Mathematics
Grade 7
15 min
Similar and congruent figures
Similar and congruent figures
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1
Introduction & Learning Objectives
Learning Objectives
Define and differentiate between congruent and similar figures.
Identify corresponding angles and sides in congruent and similar figures.
Determine if two figures are congruent using transformations (translation, rotation, reflection).
Determine if two figures are similar by checking angle equality and side proportionality.
Calculate the scale factor between two similar figures.
Use properties of similar figures to find unknown side lengths.
Have you ever noticed how a small toy car looks exactly like a real car, just shrunken down? 🚗 Or how two identical puzzle pieces fit together perfectly? 🤔
In this lesson, we'll explore the fascinating world of geometric figures that are either exactly alike or perfectly scaled versions of each other. Understa...
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Key Concepts & Vocabulary
TermDefinitionExample
Figure (or Shape)Any two-dimensional closed shape, such as a triangle, square, rectangle, or circle.A square, a circle, a pentagon.
Congruent FiguresTwo figures are congruent if they have the exact same size and the exact same shape. One figure can be transformed into the other using only translations, rotations, or reflections.Two identical squares, two puzzle pieces that fit together perfectly.
Similar FiguresTwo figures are similar if they have the exact same shape but can be different sizes. One figure can be transformed into the other using translations, rotations, reflections, and/or dilations.A small photograph and a larger print of the same photograph; a miniature car and a real car.
Corresponding PartsThe matching angles and sides in two or more figures. If...
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Core Formulas
Congruence Rule
If two figures are congruent, then their corresponding angles are equal, and their corresponding side lengths are equal.
This rule helps you identify if two figures are exactly the same. You can use it to find missing angle measures or side lengths if you know the figures are congruent. For example, if $\triangle ABC \cong \triangle DEF$, then $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$, and $AB = DE$, $BC = EF$, $AC = DF$.
Similarity Rule
If two figures are similar, then their corresponding angles are equal, and their corresponding side lengths are proportional.
This rule is key for understanding shapes that are scaled versions of each other. The proportionality means that the ratio of any pair of corresponding sides is constant. For...
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Challenging
Based on the concepts in the tutorial, which of the following statements is ALWAYS true?
A.If two figures are similar, they are also congruent.
B.If two figures are congruent, they are also similar.
C.All rectangles are similar to each other.
D.All isosceles triangles are similar to each other.
Challenging
Triangle PQR is similar to triangle STU. The area of $\triangle PQR$ is 25 cm². The ratio of the length of side PQ to the length of side ST is 5:3. What is the area of $\triangle STU$?
A.15 cm²
B.10 cm²
C.9 cm²
D.5 cm²
Challenging
A triangle with vertices A(1,1), B(4,1), and C(1,5) undergoes a sequence of two transformations. First, it is reflected across the y-axis. Second, it is dilated by a scale factor of 3 centered at the origin. Is the final figure congruent to the original triangle?
A.Yes, because reflection is a rigid transformation.
B.Yes, because the shape is preserved.
C.No, because the reflection changes its orientation.
D.No, because the dilation changes its size.
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