Unit 1, Lesson 11
Congruence
Accelerated 7
11.1
Warm-up
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 under a translation.
11.2
Activity
For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.
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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).
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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).
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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).
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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).
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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).
11.3
Activity
Here are two congruent shapes with some corresponding points labeled:
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On the bottom figure, draw the points corresponding to , , and , and label them , , and .
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Draw line segments and and measure them. Do the same for segments and and for segments and . What do you notice?
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Do you think there could be a pair of corresponding segments with different lengths? Explain.
Student Lesson Summary
How do we know if two figures are congruent?
- 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.
- Distances between corresponding points on congruent figures are always equal, even for curved shapes. For example, corresponding segments and on these congruent ovals have the same length:
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 there is no correspondence between the figures where the parts have equal measure, that shows that the two figures are not congruent.
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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.
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If two polygons have the same side lengths, but not in the same order, the polygons can’t be congruent.
For example, rectangle can’t be congruent to quadrilateral . 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 can’t be congruent to rectangle . Even though they have the same side lengths in the same order, the angles are different. All angles in are right angles. In , angles and are less than 90 degrees and angles and are more than 90 degrees.
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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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