Question

example 6
decide whether abcde and klqrp are sim
solution
corresponding sides of the pentagons are proportional with a scale factor of $\frac{2}{3}$.
however, this does not necessarily mean the pentagons are similar. use a
similarity transformation to decide whether the pentagons are similar.
a dilation with center a and scale factor $\frac{2}{3}$ moves abcde onto afghj.
then a reflection moves afghj onto klmnp.

klmnp does not exactly coincide with klqrp, because not all the corresponding
angles are congruent. (only $\angle a$ and $\angle k$ are congruent.)
▶ because angle measure is not preserved, the two pentagons are not similar.

self-assessment
1 i dont understand yet. 2 i can do it with help. 3 i can do it on m
refer to the floor tile designs below. in each design, the red outer shape is a regular he

tile design 1
tile design 2

1. decide whether the hexagons in tile design 1 are similar. explain.
2. decide whether the hexagons in tile design 2 are similar. explain.

672 chapter 8 similarity

Explanation:

for Question 7:

Step1: Check side proportionality

The red outer shape is a regular hexagon (all sides congruent, all angles congruent). The inner hexagon has sides marked with congruence symbols matching the ratio of the outer hexagon's sides, so corresponding sides are proportional.

Step2: Check angle congruence

All angles of a regular hexagon are $120^\circ$, and the inner hexagon's marked angles show they are congruent to the outer hexagon's angles.

Step3: Verify similarity condition

For polygons to be similar, all corresponding angles must be congruent, and all corresponding sides must be in proportion. Both conditions are satisfied.

for Question 8:

Step1: Check side proportionality

The red outer shape is a regular hexagon (all sides congruent). The inner octagon has sides marked with congruence symbols, so corresponding sides are proportional to the outer hexagon's sides.

Step2: Check angle congruence

The outer shape is a regular hexagon, so its interior angles are $120^\circ$. The inner shape is an octagon, whose interior angles are $135^\circ$ (calculated by $\frac{(8-2)\times180^\circ}{8}=135^\circ$). Corresponding angles are not congruent.

Step3: Verify similarity condition

Since corresponding angles are not congruent, the polygons cannot be similar.

Answer:

1. The hexagons in Tile Design 1 are similar. All corresponding angles are congruent (all $120^\circ$ for regular hexagons) and all corresponding sides are proportional, satisfying the polygon similarity criteria.
2. The shapes in Tile Design 2 are not similar. The outer shape is a hexagon with $120^\circ$ interior angles, while the inner shape is an octagon with $135^\circ$ interior angles, so corresponding angles are not congruent, violating the similarity condition.