Unit 1, Lesson 11
Congruence
Accelerated 7
11.1
Warm-up
Translated Images
All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle
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All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle
For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.
Two figures E F G H and A B C D on a coordinate plane. Figure H G F E is the image of figure A B C D after rotation of 180 degrees, followed by a translation left 3 and down 2 units. A B C D has the coordinates A(5 comma 0), B(4 comma 2), C(3 comma 1) and D(1 comma 1). H G F E has the coordinates H(negative 6 comma 0), G(negative 5 comma negative 2), F(negative 4 comma negative 1 and E(negative 2 comma negative 1).
Two pentagons on a coordinate plane. Pentagon ABCDE is non-convex and has coordinates A(negative 4 comma 0), B(negative 2 comma 1), C(negative two comma 4), D(negative 3 comma 3) and E(negative 4 comma 3). Pentagon FGHIJ is convex and has coordinates F(1 comma 2), G(3 comma 1), H(4 comma 2), I(4 comma 4) and J(2 comma 4).
Two triangles D E F and A B C on a coordinate plane. Triangle D E F is the image of triangles A B C after rotation of 90 degrees, followed by a translation left 3 and down 2 units. Triangle A B C has the coordinates A(2 comma negative 2), B(6 comma 0) and C(6 comma 2). Triangle D E F has the coordinates D(negative 1 comma 0), E(negative 3 comma 4) and F(negative 5 comma 4).
Two figures A and B on a coordinate plane. Figure A has points G(negative 4 comma 3), H(negative 1 comma 3), I(negative 1 comma 0) and J(negative 3 comma 0). Figure B has points P(3 comma negative 1), Q(3 comma 2), R(6 comma 2) and S(6 comma 1).
Two figures, square A and rhombus B on a coordinate plane. Figure A is a square with side lengths 3 units and has points G(negative 5 comma 1), H(negative 2 comma 1), I(negative 2 comma 4) and J(negative 5 comma 4). Figure B is a rhombus with side lengths near 2 point 8 units and has points P(3 comma 3), Q(5 comma 5), R(7 comma 3) and S(5 comma 1).
Here are two congruent shapes with some corresponding points labeled:
On the bottom figure, draw the points corresponding to
Draw line segments
Do you think there could be a pair of corresponding segments with different lengths? Explain.
Two congruent ovals on a square grid. In the first oval, two points on opposite longer sides of the oval are labeled A and B and are connected by a line segment. In the second oval, two points on opposite longer sides of the oval are labeled A prime and B prime and are connected by a line segment. Let the lower left corner be (0 comma 0). Then point A is near (5 comma 4 point 75) and point B is near (1 comma 3 point 7 5). Point A prime is near (9 comma 5) and point B prime is near (10 point 5 comma 1 point 3).
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
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
If two figures have different corresponding distances, they can’t be congruent.
For example on both ovals, the longest distance is 5 units across, and the longest distance from top to bottom is 4 units. The line segment from the highest to lowest point is in the middle of the left oval, but in the right oval, it’s 2 units from the right end and 3 units from the left end. This shows they are not congruent.
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