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Lesson 3.4.6Lesson 3.4.6
Congruence andSimilarity
Congruence andSimilarity
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Lesson
3.4.6Congruence and SimilarityCongruence and Similarity
California Standard:Measurement and Geometry 3.4Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
What it means for you:You’ll learn the meaning of the terms congruent and similar. You’ll find out how to tell if two shapes are congruent, similar, or neither.
Key words:• congruent• similar• size• shape• scale factor
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Congruence and SimilarityCongruence and SimilarityLesson
3.4.6
Congruent figures are shapes that are exactly the same size and shape as each other.
That means that if you could lift them off the page, there would always be a way to make them fit exactly on top of each other, just by flipping them over or turning them around.
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Congruence and SimilarityCongruence and Similarity
Congruent Shapes Have the Same Size and Shape
Lesson
3.4.6
Two figures are congruent if they match perfectly when you place them on top of each other.
These pairs of shapes are all congruent.
They can be turned around or flipped over, but they always have the same size, shape, and length of each dimension.
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In pairs 1 and 4, each shape is identical to the other, but upside down.
Congruence and SimilarityCongruence and Similarity
Example 1
Solution follows…
Lesson
3.4.6
Which of these pairs of shapes are congruent? Which are not, and why?
1. 2. 3. 4.
Solution
So pairs 1 and 4 are congruent.
Solution continues…
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Pair 2 is also congruent, as each shape is a mirror image of the other.
Congruence and SimilarityCongruence and Similarity
Example 1
Lesson
3.4.6
Which of these pairs of shapes are congruent? Which are not, and why?
1. 2. 3. 4.
Solution (continued)
The rectangles in pair 3 are the same shape but they’re not the same size, so they’re not congruent.
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 1–4, say whether or not each pair of shapes is congruent. If they are not, give a reason why not.1. 2.
3. 4.
Yes No – different sizes
No – different shapes
Yes
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 5–8, say whether or not each pair of shapes is congruent. If they are not, give a reason why not.5. 6.
7. 8.
No — different shapes No – different shapes
Yes No — different shapes
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Congruence and SimilarityCongruence and Similarity
Congruent Polygons Have Matching Sides and Angles
Lesson
3.4.6
Sometimes two polygons might look quite alike.
You can tell for sure if they’re congruent if you know the measures of their sides and angles.
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Congruence and SimilarityCongruence and Similarity
Example 2
Solution follows…
Lesson
3.4.6
Which two of these quadrilaterals are congruent?
10 cm
10 cm
6.1 cm10 cm
6.4 cm
10 cm
6.4 cm
10 cm
10 cm
10.7 cm 10.6 cm 10.6 cm
69°
111°
70°
110°
70°
110°
1. 2. 3.
Solution
Quadrilaterals 1 and 2 look alike, but you can see from the angle measures and side lengths that they’re not identical.
Solution continues…
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The angle measures tell you that Quadrilateral 3 is a mirror image of Quadrilateral 2.
Congruence and SimilarityCongruence and Similarity
Example 2
Lesson
3.4.6
Which two of these quadrilaterals are congruent?
10 cm
10 cm
6.1 cm10 cm
6.4 cm
10 cm
6.4 cm
10 cm
10 cm
10.7 cm 10.6 cm 10.6 cm
69°
111°
70°
110°
70°
110°
1. 2. 3.
Solution (continued)
So Quadrilaterals 2 and 3 are congruent.
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 9–10, say which two out of each group of shapes are congruent. Give a reason why the other one is not.
9.
10. a & c — b has different side lengths
b & c — a has different angles (and different side lengths)
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 11–12, say which two out of each group of shapes are congruent. Give a reason why the other one is not.
11.
12.
a & b — all have the same angles, but in c they are in a different order, giving different side lengths.
a & c — b is a different size
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Congruence and SimilarityCongruence and Similarity
Similar Figures Can Be Different Sizes
Lesson
3.4.6
Similar figures have angles of the same measure and have the same shape as each other, but they can be different sizes.
So two figures are similar if you can apply a scale factor and get a congruent pair.
These pairs of shapes are all similar.
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Pair 2 is not a similar pair. The shapes are different — they have different angles.
Congruence and SimilarityCongruence and Similarity
Example 3
Solution follows…
Lesson
3.4.6
Which of these pairs of shapes are similar?
Solution
Pair 1 is a similar pair. They are both squares, and the only difference is the size.
1. 2. 3. 4.
Solution continues…
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Pair 3 is not a similar pair. You can’t multiply either of them by a scale factor to get a rectangle congruent to the other.
Congruence and SimilarityCongruence and Similarity
Example 3
Lesson
3.4.6
Which of these pairs of shapes are similar?
Solution (continued)
Pair 4 is a similar pair. If you multiply the smaller triangle by a scale factor of 2, you will get a triangle congruent to the larger one.
1. 2. 3. 4.
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 13–15, say whether or not each pair of shapes is similar.
13. 14. 15.
Yes No — different shapes No — different shapes
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Congruence and SimilarityCongruence and Similarity
Guided Practice
Solution follows…
Lesson
3.4.6
In Exercises 16–18, say whether or not each pair of shapes is similar.
16. 17. 18.
Yes No — different angles Yes
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Congruence and SimilarityCongruence and Similarity
Independent Practice
Solution follows…
Lesson
3.4.6
Use the triangles shown to answer Exercises 1–4.
1. Which triangle is congruent to triangle 1?
2. Which triangle is similar to triangle 6?
1 2 3
45
6 7
8
9
3. Which triangle is congruent to triangle 4?
4. Which two triangles are similar to triangle 3?
Triangle 9
Triangle 8
Triangle 2
Triangles 1 and 9
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Congruence and SimilarityCongruence and Similarity
Independent Practice
Solution follows…
Lesson
3.4.6
In Exercises 5–6, identify each pair of shapes as congruent, similar, or neither. Explain your answers.
5. 6.
Congruent. They are the same size and shape.
Neither. The triangles are not the same shape or size.
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Congruence and SimilarityCongruence and Similarity
Independent Practice
Solution follows…
Lesson
3.4.6
In Exercises 7–8, identify each pair of shapes as congruent, similar, or neither. Explain your answers.
7. 8.
Similar. The sides are in proportion to one another (and the angles are the same)
Congruent. The triangles are the same size and shape.
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Congruence and SimilarityCongruence and Similarity
Independent Practice
Solution follows…
Lesson
3.4.6
9. Explain the difference between congruency and similarity when examining two figures.
10. Triangle ABC has sides measuring 5 in, 6 in, and 8 in. Write the side lengths of a triangle that would be similar to ABC.
Congruent shapes have the same shape and size. Similar shapes have the same shape, but not the same size.
Any lengths that are multiples of (5, 6, 8), like (10, 12, 16) or (0.5, 0.6, 0.8).
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Congruence and SimilarityCongruence and Similarity
Independent Practice
Solution follows…
Lesson
3.4.6
11. "You can tell whether two shapes are congruent just by looking at the lengths of the sides. It is not necessary to look at the measures of the angles."
Is this statement true or false? Give a reason why.False (except for triangles). For example, a rectangle and a parallelogram could have the same lengths, but these two figures are not congruent.
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Congruence and SimilarityCongruence and Similarity
Round UpRound Up
Lesson
3.4.6
You’ll learn more about congruence and similarity — particularly with triangles — in later grades.
For now, make sure you know what each term means, and don’t forget which is which.
