1. 4.1 Detours & Midpoints Obj: Use detours in proofs Apply the midpoint formulas Apply the midpoint formulas

2. Detour Proofs:  Detour Proofs: used when you need to prove 2 pairs of s  to solve a case. Ex:1 A E Given: AB  AD A E Given: AB  AD BC  CD BC  CD B D Prove: ABE  ADE Do we have enough info? Do we have enough info? We only have sides AB  AD & AE  AE We need an angle. C

3. Prove ABC  ADC First by SSS Statements 1.(S) AB  AD 2.(S) BC  DC 3.(S) AC  AC 4. ABC  ADC 5.(A)  BAC   ADC 6.(S) AE  AE 7. ABE  ADE EX.1 cont. Reasons 1.Given 2.Given 3.Reflexive Property 4.SSS (1,2,3) 5.CPCTC 6.Reflexive Property 7.SAS (1,5,6)

4. Procedure for Detour Proofs 1.Determine which triangles you must prove to be congruent to reach the required conclusion. 2.Attempt to prove that these triangles are congruent. If you cannot do so for lack of enough information, take a detour. 3.Identify the parts that you must prove to be congruent to establish the congruence of the triangles.

5. 4.Find a pair of triangles that 1.You can readily prove to be congruent. 2.Contain a pair of parts needed for the main proof. 5.Prove that the triangles found in step 4 are congruent. 6.Use CPCTC and complete the proof planned in step 1. Procedure for Detour Proofs

6. Midpoint formula: for the midpoint of a line take the average of two given points. X m = X 1 + X 2 EX.2: Find the midpoint of line segment AB equal distance, hence midpoint equal distance, hence midpoint X = -2 + 8 2 = 6 2 =3 2 A B A B -28 X3X3X3X3

7. Midpoint formula for segment on the coordinate plane: Find the midpoint of (1, 4) and (6, 2).Find the midpoint of (1, 4) and (6, 2). 1 + 6, 4 + 2 2 21 + 6, 4 + 2 2 2 ( 7 / 2, 6 / 2 )( 7 / 2, 6 / 2 ) (3.5, 3)(3.5, 3) ( )