In class today we worked on a mathcounts problem set and continued to study the triangle congruence shortcuts. In 5th period students also looked at an example of how triangle congruence can be used to prove/discover new ideas in geometry. So, I helped students to explain why the opposite sides and angles have to be congruent in a parallelogram.
Assignment. For Tuesday, students should have completed
- 4.5 on p. 227: 1-17
- 2+ Study Questions
- Sentence form summary of the topic of the assignment
Question
- What are you learning in Geometry about triangle congruence?
- How is triangle congruence related to the properties of a parallelogram?
Answers
- We are studying relationships relating to triangles. We have learned that there are shortcuts for determining when two triangles are congruent – SSS, SAS, ASA, and AAS are the triangle congruence shortcuts we have discussed. Also, students should be able to explain why ASS and AAA are not shortcuts. Again, chuckles, but really there is a reason to write these acronyms this way. For example, SAA congruence is the idea that if two separate triangles can be shown to have one pair of corresponding sides and two pairs of corresponding angles that are congruent then the triangles are congruent. In this case it is important that the position of the sides and angles correspond exactly.
- Triangle congruence can be used to prove the properties of a parallelogram if we assume only the definition of a parallelogram (quadrilateral with 2 pairs of opposite, parallel sides). We might notice that two triangles can be formed if we draw one of the diagonals of the parallelogram. And by proving these two triangles to be congruent, we can explain why the opposite sides and angles are congruent.
Now, we with each pair of parallel sides we can find a set of congruent alternate interior angles
So we can see that in 
and 
the segment 
. So we have 
by ASA.
Since these two triangles are congruent, we can see that 
and 
and therefore that the opposite sides of a parallelogram are congruent.
Also, we can see that 
and that 
and 
must be congruent because they are made from congruent parts. So, the opposite angles of a parallelogram are congruent.
So, we can use triangle congruence to learn about shapes beyond triangles. And students will find that much of our knowledge of Geometry is learned through the application of triangle relationships.
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What better way to spend the evening and the next six days off than to read and re-read each of these updates together?! Have a good break, see everyone on Tuesday.