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Transcript
Congruent shapes Congruent shapes have the same size and the same shape. In other words, if you place an object in front of a mirror, the image that you see is congruent or " equal " to the object When shapes are congruent, all corresponding sides and angles are also congruent. Look at the following two triangles.You should notice that some sides and some angles have one marking. Others have more than one The side that has one marking corresponds to the other side that also has one marking. and so forth... The angle that has one marking corresponds to the other angle that also has one marking. And so forth... Having said that the best way to know if two figures are congruent is to compare the corresponding sides and corresponding angles If these are equal, the figures are congruent. Therefore, be careful when doing your geometry problems about congruence. Two figures may appear to be congruent yet one side or one angle in one shape may be slightly different than the other shape. Markings on shapes are the best way to compare sides and angles to see if they are equal or congruent Circles too can be congruent. All circles have congruent angles since a complete turn measures 360 degrees for all circles Therefore, it is up to the size of the circle and they will have the same size if they have the same radius Other examples of congruent shapes or figures For the quadrilaterals, I used different markings for each side and each angles. Again, corresponding sides and angles have the same amount of markings For the rectangles, I did not use four different markings although we have four sides. Since opposite sides are equal, I used the same marking for equal sides The hexagon is a different story. I used the same marking for all angles and sides to keep the figure easy to read. However, at this point, you should be able to recognize corresponding sides and angles without major problems Congruent triangles Congruent triangles have the same size and the same shape. When triangles are congruent, all corresponding sides and corresponding angles are also congruent or equal For examples, the two triangles below are congruent Corresponding angles are angles in the same position Corresponding sides are sides that are in the same position The two triangles above have a side with 3 markings. These sides are at the same position and thus are corresponding Congruent sides are sides that have equal measures Congruent angles are angles that have equal sides and equal measures In the triangle above, if we pull out the side with one and three markings and the included angle, we get the following: The above 45 degrees angle is a good example of congruent angles because the sides are equal and the angles are equal Included side: A side between two angles Included angle: An angle between two sides There are three postulates and two theorems that are used to identify if two triangles are congruent With these postulates and theorems, you don't have to check if all corresponding angles and all sides are congruent If the triangles meet the condition of the postulate or theorem, then, you have congruent triangles. They are the SSS postulate, SAS postulate, ASA postulate, AAS theorem, and Hypotenuse-Leg theorem SSS postulate: If three sides of a triangle are congruent to three sides of a second triangle, then the two triangles are congruent Example: ASA postulate: If two angles and the side between these two angles (included side) of one triangle are congruent to the corresponding angles and the included side of a second triangle, then the two triangles are congruent Example: SAS postulate: If two sides and the angle between these two sides (included angle) of one triangle are congruent to the corresponding two sides and the included angle of a second triangle, then the two triangles are congruent Example: AAS theorem: If two angles and a side not between these two angles of one triangle are congruent to two angles and the corresponding side not between these two angles of a second triangle, then the two triangles are congruent Example: Hypotenuse-Leg theorem: If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and corresponding leg of a second right triangle, then the two triangles are congruent Example: