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4.4 Perpendicular Lines Date:
1) Construct Perpendicular Bisector
2) Construct a perpendicular line through a point not on the line
3) Recall: Construct a line parallel to the given line through the given point- Use any method!
1) Proving the Perpendicular Bisector Theorem using Transformations
2) Proving the Converse of the Perpendicular Bisector Theorem using Indirect Reasoning (or Proof by Contradiction)
Given: 𝑃 is on the perpendicular bisector 𝑚 of 𝐴𝐵̅̅ ̅̅ Prove: PA= PB Consider the reflection across _________.
Then the reflection of point P across
line m is also ___________ because point P lies on
_________, which is the line of reflection. Also, the
reflection of _______ across line m is B by the definition of
__________. Therefore PA = PB because __________
preserves distance.
Reflect: What can you conclude about ∆𝐾𝐿𝐽?
Given: PA = PB Prove: 𝑃 is on the perpendicular
bisector 𝑚 of 𝐴𝐵̅̅ ̅̅ Step 1: Assume what you are Trying to prove is false. Assume
Then when you draw a perpendicular line from P
containing A and B, it intersects 𝐴𝐵̅̅ ̅̅ at point Q, which is not
___________________.
PQ forms two right triangles. __________ and __________
So by Pythagorean theorem,
Contradiction because
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Proving Theorems about Right Angles
If two lines intersect to form 1 right angle, then they are perpendicular and they intersect to form 4 right angles Given: 𝑚∠1 = 90° Prove: 𝑚∠2 = 90°, 𝑚∠3 = 90°, 𝑚∠4 = 90°
If two intersecting lines form a linear pair of angles with equal measures, then the lines are perpendicular. Given: 𝑚∠1 = 𝑚∠2 Prove: 𝑙 ⊥ 𝑚 Reflect: State the converse of the conditional. Is the converse true? Justify your answer.
Write an indirect proof for each: 1) Given: ∠1 𝑎𝑛𝑑 ∠2 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 Prove: ∠1 𝑎𝑛𝑑 ∠2 𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑜𝑡ℎ 𝑏𝑒 𝑜𝑏𝑡𝑢𝑠𝑒
2) Use indirect reasoning to prove that an obtuse triangle cannot have a right angle.
Determine the unknown values:
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