Quadrilateral ABCD is similar to quadrilateral EFGH. Given AB=4cm and EF=6cm, FG=4.5cm, find the ratio of similarity and length of BC. Also determine if all equilateral triangles are similar and if al

Given that quadrilateral $ABCD$ is similar to quadrilateral $EFGH$ ($ABCD \sim EFGH$).
This means their corresponding angles are equal, and the ratio of their corresponding sides is constant.

From the image:
Side $AB = 4 \text{ cm}$
Side $EF = 6 \text{ cm}$
Side $FG = 4.5 \text{ cm}$

• a. All equilateral triangles are similar.

  • True. All equilateral triangles have three $60^\circ$ angles. Since all corresponding angles are equal, and the ratio of sides is constant, they are similar.

• b. All isosceles triangles are similar.

  • False. Isosceles triangles can have different angle measures (e.g., $80^\circ, 80^\circ, 20^\circ$ vs. $50^\circ, 50^\circ, 80^\circ$), so they are not necessarily similar.

• c. All isosceles right triangles are similar.

  • True. All isosceles right triangles have angles $45^\circ, 45^\circ, 90^\circ$. Since all corresponding angles are equal, they are similar.

• d. All rectangles are similar.

  • False. While all rectangles have four $90^\circ$ angles, their side ratios (length to width) can differ (e.g., a $2 \times 4$ rectangle vs. a $2 \times 2$ square).

• e. All rhombuses are similar.

  • False. While all rhombuses have four equal sides, their angles can differ (e.g., a square vs. a rhombus with $60^\circ$ angles).

• f. All squares are similar.

  • True. All squares have four $90^\circ$ angles and four equal sides. Their corresponding angles are equal, and the ratio of their sides is constant.

• g. All congruent polygons are similar.

  • True. Congruent polygons have the same shape and size. This means their corresponding angles are equal, and the ratio of their corresponding sides is $1:1$, which satisfies the conditions for similarity.

• h. All similar polygons are congruent.

  • False. Similar polygons have the same shape but not necessarily the same size. For example, a $2 \times 2$ square and a $4 \times 4$ square are similar but not congruent.

• i. All regular pentagons are similar.

  • True. All regular pentagons have five equal sides and five equal interior angles of $108^\circ$. Since all corresponding angles are equal, and the ratio of sides is constant, they are similar.