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Chapter 5Properties of Triangles
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Learning Targets (KRSP):I can……
1. Identify a perpendicular and angle bisector, with the knowledge of equidistant, create a drawing that represents it, then solve problems involving a perpendicular bisector including proving the perpendicular bisector theorem;
2. Recognize and demonstrate perpendicular and angle bisectors in a triangle and label their points of concurrency as circumcenters and incenters respectively;
3. Solve problems with triangles involving perpendicular and angle bisectors;4. Create medians and altitudes of a triangle and label their points of
concurrency as centroids and orthocenters respectively;5. Construct midsegments of a triangle on a graph;6. Apply the triangle inequality to determine characteristics of a triangle.
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Learning Target #1Identify a perpendicular bisector, with the knowledge of equidistant, create a drawing that represents it, then solve problems involving a perpendicular bisector
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Perpendicular Bisector
A B
equidistant
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Perpendicular Bisector Theorem
BA
C
M
Prove it:Given: CM is bisector of ABProve: CA = CB
Statements Reasons1. CM is bisector of AB 1. Given2. AM = BM 2. def. of bisector3. ___________________ 3. def. of ≅ segments4. ___________________ 4. 2 lines, 4 rt. 5. CMA ≅ CMB 5. ___________________6. ___________________ 6. ___________________7. ΔCMA ≅ ΔCMB 7. ___________________8. CA ≅ CB 8. ___________________9. ___________________ 9. ___________________
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12
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T
M N
S
Q
ex: In the diagram shown, MN is the bisector of ST a. What segments are equal?
b. Explain why Q is on MN.
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Recall: Angle Bisector
Angle Bisector Theorem
D C
A
B
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Learning Target #2Recognize and demonstrate perpendicular and
angle bisectors in a triangle and label their points of concurrency as circumcenters and incenters
respectively
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Triangle ActivityGet into 4 groups
With your triangle, fold the bisectors of each side (2 members fold the angle bisectors, 2 members fold the perpendicular bisectors).
Then trace the bisectors with your straight edge. What do you observe?
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ASSIGNMENTp. 267 #313, 1626,
3335
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Concurrency
Concurrent lines
Point of Concurrency (P.O.C.)
**THE THREE BISECTORS OF A TRIANGLE ARE CONCURRENT!
P
P
P
Circumcenter
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P
AB
C
The bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle
PA = PB = PC
Circumscribed
= bisector
= radius of P
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Angle Bisector of a Triangleangle bisectors are concurrent Incenter
P
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The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle
P
B
A
C
D
E
F
Inscribed
PD = PE = PF
= angle bisector= radius of P
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Learning Target #3Solve problems with triangles
involving perpendicular and angle bisectors
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The perpendicular bisectors of ΔABC meet at point P.
Which segments are congruent?
Find PC.
P
AB
C
85
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D
E F
Find BD.
Find AB.
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The angle bisectors of ΔABC meet at point P.
Which segments are congruent?
Find PF.
Find BP.
P
B
A
C
D
E
F
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8
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Find AF.
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The angle bisectors of ΔBRO intersect at point M.
Which segments are congruent?
Find ME.
B
R
O
M
T
A
E
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ASSIGNMENTp. 275 #3, 4, 1017
