Are these triangles similar? If YES, which theorem proves them similar?

The base of a ladder 20 feet long is placed on the ground 5 feet from the side of a house. How high up on the house does the top of the ladder reach?

What is the inverse of the following statement? If a figure has circular bases that are parallel, then it is a cylinder.

What is the perimeter of this triangle?

PQRS is a square. PR=16. What is the length of RS?

An aquarium curator is expanding her facility. Her old aquarium is a cylinder that is 20 feet tall with a diameter of 50 ft. The new aquarium is a similar cylinder with diameter of 100 feet. How much more water will the new one hold?

A triangular garden is dug, making a 90 degree angle at one end of a house and a 30 degree angle at the other end. If the house is 30√3 feet wide, what is the perimeter of the garden?

For parallelogram JKLM, find the value of x.

Given segment x is parallel to segment y, find the value of x.

Write a congruency statement for these triangles.

Find the value of x.

What is the lateral area of the cone? Put your answer in π form.