=\frac{1}{2}\text{AC}$
F is the midpoint of AC Then, $\text{DE}=\Big(\frac{1}{2}\Big)\text{AC}=\text{CF}$ In a Quadrilateral DECF$\text{DE}||\text{AC},\ \text{DE}=\text{CF}$
Hence DECF is a parallelogram.$\therefore\angle\text{C}=\angle\text{D}=70^\circ$ [Opposite sides of a parallelogram]
Similarly BEFD is a parallelogram, $\angle\text{B}=\angle\text{F}=60^\circ$ ADEF is a parallelogram, $\angle\text{A}=\angle\text{E}=50^\circ$$\therefore$ Angles of $\triangle\text{DEP}$ are:
$\angle\text{D}=70^\circ,\angle\text{E}=50^\circ,\angle\text{F}=60^\circ$
D and E are mid points of AB and BC. By Mid point theorem,$\text{DE}||\text{AC},\ \text{DE}=\frac{1}{2}\text{AC}$F is the midpoint of AC Then, $\text{DE}=\Big(\frac{1}{2}\Big)\text{AC}=\text{CF}$ In a Quadrilateral DECF$\text{DE}||\text{AC},\ \text{DE}=\text{CF}$
Hence DECF is a parallelogram.$\therefore\angle\text{C}=\angle\text{D}=70^\circ$ [Opposite sides of a parallelogram]
Similarly BEFD is a parallelogram, $\angle\text{B}=\angle\text{F}=60^\circ$ ADEF is a parallelogram, $\angle\text{A}=\angle\text{E}=50^\circ$$\therefore$ Angles of $\triangle\text{DEP}$ are:
$\angle\text{D}=70^\circ,\angle\text{E}=50^\circ,\angle\text{F}=60^\circ$
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