SAS, SSS, ASA, and HL Congruence Theorems

Similarity and Congruent Theorems

by Alyssa Rybacki

Open Ended

What is the difference between similar and congruent figures? Provide a real life example of each. 

​Similar Figures

• ​ In similar figures all sides must be proportional and all angles must be congruent.

• ​Similar figures may also be congruent if they are the same size but they dont have to be.

​In this picture the corresponding sides are proportional :

​12/3=4 and 24/6=4

​All three angles are congruent.

​Therefore, these two triangles are similar.

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​Similarity Theorems

• ​SSS ( side-side-side)

​If three sides of one triangle are proportional to the corresponding sides of another triangle then they are similar triangles.

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• ​AA (angle-angle)

​If two angles of one triangle are congruent to the corresponding angles of another triangle then they are similar.

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• ​SAS ( side-angle-side)

​If two sides of one triangle are proportional to the corresponding sides of another triangle and if the angle between those sides are congruent then they are similar

​Congruent Figures

• Congruent figures are figures having the same size and the same shape.

• ​If the three sides of one triangle are congruent to the three sides of another triangle and if the three angles of one triangle are congruent to the three angles of another triangle, then the triangles are congruent.

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Notice how I mark the sides and angles of the triangles

to show which ones are congruent.

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​But.....this is a lot of steps to simply show that two triangles are congruent. Therefore, triangle congruence theorems were invented.

1. SSS (side-side-side)

2. ASA (angle-angle-side)

3. SAS (side-angle-side)​

4. ​HL (hypotenuse-leg)

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​We will take each of these separately.....

​SSS Congruence Theorem

​If three sides of one triangle are congruent to three sides of another triangle, then you can conclude that the two triangles are congruent.

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​*Notice that I did not need to say anything about the angles.

​ASA Congruence Theorem

​If two angles of one triangle are congruent to two angles of another triangle and if the side between the two congruent angles are also congruent, then the triangles are congruent.

​SAS Congruence Theorem

​If two sides of one triangle are congruent to two sides of another triangle and if the angle between the congruent sides are also congruent, then the triangles are congruent.

​HL Congruence Theorem

​In right triangles, if the hypotenuse and leg of one right triangle is congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.

​*Notice that this theorem is only true for right triangles.