Unit 1 Rigid Transformations And Congruence — Unit Plan

Title	Takeaways	Assessment
Lesson 1 Moving in the Plane	—	Frame to Frame (1 problem) Here are positions of a shape: Describe how the shape moves from: Frame 1 to Frame 2. Frame 2 to Frame 3. Frame 3 to Frame 4. Show Solution Slide down Turn counterclockwise 90 degrees (or one quarter of a full turn) Slide up
Lesson 2 Naming the Moves	—	Is It a Reflection? (1 problem) What type of move takes Figure A to Figure B? Explain your reasoning. Show Solution Sample responses: The move is 1 rotation. If Figure A is turned around the point shared by Figures A and B, it can land on Figure B. The move is 2 reflections. If Figure A is flipped over line $\ell$ and then flipped over again so that the shared points and angle line up, then it can land on Figure B.
Lesson 3 Grid Moves	—	Triangle Images (1 problem) Translate triangle $ABC$ so that $B$ goes to $B’$ . Reflect triangle $ABC$ over line $\ell$ . Show Solution
Lesson 4 Making the Moves	—	What Does It Take? (1 problem) For each description of a transformation, identify what information is missing. Translate triangle $ABC$ to the right. Rotate triangle $ABC$ $90^\circ$ around point $C$ . Reflect triangle $ABC$ over a line. Show Solution Sample responses: Distance—how many units to the right Direction—clockwise or counterclockwise A drawing or description of where the line is
Lesson 5 Coordinate Moves	—	Rotation or Reflection (1 problem) One of the triangles pictured is a rotation of triangle $ABC$ and one of them is a reflection. Triangle A B C reflected on a coordinate plane, origin O. Horizontal axis scale negative 6 to 6 by 1’s. Vertical axis scale negative 5 to 5 by 1’s. Triangle A B C is blue and has coordinates: A(1 comma 1), B(3 comma 2) and C(2 comma 5). The green triangle has coordinates: (negative 1 comma 1), (negative 2 comma 3) and (negative 5 comma 2). The red triangle has coordinates: (1 comma negative 1), (3 comma negative 2) and (2 comma negative 5). Label the rotated image $PQR$ . Label the reflected image $XYZ$ . Show Solution
Lesson 6 Describing Transformations	—	Describing a Sequence of Transformations (1 problem) Triangle T' is the image of Triangle T. Han gave this information to Jada to describe the sequence of transformations. Triangle T is reflected over line $\ell$ . Triangle T is translated 2 units to the left. The order of the sequence of transformations is translation, then reflection. Which of these figures shows the correct Triangle T'? Figure 1 Figure 2 Show Solution Figure 2
Section A Check Section A Checkpoint		Problem 1 Here is line segment $AB$ and a point $C$ . Reflect line segment $AB$ across the $x$ -axis. What are the coordinates of the new endpoints? Point $C$ is translated 3 units to the left and 2 units up. Plot this point on the grid and label it $C’$ . Show Solution The image of $A$ is at $(\text-4,\text-5)$ and the image of $B$ is at $(3, \text-2)$ . Problem 2 Here are 2 figures. Describe a sequence of transformations that takes triangle $ABC$ to triangle $DEF$ . Show Solution Sample response: Translate triangle $ABC$ so that $A$ moves to $D$ . Rotate 90 degrees counterclockwise around point $D$ .

Lesson 7 No Bending or Stretching	—	Translated Trapezoid (1 problem) Trapezoid $A’B’C’D’$ is the image of trapezoid $ABCD$ under a rigid transformation. Trapezoid A B C D and its image, trapezoid A prime B prime C prime and D prime. Angle A is 130 degrees, angle B is 50 degrees and angles D and C are right angles. Side A prime D prime is 6 units and side D prime C prime is 4 units. Label all vertices on trapezoid $A’B’C’D’$ . On both figures, label all known side lengths and angle measures. Show Solution
Lesson 8 Rotation Patterns	—	Is It a Rotation? (1 problem) Triangle $ABC$ is rotated $180^\circ$ around point $C$ . Will the image line up with triangle $CDE$ ? Explain how you know. Show Solution No. Sample response: If triangle $CDE$ was a $180^\circ$ rotation of triangle $ABC$ , then line segment $AB$ would be parallel to line segment $DE$ .
Lesson 9 Moves in Parallel	—	Finding Unknown Measurements (1 problem) Points $A’$ and $B’$ are the images of $A$ and $B$ after a $180^\circ$ rotation around point $O$ . Answer each question and explain your reasoning without measuring segments or angles. Name a segment whose length is the same as segment $AO$ . What is the measure of angle $A'OB'$ ? Show Solution Segment $A’O$ , because $A’$ is the image of $A$ after a $180^\circ$ rotation with center at $O$ . This rotation preserves distances and takes segment $AO$ to segment $A’O$ . $79^\circ$ , the same measure as $\angle AOB$ , because the $180^\circ$ rotation with center at $O$ takes $\angle AOB$ to $\angle A’OB’$ . The rotation preserves angle measures.
Lesson 10 Composing Figures	—	Identifying Side Lengths and Angle Measures (1 problem) Two quadrilaterals, A B C D on the left and A B C E on the right. Both have segment A C, and quadrilateral A B C E has midpoint M on A C. Quadrilateral A B C D has side A B length 2 point 7, angle B is 64 point 3 degrees and side B C is 3 point 2. Quadrilateral has A B C E has side A B length 2 point 7, angle B is 64 point 3 degrees and side B C is 3 point 2. Point E on quadrilateral A B C E is about the same level as the midpoint M and angle B. Point D on quadrilateral A B C D is above the level of the angle B and the midpoint of segment A C. Here is a diagram showing triangle $ABC$ and some transformations of triangle $ABC$ . On the left side of the diagram, triangle $ABC$ has been reflected across line $AC$ to form quadrilateral $ABCD$ . On the right side of the diagram, triangle $ABC$ has been rotated $180^\circ$ using midpoint $M$ as a center to form quadrilateral $ABCE$ . Using what you know about rigid transformations, side lengths and angle measures, label as many side lengths and angle measures as you can in quadrilaterals $ABCD$ and $ABCE$ . Show Solution
Section B Check Section B Checkpoint		Problem 1 Here is a line segment $CD$ with midpoint $M$ . Rotate segment $CD$ $90^\circ$ clockwise around point $M$ , and label the image as $FG$ . Rotate segment $CD$ $180^\circ$ around point $E$ and label the new segment $HJ$ . Which segment is parallel to segment $CD$ ? Show Solution See image See image $HJ$ is parallel to $CD$ Problem 2 Triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation. Describe a rigid transformation that takes $ABC$ to $EDC$ . Name 2 angles that have the same measure and explain how you know. Name 2 side lengths that must be the same and explain how you know. Show Solution Sample responses: Rotate triangle $ABC$ $180^\circ$ around point $C$ . Angle $A$ is the same as angle $E$ , or angle $B$ is the same as angle $D$ , or angle $ACB$ is the same as angle $ECD$ ; since triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation, the corresponding angles are the same measure. $AB$ is the same length as $ED$ , or $BC$ is the same length as $DC$ , or $AC$ is the same length as $EC$ ; since triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation, the corresponding side lengths are the same.

