Fill in the blanks of the angles: D, U

Let’s calculate the angles.

Angle D

Since triangle \( \triangle ADU \) is an isosceles triangle, angles \( \angle DUA \) and \( \angle DAU \) are equal. Therefore, \( \angle DUA = \angle DAU = 69^\circ \).

The sum of the angles in a triangle is \( 180^\circ \), so:

\[\angle D + \angle DUA + \angle DAU = 180^\circ\]\[\angle D + 69^\circ + 69^\circ = 180^\circ\]\[\angle D + 138^\circ = 180^\circ\]\[\angle D = 180^\circ - 138^\circ\]\[\angle D = 42^\circ\]

Angle U

Angles \( \angle AUE \) and \( \angle DUA \) form a straight angle, so their sum is \( 180^\circ \):

\[\angle AUE + \angle DUA = 180^\circ\]\[\angle AUE + 69^\circ = 180^\circ\]\[\angle AUE = 180^\circ - 69^\circ\]\[\angle AUE = 111^\circ\]

Given \( \angle E = 55^\circ \), we can find \( \angle U \) in \( \triangle AEU \):

\[\angle U + \angle E + \angle EAU = 180^\circ\]

We know that \( \angle EAU = 90^\circ \) because it’s a right angle:

\[\angle U + 55^\circ + 90^\circ = 180^\circ\]\[\angle U + 145^\circ = 180^\circ\]\[\angle U = 180^\circ - 145^\circ\]\[\angle U = 35^\circ\]

Answer: D = 42°, U = 69°.