# Tuesday, Nov. 22nd

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?

• 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 $\dpi{300}\inline \triangle HJO$ and $\dpi{300}\inline \triangle JHN$ the segment $\dpi{300}\inline \overline{HJ} \cong \overline{HJ}$. So we have $\dpi{300}\inline \triangle HJO \cong \triangle JHN$ by ASA.

Since these two triangles are congruent, we can see that $\dpi{300}\inline \overline{OH} \cong \overline{NJ}$ and $\dpi{300}\inline \overline{OJ} \cong \overline{NH}$ and therefore that the opposite sides of a parallelogram are congruent.

Also, we can see that $\dpi{300}\inline \angle O \cong \angle N$ and that $\dpi{300}\inline \angle NJO$ and $\dpi{300}\inline \angle OHN$ 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.

Want to re-read these emails all in one place? Each email gets posted automatically to the class update blog. The little ;start ;end tags help with blog post formatting. You can see all Geometry updates at

http://teacher.nickhershman.com/blog/category/geometry/

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.