Download - Chapter 4 Congruent Triangles
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Chapter 4Congruent Triangles
• Identify the corresponding parts of congruent figures
• Prove two triangles are congruent
• Apply the theorems and corollaries about isosceles triangles
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4.1 Congruent Figures
Objectives
• Identify the corresponding parts of congruent figures
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What we already know…
• Congruent Segments– Same length
• Congruent Angles– Same degree measure
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Congruent Figures
Exactly the same size and shape. Don’t ASSume !
D
C
A
B
EF
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Definition of Congruency
Two figures are congruent if corresponding vertices can be matched up so that:
1. All corresponding sides are congruent
2. All corresponding angles are congruent.
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What does corresponding mean again?
• Matching
• In the same position
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Volunteer
• Draw a large scalene triangle (with a ruler)
• Cut out two congruent triangles that are the same
• Label the Vertices A, B, C and D, E, F
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You can slide and rotate the triangles around so that they MATCH up perfectly.
ABC DEF
A
BC
F
E
D
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The order in which you name the triangles mattersmatters !
ABC DEF
A
BC
F
E
D
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Based on the definition of congruency….
• Three pairs of corresponding angles
• Three pairs of corresponding sides
1. A D
3. C F
2. B E
1. AB DE
3. CA FD
2. BC EF
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It is not practical to cut out and move the triangles around
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ABC XYZ
• Means that the letters X and A, which appear first, name corresponding vertices and that X A.
• The letters Y and B come next, so – Y B and–XY AB
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CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason– Shared sides, shared angles, vertical angles,
parallel lines
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White Boards
• Suppose TIM BER
IM ___
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White Boards
• Suppose TIM BER
IM ER , Why ?
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White Boards
• Corresponding Parts of Congruent Triangles are Congruent
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White Boards
• Suppose TIM BER
___ R
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White Boards
• Suppose TIM BER
M R, Why?
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White Boards
• Corresponding Parts of Congruent Triangles are Congruent
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White Boards
• Suppose TIM BER
MTI ____
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White Boards
• Suppose TIM BER
MTI RBE
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White Boards
• If ABC XYZm B = 80m C = 50
Name four congruent angles
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White Boards
• If ABC XYZm B = 80m C = 50
A, C , X, Z
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White Boards
• If ABC XYZ
Write six congruences that must be correct
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White Boards
• If ABC XYZ
1. A X
3. C Z
2. B Y
1. AB XY
3. CA ZX
2. BC YZ
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Remote time
A. Always
B. Sometimes
C. Never
D. I don’t know
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• An acute triangle is __________ congruent to an obtuse triangle.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• A polygon is __________ congruent to itself.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• A right triangle is ___________ congruent to another right triangle.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• If ABC XYZ, A is ____________ congruent to Y.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• If ABC XYZ, B is ____________ congruent to Y.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• If ABC XYZ, AB is ____________ congruent to ZY.
A. AlwaysB. SometimesC. NeverD. I don’t know
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4.2 Some Ways to Prove Triangles Congruent
Objectives
• Learn about ways to prove triangles are congruent
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Don’t ASSume
• Triangles cannot be assumed to be congruent because they “look” congruent.
and
• It’s not practical to cut them out and match them up
so,
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We must show 6 congruent pairs
• 3 angle pairs and
• 3 pairs of sides
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WOW
• That’s a lot of work
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Spaghetti Experiment
• Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle.
• Be careful, IT’S SPAGHETTI, and it will break.
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• Compare your spaghetti triangle to your neighbors
• Compare your spaghetti triangle to my spaghetti triangle.
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We are lucky…..
• There is a shortcut– We don’t have to show
• ALL pairs of angles are congruent and
• ALL pairs of sides are congruent
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SSS Postulate
If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
A
B E
C DF
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Patty Paper Practice
5 inches
2.5 inches
3 inches
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Volunteer
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SAS PostulateIf two sides and the included angle are congruent
to the corresponding parts of another triangle, then the triangles are congruent.
B E
C DF
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ASA PostulateIf two angles and the included side of one triangle
are congruent to the corresponding parts of another triangle, then the triangles are congruent.
A
B
C D
E
F
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The order of the letters MEAN something
• Is SAS the same as SSA or A$$ ?
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Construction 2Given an angle, construct a congruent angle.
