Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. The triangle so formed is an isosceles triangle.If true enter 1 else 0.

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Solution

Consider $∠ X Y Z$ , $O Y$ bisects $∠ X Y Z$ , $O P ∥ Y Z$ and $O B ∥ Y Z$
$O Y$ bisects $∠ X Y Z$
Hence, $∠ P Y O = ∠ O Y Z$ (1)
Now, $P O ∥ Y Z$
Hence, $∠ O Y Z = ∠ P O Y$ (Alternate angles) (2)
Thus, from (1) and (2)
$∠ P Y O = ∠ P O Y$
or $△ P O Y$ is an Isosceles triangle
Similarly, $△ O B Y$ is an Isosceles triangle.

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Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

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Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. The triangle so formed is an isosceles triangle.If true enter 1 else 0.

Consider ∠XYZ, OY bisects ∠XYZ, OP∥YZ and OB∥YZOY bisects ∠XYZHence, ∠PYO=∠OYZ (1)Now, PO∥YZHence, ∠OYZ=∠POY (Alternate angles) (2)Thus, from (1) and (2)∠PYO=∠POYor △POY is an Isosceles triangleSimilarly, △OBY is an Isosceles triangle.