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

Z

A

B

C B

And key Job . Ntt10/20/16

" %t*toTammy

.mn#*BYYedisian*tnabdiYrtYutms*endpts.(Maethanhalfway)tBiseokrTheorem

:Converse : lfaptiseqvidistantfrmendpts .

Ifapointisanthebbiseotorofasegment , qasegmentthenitismtbisector .

then it is equidistant famendpts. of indirect : Assumewnatyaiwanttopweis

of the segment false agettoa contradiction

÷inempisnotmthesbiseutrm AATB

plinen

A themidptoftbreflection reflection OPQA OPQB

PQ2tAQ2=pA2 And POITQBYPBZPQrtAQ2=PQ2tQB2

okljisisosaleb NANNY

AQ=QBfHMJ⇒lMJ because okmjmappedtoowp We said QWASNOIMIDPHOFAB .

by a reflection. Somyopotc

.ws#wwhsqPmustbemtbseCtNMofABmaHsitisosaksby

definition

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:

T

#Ll .

1.givenStateMentf2.L1aL2areLP2.deflP1.Lk9001.giVenlPthmidmrmnpi9hYn.emYi9IIEIkdiPYMwstitwiIkikeYmIIWitI7.L1-9007.division4.LltL2sl804.defsupP.ifeinesaret.thenthe2intersldAnglinls5.90tL2-l805.substit.v6.L2-qooformaLpw1equhlmlaswk6.subtractim7j.lHL4lldBveAp7.Vert.def.YeHoNywaytotlinesistohaVeAHlisL2-L4.LtL38.Vert.listhMhavingeqvalmeaswkof90ov9.L4.9o9FL39.substitVtiohAssumewhatyouarepnvingisfoAssvme4aL2canbothbeobtuteds.gPYthag0reanThAtmeansU790andL2T900.Byn82tX41224rsadditimiLltL27lRifx8oCohtradiCtian-givenLltL2aresupp.whioh166Nrmlansthey@Ho0.1aBC-4BOS0iLlo.L

't cannot both be obtuse.

289

¥4

17 152+1/2=17215

zzstXst28x64AssumeobtuseLcanhavearightangk.Tnismlans4tL2tL3-l8ooLbecauseitjf7gCQ@aLfandL179Odbtusxdl2-9OCnghtlfSo9Pt900tL3.1so0L3-yoContradiction-LiscantbenegativeSo.an

obtuse scan 't have anobtwsel