Given:
Construct:
Steps:
ABCCDE ABC
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Construction 3Given an angle, construct the bisector of the angle
Given:
Construct:
Steps:
ABCbisector of ABC
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CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason– Shared sides, shared angles, vertical angles,
parallel lines
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Remote Time
Can the two triangles be proved congruent? If so, what postulate can be used?
A. SSS Postulate
B. SAS Postulate
C. ASA Postulate
D. Cannot be proved congruent
E. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know
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White Board
• Decide Whether you can deduce by the SSS, SAS, or ASA Postulate that the two triangles are congruent. If so, write the congruence ( ABC _ _ _ ). If not write not congruent.
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D
C
B
A
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DBC ABCSSS
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A
DC
B
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No Congruence
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Construction 7Given a point outside a line, construct a line parallel to the given line through the point.
Given:
Construct:
Steps:
line l with point A to l through A
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4.3 Using Congruent Triangles
Objectives
• Use congruent triangles to prove other things
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Our Goal
• In the last section, our goal was to prove that two triangles are congruent.
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The Reason
• If we can show two triangle are congruent, using the SSS, SAS, ASA postulates, then we can use the definition of Congruent Triangles to say other parts of the triangles are congruent. – Corresponding Parts of Congruent Triangles are
Congruent.
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This is an abbreviated way to refer to the definition of congruency with respect to triangles.
C orresponding
P arts of
C ongruent
T riangles are
C ongruent
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Basic Steps
1. Identify two triangles in which the two segments or angles are corresponding parts.
2. Prove that those two triangles are congruent
3. State that the two parts are congruent using the reason CPCTC.
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Given: m 1 = m 2 m 3 = m 4Prove: M is the midpoint of JK
L
MJ K
3 4
21
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
LM = LM
m J = m K If 2 ’s of 1 are to 2 ’s of another , then the third ’s are .
Reflexive Property
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
LM = LM
m J = m K If 2 ’s of 1 are to 2 ’s of another , then the third ’s are .
Reflexive Property
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
JLM KLM ASA
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
5. M is the midpoint of JK
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
4. JM = KM
5. M is the midpoint of JK
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
3. JLM KLM
4. JM = KM 4. CPCTC
5. M is the midpoint of JK
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
2. LM = LM 2. Reflexive Property
3. JLM KLM 3. ASA Postulate
4. JM = KM 4. CPCTC
5. M is the midpoint of JK
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
2. LM = LM 2. Reflexive Property
3. JLM KLM 3. ASA Postulate
4. JM = KM 4. CPCTC
5. M is the midpoint of JK 5. Definition of midpoint
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Given: m 1 = m 2 m 3 = m 4
Prove: M is the midpoint of JK
L
MJ K
3 4
21
Statements Reasons
1. m 1 = m 2 m 3 = m 4
1. Given
2. LM = LM 2. Reflexive Property
3. JLM KLM 3. Postulate
4. JM = KM 4. CPCTC
5. M is the midpoint of JK 5. Definition of midpoint
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
3 4 Definition of bisector
JK JK Reflexive Property
MKJ OKJ SAS Postulate
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK;
KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK;
KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
6. JK bisects MJO 6.
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK;
KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
5. 1 2 5. CPCTC
6. JK bisects MJO 6.
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK;
KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
4. MKJ OKJ 4. SAS Postulate
5. 1 2 5. CPCTC
6. JK bisects MJO 6.
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK;
KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
4. MKJ OKJ 4. SAS Postulate
5. 1 2 5. CPCTC
6. JK bisects MJO 6. Def of bisector
K
O
J
M
1
2
34
K
O
J
M
1
2
34
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4.4 The Isosceles Triangle Theorem
Objectives
• Apply the theorems and corollaries about isosceles triangles
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Isosceles TriangleBy definition, it is a triangle with two
congruent sides called legs.X
Y Z
Base
Base Angles
Legs Vertex Angle
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Experiment - Goal
• Discover Properties of an Isosceles Triangle
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Supplies
• Blank sheet of paper
• Ruler
• Pencil
• Scissors
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Procedure
1. Fold a sheet of paper in half.
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Procedure
2. Draw a line with the ruler going from the folded edge (very important) to the corner of the non folded edge.
Folded edge
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Procedure
3. Cut on the red line
Cut here
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Procedure
4. Open and lay flat. You will have a triangle
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Procedure
5. Label the triangleP
Q
SR
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Procedure6. Since PRQ fits exactly over PSQ
(because that’s the way we cut it),
PRQ PSQ P
Q
SR
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Procedure7. What conclusions can you make? Be
careful not to ASSume anything. P
Q
SR
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Conclusions
P
Q
SR
1. PRS PSR
2. PQ bisects RPS
3. PQ bisects RS
4. PQ RS at Q
5. PR PS
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These conclusions are actually
• Theorems and corollaries
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TheoremThe base angles of an isosceles triangle are
congruent.
