Exercise 3 Page 652

b Yes, it is a congruence transformation.

Corresponding sides	Corresponding angles
XZ ≅ JL	∠ X ≅ ∠ J
XY ≅ JK	∠ Y ≅ ∠ K
YZ ≅ KL	∠ Z ≅ ∠ L

Practice makes perfect

a Examining the diagram, we notice that the orientation of △ XYZ is upside down when compared to △ JKL. Therefore, to give the triangles the same orientation, we should reflect △ JKL in the x-axis which turns it upside down. When the triangles have the same orientation, we can perform a translation to map the triangles to each other.

Reflection in the x-axis

If (a,b) is reflected in the x-axis, then its image is the point (a,- b). preimage (a,b) → image (a,- b) Using this rule, we can determine the coordinates of our image.

Point	(a,b)	(a,- b)
J	(- 3,2)	(- 3,- 2)
K	(- 2,4)	(- 2,- 4)
L	(0,2)	(0,- 2)

Now we can graph the image of △ JKL after reflecting it in the x-axis.

Translation

The coordinates of △ J'K'L' has the same vertical position as △ XYZ. Therefore, we only have to translate △ J'K'L' horizontally by 4 units to the right: preimage (a,b) → image (a+4,b) By performing this translation, we can map the triangles to each other.

The composition of transformations that maps △ JKL to △ XYZ is: Reflection:& in the $x-$axis. Translation:& (a,b) → (a+4,b)

b Another name for a rigid motion or a combination of rigid motions is a congruence transformation. Since both reflections and translations preserves angle and side measures, these transformations are in fact rigid motions. Therefore, we know that the composition is a congruence transformation. Let's identify the corresponding parts

&Corresponding sides &&Corresponding angles &XZ ≅ JL && ∠ X ≅ ∠ J &XY ≅ JK && ∠ Y ≅ ∠ K &YZ ≅ KL && ∠ Z ≅ ∠ L

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Exercises p.652-653

Exercise 3 Page 652

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Reflection in the x-axis

Translation