4.1 Detours

1
4.1 Detours Midpoints

• Obj Use detours in proofs
• Apply the midpoint formulas

2
Detour Proofs used when you need to prove 2
pairs of s ? to solve a case.

• Ex1
• A E Given AB ? AD
• BC ?
  CD
• B D Prove ABE ?
  ADE
• 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
EX.1 cont.

• Statements
• (S) AB ? AD
• (S) BC ? DC
• (S) AC ? AC
• ABC ? ADC
• (A) ? BAC ? ?DAC
• (S) AE ? AE
• ABE ? ADE

• Reasons
• Given
• Given
• Reflexive Property
• SSS (1,2,3)
• CPCTC
• Reflexive Property
• 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
Procedure for Detour Proofs

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

6
Midpoint formula for the midpoint of a line take
the average of two given points. Xm X1
X2

• EX.2 Find the midpoint of line segment AB
• 
  
  
• equal distance, hence midpoint

2
X -2 8 2 6 2 3
A B
X3
-2
8
7
Midpoint formula for segment on the coordinate
plane

• Find the midpoint of (1, 4) and (6, 2).
• 1 6, 4 2 2 2
• (7/2, 6/2)
• (3.5, 3)

(
)