Lesson 11 What Is the Same?	—	Mirror Images (1 problem) Figure B is the image of Figure A when reflected across line $\ell$ . Are Figure A and Figure B congruent? Explain your reasoning. Show Solution Yes, they are congruent. There is a rigid transformation that takes one figure to the other, so they are congruent.
Lesson 12 Congruent Polygons	—	Moving to Congruence (1 problem) Describe a sequence of reflections, rotations, and translations that shows that quadrilateral $ABCD$ is congruent to quadrilateral $EFGH$ . Two figures, trapezoids A B C D and E F G H on a square grid. Let the lower left corner be (0 comma 0), Then trapezoid A B C D has points A(1 comma 6), B(1 comma 2), C(6 comma 2) and D(2 comma 6). Trapezoid E F G H has points E(7 comma 6), F(11 comma 6), G(11 comma 1) and H(7 comma 6). Show Solution Sample response: Translate $ABCD$ down 1 and 5 to the right. Then reflect over line $GH$ .
Lesson 13 Congruence	—	Explaining Congruence (1 problem) Are Figures A and B congruent? Explain your reasoning. Show Solution These figures are not congruent. Sample reasoning: If they were congruent, the longest horizontal distances between two points would be the same. However, for A it is less than 4 units, and for B it is about 4 units.
Section C Check Section C Checkpoint		Problem 1 Which shape is congruent to Shape A? Describe a rigid transformation that takes A to that figure. Which shape is not congruent to Shape A? Explain how you know. Show Solution Shape B is congruent to Shape A. Sample response: Translate Shape A 4 units right, then rotate $90^\circ$ clockwise around $(2, 1)$ . Shape C is not congruent to Shape A. Sample response: Shape C has 2 side lengths of 3, but Shape A has no side lengths of 3, so they cannot be congruent.

Lesson 14 Alternate Interior Angles	—	All the Rest (1 problem) The diagram shows two parallel lines cut by a transversal. One angle measure is shown. Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angles are marked in clockwise order as a degrees, b, degrees, c degrees, and 54 degrees. At the second intersection, angles are marked in clockwise order as e degrees, f degrees, g degrees, and d degrees. Find the values of $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ . Show Solution $a$ : $126$ , $b$ : $54$ , $c$ : $126$ , $d$ : $54$ , $e$ : $126$ , $f$ : $54$ , $g$ : $126$
Lesson 15 Adding the Angles in a Triangle	—	Three Angles (1 problem) Tyler has 3 right angles. Can he use them to make a triangle? Explain your reasoning. Show Solution No. Sample reasoning: 3 right angles sums to more than 180 degrees, since $3\boldcdot90=270$ .
Lesson 16 Parallel Lines and the Angles in a Triangle	—	Angle Sum (1 problem) What is the sum of the angle measures of triangle $ABC$ ? How do you know? Show Solution $180^\circ$ . Sample response: Since the base and vertex lie on grid lines, we can see the line parallel to $BC$ through $A$ is a straight line. The three angles around point $A$ add up to a straight angle. Using alternate interior angles, two angles are congruent to angle $B$ and angle $C$ , and the third angle is the same as angle $A$ . So angles $A$ , $B$ , and $C$ add up to $180^\circ$ .
Section D Check Section D Checkpoint		Problem 1 Line $FG$ is parallel to line $HJ$ and cut by transversal $m$ . Find each angle measure: $a$ $b$ $c$ $d$ Show Solution $45^\circ$ $135^\circ$ $45^\circ$ $135^\circ$ Problem 2 Line $AB$ is parallel to line $CD$ . Explain how you know that the sum of the angles of triangle $ABC$ is $180^\circ$ . Show Solution Sample response: Angles $ECA$ , $ACB$ , and $BCD$ form a straight angle, which is $180^\circ$ . Angle $ECA$ is congruent to angle $CAB$ because they are alternate interior angles. Angle $BCD$ is congruent to angle $CBA$ because they are alternate interior angles. So the angles in triangle $ABC$ are congruent to the angles that make a straight angle and must also sum to $180^\circ$ .

Lesson 17 Rotate and Tessellate	—	No cool-down
Unit 1 Assessment End-of-Unit Assessment