# Triangle XYZ is an isosceles right triangle, right angled at Y.M and N are mid points of XY and XZ respectively. find the measure of angle XNM.

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triangle XYZ is an isosceles right triangle, right at Y. M and N are mid points of XY and XZ. so angle ZYX is of 90 degree

so, angle yzx is of 45 degree

and angle yxz is also of 45 degree

so line segment MN will be perpendicular to segment YZ.

thus angle XNM will be of 45 degree

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It is 45 deg. = π/4.

Since, XYZ is a right angle triangle at Y, angle Y = π/2.

Since it is an isosceles triangle: angle X = angle Z

sum of angles X + angle Z = π - angle Y = π - π/2 = π/2

hence angle X =*angle Z = π/4*

Now, the triangle XNM is also an isosceles right angle triangle. The reasons:

1. NM is a line joining the mid points of sides XY and XZ respectively. Hence* NM is parallel to the third side YZ*. also, the length of NM is 1/2 * YZ

2. As NM || YZ, the transversal XZ cuts the two parallel lines at Z and N. The angles XZY and XNM are equal. They are included angles on the same side of transversal.*Hence, angle XNM = angle XZY = π/4.*

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further more:

3. As NM || YZ and YZ is perpendicular to XY, NM is perpendicular to XY. Hence angle at N in triangle XNM is also a right angle.

Since, XYZ is a right angle triangle at Y, angle Y = π/2.

Since it is an isosceles triangle: angle X = angle Z

sum of angles X + angle Z = π - angle Y = π - π/2 = π/2

hence angle X =

Now, the triangle XNM is also an isosceles right angle triangle. The reasons:

1. NM is a line joining the mid points of sides XY and XZ respectively. Hence

2. As NM || YZ, the transversal XZ cuts the two parallel lines at Z and N. The angles XZY and XNM are equal. They are included angles on the same side of transversal.

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further more:

3. As NM || YZ and YZ is perpendicular to XY, NM is perpendicular to XY. Hence angle at N in triangle XNM is also a right angle.