Homework 22 Worksheet
Properties of Isometries
1. Prove that isometries preserve betweenness of points, i.e., if A * B * C, then A' * B' * C'.
2. To prove that an isometry T preserve line segments, we must show that 𝑇 𝐴𝐵 (i.e., the set of
points that 𝐴𝐵 lands on after getting mapped by T, or {𝑃! |𝑃 ∈ 𝐴𝐵 and 𝑇 𝑃 = 𝑃! }) is the
line segment 𝐴′𝐵′. Both 𝑇 𝐴𝐵 and 𝐴′𝐵′ are sets, and the way that equality of sets is usually
proved is through a two part proof. In the first part, we arbitrarily pick an element from one
set and show that it is also an element of the second set. In the second part of the proof, we
arbitrarily pick an element of the second set and show that it is also in the first set.
a. Assume that P' ∈ 𝑇 𝐴𝐵 . Show that P' ∈ 𝐴′𝐵′. (Hint: P' ∈ 𝑇 𝐴𝐵 implies that there is a
point P ∈ 𝐴𝐵 such that T(P) = P'. Use Thm 10.1.7 pts 1-2 on P, A, and B to help you
conclude that P' ∈ 𝐴′𝐵′.)
b. Assume that P' ∈ 𝐴′𝐵′. Show that P' ∈ 𝑇 𝐴𝐵 . (Hint: Use the fact that T-1 is also an
isometry to conclude that A' * P' * B' forces A * P * B, so P is on 𝐴𝐵, which means that
P' ∈ 𝑇 𝐴𝐵 .)
3. Prove that an isometry preserves rays, i.e., that 𝑇(𝐴𝐵) = 𝐴′𝐵′. (Hint: Use the two-part proof
idea from #2 and the definition of ray which says that 𝑃𝑄 = 𝑃𝑄 ∪ {𝑅|𝑃 ∗ 𝑄 ∗ 𝑅}. The way a
ray is defined suggests that you are going to have to consider two cases: When R is on the
segment, and when R is on the part of the ray beyond the segment.}
4. Prove that an isometry preserves lines, i.e., that 𝑇 𝐴𝐵 = 𝐴′𝐵′. (Hint: Use the two-part proof
idea again along with an alternate characterization of the points on a line, namely 𝑃𝑄 =
{𝑅|𝑅 ∗ 𝑃 ∗ 𝑄} ∪ 𝑃𝑄.)
5. Prove that an isometry preserves angles, i.e., that T(∠ABC) = ∠A'B'C', and that ∠𝐴𝐵𝐶 ≅
∠𝐴′𝐵′𝐶′. (Hint: To show the equality, use the idea that an angle consists of two rays with a
common endpoint. To show congruence, consider the triangles ABC and A'B'C'.)
6. Prove that an isometry preserves triangles, i.e., that T(ABC) = A'B'C' and that ABC ≅
A'B'C'.