Chapter 7: 7.1 and 7.2

12.1.10

• 7.1 Triangle Applications Theorems
• 7.2 Two Proof-Oriented Triangle Theorems

7.1

3 new theorems

1. The sum of the measures of the 3 angles of a triangle is 180⁰

 http://www.wyzant.com/Help/Math/Geometry/Triangles/Angles_of_Triangles.aspx

This can be proven using the parallel postulate, saying that line M is parallel to NS, and measure 180⁰, being a straight line and all.  Like Mr. Wilhelm demonstrated for us in class, when the three angles are lined up along that straight line they complete the 180⁰.

http://www.tutorvista.com/topic/parallel-lines-triangle-angle-sum-theorem

2.      The measure of any exterior angle is equal to the sum of the measures of remote interior angles.

http://schoolwires.henry.k12.ga.us/4376205615266500/blank/browse.asp?A=383&BMDRN=2000&BCOB=0&C=66757

An exterior angle is formed by extending on of the sides to form an angle that is exterior to the polygon and is supplementary to its corresponding interior angle.  It is, by definition, adjacent and supplementary to an interior angle of the polygon. 

Since all the angles of a triangle add up to 180⁰, then any interior angle is supplementary to the two other angles combined.  And if the exterior angle is supplementary to the interior angle, then by a loose use of the transitive property the exterior angle must be equal to the two remote interior angles. 

3.      The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of that third side. 

http://www.tutorvista.com/topic/proof-of-midpoint-formula

7.2

2 new theorems

1.      If two angles of a triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No-Choice Theorem)

Proof: since the sum of all the angels must equal 180, the sums must be set equal to each other.  If we then apply the Subtraction Property, we see that the last two angles must be equal, and therefore congruent

DISCLAIMER!!!: the two triangles need not be congruent for us to apply the No- Choice Theorem.

Also, can be written ass AA→ AAA

2.      Proving AAS.  If two angles and one side, not between those two angles, are all congruent between triangles, then the triangles are congruent. 

This can be proven using the No- Choice Theorem to say first that each pair of angles is congruent, then that the third angle is congruent, then that the sides are congruent.  Technically its AAA or even ASA, but it’s one less step to say AAS.

http://hotmath.com/hotmath_help/topics/AAS-postulate.html

And thats what we learned today! homework was:

7.1: 9,12,15-19

7.2: 18

blessings, Em J