November 17, 2015
Steps of indirect proof:
1. Assume the opposite of the
PROVE statement. ( WARNING!
WARNING! NEVER MESS WITH
THE GIVEN STATEMENT. IT WAS
WRITTEN BY THE MATH GODS!)
2. WORK until you reach a
contradiction.
3. State "this is a contradiction to the
known theorem, definition, etc.
4. Therefore, what we assumed is
false and ___________ is true.
November 17, 2015
B
2
1
A
D
3
C
Proof of thm 5.10
Given BC > AB
Prove m ∠BAC > m ∠C
1. Given BC > AB
1. Given
2. Draw auxillary line AD so that AB = DB 2.
3. <1=<2
3. Base Angles Theorem
4. m<BAC=m<1+m<3
4. Angle Addition Postulate
5. m<1=m<2
5. Definition of Congruence
6. m<BAC=m<2+m<3
6. Substitution
7. M<BAC > m<2
8. m<2 = m<3 +m<C
8. Exterior Angle Theorem
9. m<2 > m<C
10. m<BAC > m<2 > m<C
11. m<BAC > m<C
Review
1. The segment that connects the vertex and midpoint in a triangle.
2. The point of concurrency formed by perpendicular bisectors.
3. The center of a circle inscribed in a triangle is at this point.
4. This is the balancing point for a ∆.
5. The segment that is through the vertex and perpendicular to the opposite side.
6. The point of concurrency formed by angle bisectors.
7. The point of concurrency that is equidistant from the sides of a ∆.
8. The segment that does NOT have to pass through the vertex.
9. The points of concurrency that are ALWAYS inside the ∆.
10. The point of concurrency that is the midpoint of the hypotenuse of a right ∆.
11. The point of concurrency that is at the right angle in a ∆.
12. The points of concurrency that are outside of a ∆.
13. The special segments that must include a midpoint.
14. The 2/3 , 1/3 segment.
15. The Super Duper Centroid Theorem.
16. The point of concurrency of medians.
17. The point of concurrency of altitudes.
18. The center of a circle circumscribed about a ∆.
November 17, 2015
1.
2.
3.
4.
5.
6.
7.
8.
9.
median
circumcenter
incenter
centroid
altitude
incenter
incenter
perpendicular bisector
incenter, centroid
10. circumcenter
11.
12.
13.
14.
15.
orthocenter
circumcenter of obtuse ∆, orthocenter of obtuse ∆
perpendicular bisector, median
median
average the vertices to find the centroid
16. centroid
17. orthocenter
18. circumcenter