Unit 1 Rigid Transformations And Congruence — Unit Plan

Here are the moves we have learned about so far:

• A translation slides a figure without turning it. Every point in the figure goes the same distance in the same direction. For example, Figure A was translated down and to the left, as shown by the arrows. Figure B is a translation of Figure A.

  

• A rotation turns a figure about a point, called the center of the rotation. Every point on the figure goes in a circle around the center and makes the same angle. The rotation can be clockwise, going in the same direction as the hands of a clock, or counterclockwise, going in the other direction. For example, Figure A was rotated $45^\circ$ clockwise around its bottom vertex. Figure C is a rotation of Figure A.

  

• A reflection places points on the opposite side of a reflection line. The mirror image is a backwards copy of the original figure. The reflection line shows where the mirror should stand. For example, Figure A was reflected across the dotted line. Figure D is a reflection of Figure A.

  

A transformation is a translation, rotation, reflection, or dilation, or a combination of these. To distinguish an original figure from its image, points in the image are sometimes labeled with the same letters as the original figure, but with the symbol $’$ attached, as in $A’$ (pronounced “A prime”).

• A translation can be described by two points. If a translation moves point $A$ to point $A’$, it moves the entire figure the same distance and direction as the distance and direction from $A$ to $A’$. The distance and direction of a translation can be shown by an arrow.

  For example, here is a translation of quadrilateral $ABCD$ that moves $A$ to $A’$.

  

• A rotation can be described by an angle and a center. The direction of the angle can be clockwise or counterclockwise.

  For example, hexagon $ABCDEF$ is rotated $90^\circ$ counterclockwise using center $P$.

  

• A reflection can be described by a line of reflection (the “mirror”). Each point is reflected directly across the line so that it is just as far from the mirror line, but is on the opposite side.

  For example, pentagon $ABCDE$ is reflected across line $m$.

  

When we do one or more moves in a row, we often call that a sequence of transformations. For example, a sequence of transformations taking Triangle A to Triangle C is to translate Triangle A 4 units to the right, then reflect over line $\ell$.

There may be more than one way to describe or perform a transformation that results in the same image. For example, another sequence of transformations that would take Triangle A to Triangle C would be to reflect over line $\ell$, then translate Triangle A​′ 4 units to the right.

We can use coordinates to describe points and find patterns in the coordinates of transformed points.

We can describe a translation by expressing it as a sequence of horizontal and vertical translations.

For example, segment $AB$ is translated right 3 and down 2.

Quadrilateral on a coordinate plane. Horizontal axis scale negative 3 to 5 by 1’s. Vertical axis scale negative 2 to 3 by 1’s. Quadrilateral A B B prime A prime has coordinates A(negative 2 comma 1), B(1 comma 2), B prime(4 comma 0) and A prime (1 comma negative 1).

Reflecting a point across an axis changes the sign of one coordinate.

For example, reflecting the point $A$ whose coordinates are $(2,\text-1)$ across the $x$-axis changes the sign of the $y$-coordinate, making its image the point $A’$ whose coordinates are $(2,1)$. Reflecting the point $A$ across the $y$-axis changes the sign of the $x$-coordinate, making the image the point $A’’$ whose coordinates are $(\text-2,\text-1)$.

3 points on a coordinate plane. Horizontal axis scale negative 3 to 5 by 1’s. Vertical axis scale negative 2 to 2 by 1’s. The points have these coordinates: A(2 comma negative 1), A prime(2 comma 1) and A double prime (negative 2 comma negative 1).

Reflections across other lines are more complex to describe.

We don’t have the tools yet to describe rotations in terms of coordinates in general. Here is an example of a $90^\circ$ rotation with center $(0,0)$ in a counterclockwise direction.

Point $A$ has coordinates $(0,0)$. Segment $AB$ is rotated $90^\circ$ counterclockwise around $A$. Point $B$ with coordinates $(2,3)$ rotates to point $B’$ whose coordinates are $(\text-3,2)$.

