Unit 1, Lesson 11
What Is the Same?
Grade 8
11.1
Warm-up
A person’s hands are mirror images of each other. In the diagram, a left hand is labeled. Shade all of the right hands.
11.2
Activity
For each pair of shapes, decide whether or not they are the same.
Five pairs of figures, A, B, C, D and E. Figure A shapes match, the shape on the right is the image of the left shape after a reflection and translation. Figure B shapes do not match, they have congruent sides but angle measures do not match. Figure C shapes match, the shape on the left is the image of the left shape after a reflection. Figure D shapes do not match, the shape on the left is bigger. Figure E shapes do not match, the puzzle shape on the right has a lower right side part tab than the same tab of the left puzzle shape.
Five pairs of figures, A, B, C, D and E. Figure A shapes match, the shape on the right is the image of the left shape after a reflection and translation. Figure B shapes do not match, they have congruent sides but angle measures do not match. Figure C shapes match, the shape on the left is the image of the left shape after a reflection. Figure D shapes do not match, the shape on the left is bigger. Figure E shapes do not match, the puzzle shape on the right has a lower right side part tab than the same tab of the left puzzle shape.
Five pairs of figures, A, B, C, D and E. Figure A shapes match, the shape on the right is the image of the left shape after a reflection and translation. Figure B shapes do not match, they have congruent sides but angle measures do not match. Figure C shapes match, the shape on the left is the image of the left shape after a reflection. Figure D shapes do not match, the shape on the left is bigger. Figure E shapes do not match, the puzzle shape on the right has a lower right side part tab than the same tab of the left puzzle shape.
Five pairs of figures, A, B, C, D and E. Figure A shapes match, the shape on the right is the image of the left shape after a reflection and translation. Figure B shapes do not match, they have congruent sides but angle measures do not match. Figure C shapes match, the shape on the left is the image of the left shape after a reflection. Figure D shapes do not match, the shape on the left is bigger. Figure E shapes do not match, the puzzle shape on the right has a lower right side part tab than the same tab of the left puzzle shape.
Five pairs of figures, A, B, C, D and E. Figure A shapes match, the shape on the right is the image of the left shape after a reflection and translation. Figure B shapes do not match, they have congruent sides but angle measures do not match. Figure C shapes match, the shape on the left is the image of the left shape after a reflection. Figure D shapes do not match, the shape on the left is bigger. Figure E shapes do not match, the puzzle shape on the right has a lower right side part tab than the same tab of the left puzzle shape.
11.3
Activity
- Which of these rectangles have the same area as Rectangle R but a different perimeter?
- Which rectangles have the same perimeter as Rectangle R but a different area?
- Which rectangles have the same area and the same perimeter as Rectangle R?
- Decide which rectangles are congruent. Shade congruent rectangles with the same color.
Student Lesson Summary
Congruent is a new term for an idea we have already been using. We say that two figures are congruent if one can be lined up exactly with the other by a rigid transformation.
For example, triangle is congruent to triangle because they can be matched up by reflecting triangle across followed by the translation shown by the arrow. Notice that all corresponding angles and side lengths are equal.
Triangle A B C, triangle A B prime C, dashed arrow from B prime to point F and triangle E F D. Triangle A B prime C is the image of triangle A B C after reflection over A C. Angle A is 63 point 4 degrees, angle B is 71 point 6 degrees and angle C is 45 degrees. Side A B is 2 point 2 4, side B C is 2 point 8 3 and side A C is 3 point 0 0. Angle E is 63 point 4 degrees, angle F is 71 point 6 degrees and angle D is 45 degrees. Side E F is 2 point 2 4, side F D is 2 point 8 3 and side D E is 3 point 0 0.
Here are some other facts about congruent figures:
- We don’t need to check all the measurements to prove two figures are congruent. We just have to find a rigid transformation that matches up the figures.
- A figure that looks like a mirror image of another figure can be congruent to it. This means there must be a reflection in the sequence of transformations that matches up the figures.
- Since two congruent polygons have the same area and the same perimeter, one way to show that two polygons are not congruent is to show that they have a different area or perimeter.
One figure is congruent to another if it can be moved with translations, rotations, and reflections to fit exactly over the other.
In this figure, Triangle A is congruent to Triangles B, C, and D.
- A translation takes Triangle A to Triangle B.
- A rotation takes Triangle B to Triangle C.
- A reflection takes Triangle C to Triangle D.