The values of the variables and the rule that proves the triangle are congruent are as follows;
1. x = 62°, y = 59°
2. x = 25
3. x = 13
4. Side Angle Angle, SAA, congruence rule
The details of the steps used to find the values of the variables and the congruence rule are as follows;
1. The base angles of the isosceles triangle indicates that we get;
x = 180 - 2 × 59
x = 62°
y = (180 - 62)/2
y = 59°
2. The equilateral triangle property indicates that we get;
2·x + 10 = 60
x = (60 - 10)/2
x = 25
3. The triangles are congruent by SSS congruence rule, therefore;
5·x + 3·x + 2 + 74 = 180
8·x + 76 = 180
x = (180 - 76)/8
x = 13
4. The law of sines indicates that the ratio of the sine of the angle with one congruent marking to the side with two congruent markings is equivalent to the ratio of the sine of the angle facing the side with one congruent marking to the length of the side with one congruent marking
Where the lengths of the sides are congruent, then the sine of the angles and therefore the angles in the equation for the law of sines relationship, therefore, the triangles can be said to be congruent by the Angle Angle Side congruent rule