The three examples below show how angle relationships and the properties of triangles can be used to find unknown angles. You can review angle properties here.
Try the 180° In a Triangle Experiment which is a 2page (be careful with the scissors) activity to demonstrate that the sum of the interior angles in a triangle is 180°
Work through the examples below with your children. Discuss alternate ways of finding the missing angles as there are often other ways of finding them.
What are the measures of the angles located at positions a, b, & c? Note: the figure is symmetrical on the vertical through angle b. 

The large triangle is an isosceles triangle. The two angles on the base are equal.  Angle a = 35° 
We now know two angles in the largest triangle. The third angle, angle b must add to these to make 180°.  35° + 35° + b = 180° b = 180°  70° b = 110° 
We now know two angles in a quadrilateral. The two unknown angles, including angle c are equal. All four angles add up to 360°.  2c + 110° + 120° = 360° 2c = 360°  230° 2c = 130° c = 65° 
What are the measures of the angles located at positions a, b, c, d, e, & f? Note: the figure is symmetrical on the vertical through angle f. 

Looking along the base of the large triangle. Angle a added to 75° and 70° equals 180°.  a + 75° + 70° = 180° a = 180°  145° Angle a = 35° 
Looking at the small triangle that includes angle b we now know that a = 35° and, since the sum of the triangles equal 180° we can find angle b.  35° + 95° + b = 180° b = 180°  130 ° b = 50° 
Angle c and the 95° angle are supplementary; they add up to 180°.  c + 95° = 180° Angle c = 85° 
Now we know c = 85° we can find angle d since the three angles in the triangle add up to 180°.  85° + 70 ° + d = 180° d = 180°  155 ° d = 25° 
The triangle in the middle is isosceles so the angles on the base are equal and together with angle f, add up to 180°.  e = 75° 75° + 75° + f = 180° f = 180°  150° f = 30° 
What are the measures of the angles located at positions a, b, c, d, e, & f? 

Angle a and the 70° angle are opposite angles so they are equal.  Angle a = 70° 
Angle b and the 135° angle are supplementary so they add up to 180°.  135° + b = 180° b = 180°  135° b = 45° 
We now know two angles in a triangle. These two angles along with angle c add up to 180°.  c + 70° + 45° = 180° c = 180°  115° c = 65° 
Angles d and b are alternate angles and, since the two lines are parallel, they are equal.  d = 45° 
Angle e and the 70° angle are corresponding angles and, since the two lines are parallel, they are equal.  e = 70° 
Angle f and angle e are supplementary. They add up to 180°.  70° + f = 180° f = 180°  70° f = 110° 
In the above examples the discovery of each angle followed on from finding other angles. The missing angles will not always be labeled a, b, c, d, etc. so the sequence in which to find angles might not be obvious. Sometimes it is best to just start finding whatever angles you can and keep going until all the ones that are required are found.
Print out the worksheets below to help your children practice finding missing angles.