In an isosceles triangle MNP with base MP, the apex angle is 56 °.

In an isosceles triangle MNP with base MP, the apex angle is 56 °. A bisector PF and a ray PK perpendicular to ME are drawn from the apex of the external angle NPE. Find the KPF angle measure.

The angle at the apex of an isosceles triangle is 56 °, let’s calculate the angle at the base:

(180 ° – 56 °): 2 = 124 °: 2 = 62 °.

The NPM angle is 62 °, so the NPE angle = 180 ° – 62 ° = 118 °.

Since PF is a bisector, the angle EPF = 118 °: 2 = 59 °.

Since the KPE angle is 90 °, the KPF angle = 90 ° – 59 ° = 31 °.

Answer: angle KPF = 31 °.