A
B
C
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Corollary• An equilateral triangle is also equiangular.
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Corollary• An equilateral triangle has angles that
measure 60.
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Corollary• The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
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TheoremIf two angles of a triangle are congruent, then
it is isosceles.
A
B
C
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Corollary
• An equiangular triangle is also equilateral.
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White Board Practice
• Find the value of x
30º
xº
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x = 75º
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White Board Practice
• Find the value of x2x - 4
x + 5
2x + 2
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x = 9
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White Board Practice
• Find the value of x
56 º 62 º
x
4142
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x = 42
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4.5 Other Methods of Proving Triangles Congruent
Objectives
• Learn two new ways to prove triangles are congruent
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Proving Triangles We can already prove triangles are congruent by the ASA, SSS
and SAS. There are two other ways to prove them congruent…
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AAS TheoremIf two angles and a non-included side of one
triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
A
B
C D
E
F
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The Right Triangle
leg
leg
hypotenuse
right angle
A
B
C
acute angles
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HL TheoremIf the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right triangle, then the triangles are congruent.
A
B
C D
E
F
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Five Ways to Prove ’s
All Triangles:ASA SSS SAS AASRight Triangles Only:HL
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White Board Practice• State which of the congruence methods can
be used to prove the triangles congruent. You may choose more than one answer.
SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem
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4.6 Using More than One Pair of Congruent Triangles
Objectives
• Construct a proof using more than one pair of congruent triangles.
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• Sometimes two triangles that you want to prove congruent have common parts with two other triangles that you can easily prove congruent.
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More Than One Pair of ’s
Given: X is the midpt of AF & CD
Prove: X is the midpt of BE
A
B
C
D
E
F
X
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Lecture 7 (4-7)
Objectives
• Define altitudes, medians and perpendicular bisectors.
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Median of a TriangleA segment connecting a vertex to the
midpoint of the opposite side.
midpoint
vertex
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Median of a TriangleEach triangle has three Medians
midpoint
vertex
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Median of a TriangleEach triangle has three Medians
midpoint
vertex
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Median of a Triangle• Notice that the three medians will meet
at one point.
If they do not meet, then you are not drawing the segments well.
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Altitude of a TriangleA segment drawn from a vertex
perpendicular to the opposite side.
vertex
perpendicular
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Altitude of a TriangleEach Triangle has three altitudes
vertex
perpendicular
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Altitude of a TriangleEach triangle has three altitudes
vertexperpendicular
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Altitude of a TriangleNotice that the three altitudes will meet at
one point.
If they do not meet, then you are not drawing the segments well.
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Special Cases - AltitudesObtuse Triangles: Two of the altitudes are drawn
outside the triangle. Extend the sides of the triangle
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Special Cases - AltitudesRight Triangles: Two of the altitudes are
already drawn for you.
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Perpendicular BisectorA segment (line or ray) that is perpendicular to and
passes through the midpoint of another segment.
Must put the perpendicular and congruent markings !
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Angle BisectorA ray that cuts an angle into two
congruent angles.
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TheoremIf a point lies on the perpendicular bisector of a segment of a
segment, then the point is equidistant from the endpoints.
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TheoremIf a point is equidistant from the endpoints of a segment, then
the point lies on the perpendicular bisector of the segment.
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Remember
• When you measure distance from a point to a line, you have to make a perpendicular line.
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TheoremIf a point lies on the bisector of an angle then the
point is equidistant from the sides of the angle.
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Construction 10Given a triangle, circumscribe a circle about the triangle.
Given:
Construct:
Steps:
ABC circumscribed about R ABC
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Construction 11Given a triangle, inscribe a circle within the triangle.
Given:
Construct:
Steps:
ABC inscribed within R ABC
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Remote Time
A. Always
B. Sometimes
C. Never
D. I don’t know
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• An altitude is _____________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• A median is ___________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• An altitude is ______________ a perpendicular bisector.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• An angle bisector is _______________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know
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• A perpendicular bisector of a segment is ___________ equidistant from the endpoints of the segment.
A. AlwaysB. SometimesC. NeverD. I don’t know