Segment A B rotated on a coordinate plane, origin O. Horizontal axis scale negative 4 to 4 by 1’s. Vertical axis scale negative 2 to 4 by 1’s. The segments have these coordinates: A(0 comma 0), B(2 comma 3) and B prime (negative 3 comma 2). Angle B A B prime is a right angle.

The transformations we’ve learned about so far, translations, rotations, reflections, and sequences of these motions, are all examples of rigid transformations. A rigid transformation is a move that doesn’t change measurements on any figure.

Earlier, we learned that a figure and its image have corresponding points. With a rigid transformation, figures like polygons also have corresponding sides and corresponding angles. These corresponding parts have the same measurements.

For example, triangle $EFD$ was made by reflecting triangle $ABC$ across a horizontal line, then translating. Corresponding sides have the same lengths, and corresponding angles have the same measures.

Earlier, we learned that if we apply a sequence of rigid transformations to a figure, then corresponding sides have equal length and corresponding angles have equal measure. These facts let us figure out things without having to measure them!

For example, here is triangle $ABC$.

We can reflect triangle $ABC$ across side $AC$ to form a new triangle:

Because points $A$ and $C$ are on the line of reflection, they do not move. So the image of triangle $ABC$ is $AB'C$. We also know that:

• Angle $B'AC$ measures $36^\circ$ because it is the image of angle $BAC$.
• Segment $AB'$ has the same length as segment $AB$.

When we construct figures using copies of a figure made with rigid transformations, we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles.

• If we copy one figure on tracing paper and move the paper so the copy covers the other figure exactly, then that suggests they are congruent.
• If we can describe a sequence of translations, rotations, and reflections that move one figure onto the other so they match up exactly, they are congruent.

• If there is no correspondence between the figures where the parts have equal measure, that shows that the two figures are not congruent.

  • If two polygons have different sets of side lengths, they can’t be congruent.

    For example, the figure on the left has side lengths 3, 2, 1, 1, 2, 1. The figure on the right has side lengths 3, 3, 1, 2, 2, 1. There is no way to make a correspondence between them where all corresponding sides have the same length.

    

  • If two polygons have the same side lengths, but not in the same order, the polygons can’t be congruent.

    For example, rectangle $ABCD$ can’t be congruent to quadrilateral $EFGH$. Even though they both have two sides of length 3 and two sides of length 5, they don’t correspond in the same order.

    

    Two figures, A B C D and E F G H. Figure A B C D is a rectangle with side length 3 and base and top length 5. Figure E F G H has base length 5, top length is 3, left side length is 3 and right side length is 5.

  • If two polygons have the same side lengths, in the same order, but different corresponding angles, the polygons can’t be congruent.

    For example, parallelogram $JKLM$ can’t be congruent to rectangle $ABCD$. Even though they have the same side lengths in the same order, the angles are different. All angles in $ABCD$ are right angles. In $JKLM$, angles $J$ and $L$ are less than 90 degrees and angles $K$ and $M$ are more than 90 degrees.

    

When two lines intersect, vertical angles are congruent, and adjacent angles are supplementary, so their measures sum to 180. For example, in this figure angles 1 and 3 are congruent, angles 2 and 4 are congruent, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.

When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles are created. (“Interior” means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.

Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.

Alternate interior angles are equal because a $180^\circ$ rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point $M$ halfway between the two intersections. Can you see how rotating $180^\circ$ about $M$ takes angle 3 to angle 5?

Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is $70^\circ$ we use vertical angles to see that angle 3 is $70^\circ$, then we use alternate interior angles to see that angle 5 is $70^\circ$, then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is  $110^\circ$ since $180 -70 = 110$. It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure $70^\circ$, and angles 2, 4, 6, and 8 measure $110^\circ$.

A $180^\circ$ angle is called a straight angle because when it is made with two rays, they point in opposite directions and form a straight line.

If we experiment with angles in a triangle, we find that the sum of the measures of the three angles in each triangle is $180^\circ$— the same as a straight angle!

Through experimentation we find:

• If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides, then the three angles form a straight angle.

• If we have a line and two rays that form three angles added to make a straight angle, then there is a triangle with these three angles.