Similar Triangles Mastery

Two flagpoles are standing, and the sun casts shadows of both poles on the ground. If the first flagpole is 15 feet tall with a shadow of 20 feet and the second flagpole has a shadow of 30 feet, how tall is the second flagpole, assuming the sun's rays are at the same angle for both?

• A) 22.5 feet

• B) 25 feet

• C) 30 feet

• D) 45 feet

In the context of the AA similarity theorem, which of the following is a correct statement?

• A) Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.

• B) Two triangles are similar if all three angles of one triangle are congruent to all three angles of another triangle.

• C) Two triangles are similar if one angle of one triangle is congruent to one angle of another triangle.

• D) Two triangles are similar if two angles of one triangle are proportional to two angles of another triangle.

Which of the following conditions would prove that two triangles are similar by the SAS similarity theorem?

• A) Two pairs of corresponding sides are proportional and the included angles are congruent.

• B) Two pairs of corresponding sides are congruent and the included angles are proportional.

• C) Two pairs of corresponding angles are congruent and the included side is proportional.

• D) Two pairs of corresponding sides are proportional and the included angles are proportional.

A photographer is using a camera tripod that casts a shadow of 4 feet when a tree nearby casts a shadow of 12 feet. If the tripod is 3 feet tall, how tall is the tree?

• A) 6 feet

• B) 9 feet

• C) 12 feet

• D) 36 feet

Which of the following is NOT a criterion for triangle similarity?

• A) AA

• B) SAS

• C) SSS

• D) SSA

On a map, two highways are represented by lines that are parallel. If two rest stops are located such that they form a pair of similar triangles with the highways, which similarity theorem is most directly applicable?

• A) AA

• B) SAS

• C) SSS

• D) None of the above

A child looking at the top of a building observes the angle of elevation to be 30 degrees. If the child is 4 feet tall and stands 6 feet away from the base of the building, can you determine the height of the building using similar triangles?

• A) Yes, using the AA similarity theorem.

• B) Yes, using the SAS similarity theorem.

• C) Yes, using the SSS similarity theorem.

• D) No, there is not enough information.

Two triangles have side lengths in the ratio 3:4:5 and 6:8:10, respectively. Are these triangles similar, and if so, by which theorem?

• A) Yes, by the AA theorem.

• B) Yes, by the SAS theorem.

• C) Yes, by the SSS theorem.

• D) No, they are not similar.

A bridge is constructed with support wires attached at different points along the height of two parallel towers. If the wires form triangles with the ground, what can be said about these triangles?

• A) They are similar by the AA theorem because the towers are parallel.

• B) They are similar by the SAS theorem because the wires are proportional.

• C) They are similar by the SSS theorem because the towers and ground form proportional segments.

• D) They are not necessarily similar.

A student notices that the shadow of a tree forms a triangle on the ground that seems to have the same shape as a triangle drawn on a piece of paper. If the angles of the paper triangle are 30, 60, and 90 degrees, which of the following must be true for the tree shadow to be similar to the triangle on the paper?

• A) The shadow must form a right angle.

• B) The shadow must have angles of 30 and 60 degrees.

• C) The shadow must have sides in the same ratio as the paper triangle.

• D) All of the above must be true.