Unit 2, Lesson 9
Side-Side-Side Triangle Congruence
Integrated Math 1
9.1
Warm-up
Construct a triangle with the given side lengths on tracing paper.
Can you make a triangle that doesn’t look like anyone else’s?
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Construct a triangle with the given side lengths on tracing paper.
Can you make a triangle that doesn’t look like anyone else’s?
Priya was given this task to complete:
Use a sequence of rigid motions to take onto . Given that segment is congruent to segment , segment is congruent to segment , and segment is congruent to segment . For each step, explain how you know that one or more vertices will line up.
Help her finish the missing steps in her proof:
is the same length as , so they are congruent. Therefore, there is a rigid motion that takes to .
Apply this rigid motion to triangle . The image of will coincide with , and the image of will coincide with .
We cannot be sure that the image of , which we will call , coincides with yet. If it does, then our rigid motion takes to , proving that triangle is congruent to triangle . If it does not, then we continue as follows.
is congruent to the image of , because rigid motions preserve distance.
Therefore, is equidistant from and .
A similar argument shows that is equidistant from and .
is the of the segment connecting and , because the is determined by 2 points that are both equidistant from the endpoints of a segment.
Reflection across the of , takes to .
Therefore, after the reflection, all 3 pairs of vertices coincide, proving triangles and are congruent.
Now, help Priya by finishing a few-sentence summary of her proof. “To prove 2 triangles must be congruent if all 3 pairs of corresponding sides are congruent . . . .”
So far, we’ve learned the Side-Angle-Side and Angle-Side-Angle Triangle Congruence Theorems. Sometimes, we don’t have any information about corresponding pairs of angle measures in triangles. In this case, use the Side-Side-Side Triangle Congruence Theorem: In two triangles, if all 3 pairs of corresponding sides are congruent, then the triangles must be congruent.
To prove that two triangles are congruent, look at the diagram and given information, and think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then check to see if all the information matches the Angle-Side-Angle, Side-Angle-Side, or Side-Side-Side Triangle Congruence Theorem.