3.4 Beyond CPCTC

1. 3.4 Beyond CPCTC • Objectives: • Identify medians and altitudes • Use auxiliary lines in proofs

2. R Q T S Median of a triangle: segment drawn from a vertex of a triangle to the midpoint of the opposite side. The point where all three medians intersect is called the centroid. The three medians of any triangle will always intersect inside the triangle. Example 1: Given the medians below, what segments are congruent? A B C

3. R Q O S Altitude of a Triangle:perpendicular segment drawn from a vertex of a triangle to the side or line containing the opposite side. The point where all three altitudes intersect is called the orthocenter. The three altitudes can intersect inside, outside, or on the triangle depending on the type of triangle it is. A C B

4. D X F E

5. O U V W

6. Every triangle has 3 medians and 3 altitudes. Can a median also be an altitude?

7. Auxiliary lines: lines, rays, or segments that do not appear in the original figure that can help in a proof. Postulate: Two points determine a line (or ray or segment).

8. S Example 1: Q T R Given Given Reflexive Property SSS ∆RQS ∆TQS R  T CPCTC

9. D Example 2: A B C Given Given ADB  CDB Reflexive Property SAS ∆ADB ∆CDB CPCTC Definition of median

10. C Example 3: B D F A E Given CBE & CDA are right angles Definition of altitude All right angles are congruent CBE  CDA Reflexive Property C C Given ∆CBE ∆CDA ASA CPCTC

11. V Example 4: Z U W T X Y Given T  X VUY VWY Given Linear Pair Postulate TUY is supp. to VUY Linear Pair Postulate XWY is supp. to VWY Congruent Supplements Thm. TUY XWY Given ASA ∆TUY ∆XWY CPCTC Definition of median