1.3.3: Congruence

Lesson

Let's find ways to test congruence of interesting figures.

Exercise \(\PageIndex{1}\): Not Just the Vertices

Trapezoids \(ABCD\) and \(A'B'C'D'\) are congruent.

• Draw and label the points on \(A'B'C'D'\) that correspond to \(E\) and \(F\).
• Draw and label the points on \(ABCD\) that correspond to \(G'\) and \(H'\).
• Draw and label at least three more pairs of corresponding points.

Figure \(\PageIndex{1}\): Two trapezoids A B C D and its image A prime B prime C prime D prime on a square grid. Trapezoid A B C D has point E on A D and point F on C D. Trapezoid A prime B prime C prime D prime has point H prime on A prime B prime and point G prime on A prime D prime. Let the lower left corner be (0 comma 0). Then the coordinates of A B C D are A(1 comma 3), B(2 comma 1), C(4 comma 1), F(4 point 5 comma 1 point 5), D(6 comma 3), E(2 comma 3). The coordinates of A prime B prime C prime D prime are A prime(7 comma 3), H prime(7 point 5 comma 2), B prime(8 comma 1), C prime(10 comma 1), D prime(12 comma 3), G prime(10 comma 3).

Exercise \(\PageIndex{2}\): Congruent Ovals

Are any of the ovals congruent to one another? Explain how you know.

Figure \(\PageIndex{2}\)

Are you ready for more?

You can use 12 toothpicks to create a polygon with an area of five square toothpicks, like this:

Can you use exactly 12 toothpicks to create a polygon with an area of four square toothpicks?

Figure has a vertex at 1 unit right, 2 units up. Another vertex at 1 unit right, 3 units up. Another vertex at 2 units right, 3 units up. Another vertex at 2 units right, 4 units up. Another vertex at 3 units right, 4 units up. Another vertex at 3 units right, 3 units up. Another vertex at 4 units right and 3 units up. Another vertex at 4 units right and 2 units up. Another vertex at 3 units right, 2 units up. Another vertex at 3 units right, 1 unit up. Another vertex at 2 units right, 1 unit up. Another vertex at 2 units right, 2 units up.

Exercise \(\PageIndex{3}\): Corresponding Points in Congruent Figures

Here are two congruent shapes with some corresponding points labeled.

Figure \(\PageIndex{4}\)

1. Draw the points corresponding to \(B\), \(D\), and \(E\), and label them \(B'\), \(D'\), and \(E'\).
2. Draw line segments \(AD\) and \(A'D'\) and measure them. Do the same for segments \(BC\) and \(B'C'\) and for segments \(AE\) and \(A'E'\). What do you notice?
3. Do you think there could be a pair of corresponding segments with different lengths? Explain.

Exercise \(\PageIndex{4}\): Astonished Faces

Are these faces congruent? Explain your reasoning.

Figure \(\PageIndex{5}\)

Summary

To show two figures are congruent, you align one with the other by a sequence of rigid transformations. This is true even for figures with curved sides. Distances between corresponding points on congruent figures are always equal, even for curved shapes. For example, corresponding segments \(AB\) and \(A'B\) on these congruent ovals have the same length:

Figure \(\PageIndex{6}\): 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).

To show two figures are not congruent, you can find parts of the figures that should correspond but that have different measurements.

For example, these two ovals don’t look congruent.

Figure \(\PageIndex{7}\)

On both, 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 proves they are not congruent.

Figure \(\PageIndex{8}\)

Glossary Entries

Definition: Congruent

One figure is congruent to another if it can be moved with translations, rotations, and reflections to fit exactly over the other.

In the 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, and a reflection takes Triangle C to Triangle D.

Figure \(\PageIndex{9}\)

Definition: Right Angle

A right angle is half of a straight angle. It measures 90 degrees.

Figure \(\PageIndex{10}\)

Practice

Exercise \(\PageIndex{5}\)

Which of these four figures are congruent to the top figure?

Figure A has dots more widely spread out, figure B has a slightly larger polygon, figure C is rotated the opposite direction and figure D has dots going the opposite direction than the first figure.

A: A

B: B

C: C

D: D

Exercise \(\PageIndex{6}\)

These two figures are congruent, with corresponding points marked.

Figure \(\PageIndex{12}\)

1. Are angles \(ABC\) and \(A'B'C'\) congruent? Explain your reasoning.
2. Measure angles \(ABC\) and \(A'B'C'\) to check your answer.

Exercise \(\PageIndex{7}\)

Here are two figures.

Figure \(\PageIndex{13}\)

Show, using measurement, that these two figures are not congruent.

Exercise \(\PageIndex{8}\)

Each picture shows two polygons, one labeled Polygon A and one labeled Polygon B. Describe how to move Polygon A into the position of Polygon B using a transformation.

1.

Figure \(\PageIndex{14}\)

2.

Figure \(\PageIndex{15}\)

3.

Figure \(\PageIndex{16}\)

(From Unit 1.1.3)