ANSWER
2nd, 3rd and 6th statements
Step-by-step explanation
We want to select all the correct options.
1. Dilations always increase the length of the line segments: A dilation is a change in the size of a figure, therefore, it is not always an increase in the size of that shape. It could be a decrease as well, so this option is false.
2. Dilations take perpendicular lines to perpendicular lines: Since a dilation does not change the shape (only the size) of a figure, perpendicular lines will be taken to perpendicular lines. This is true.
3. Dilations of an angle are congruent to the original angle: Because the shape of an object does not change when dilated, the angle will be congruent to the original angles. This is true.
4. Dilations increase the measure of angles: As earlier stated, the shape of a figure does not change when dilated, so the angle of the figure does not change. This option is false.
5. Dilations of a triangle are congruent to the original triangle: For a figure to be congruent to another figure, it means that they must have the same shape and the exact same size. A dilation changes the size of the figure, so this is false.
6. Dilations of a triangle are simlar to the original shape: For a figure to be similar to another figure, it must have the same shape but not necesarily the same size as that figure. Dilation changes the size but not the shape of a figure, so this is true.
The true statements are the 2nd, 3rd and 6th statements.