chapter 4 congruent triangles identify the corresponding parts of congruent figures prove two...
TRANSCRIPT
Chapter 4Congruent Triangles
• Identify the corresponding parts of congruent figures
• Prove two triangles are congruent
• Apply the theorems and corollaries about isosceles triangles
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.
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
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
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
CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason– Shared sides, shared angles, vertical angles,
parallel lines
• An acute triangle is __________ congruent to an obtuse triangle.
A. AlwaysB. SometimesC. NeverD. I don’t know
• A right triangle is ___________ congruent to another right triangle.
A. AlwaysB. SometimesC. NeverD. I don’t know
4.2 Some Ways to Prove Triangles Congruent
Objectives
• Learn about ways to prove triangles are congruent
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,
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.
• Compare your spaghetti triangle to your neighbors
• Compare your spaghetti triangle to my spaghetti triangle.
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
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
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
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
Construction 3Given an angle, construct the bisector of the angle
Given:
Construct:
Steps:
ABCbisector of ABC
CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason– Shared sides, shared angles, vertical angles,
parallel lines
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
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.
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
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.
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
4.4 The Isosceles Triangle Theorem
Objectives
• Apply the theorems and corollaries about isosceles triangles
Isosceles TriangleBy definition, it is a triangle with two
congruent sides called legs.X
Y Z
Base
Base Angles
Legs Vertex Angle
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
Corollary• The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
4.5 Other Methods of Proving Triangles Congruent
Objectives
• Learn two new ways to prove triangles are congruent
Proving Triangles We can already prove triangles are congruent by the ASA, SSS
and SAS. There are two other ways to prove them congruent…
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
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
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
4.6 Using More than One Pair of Congruent Triangles
Objectives
• Construct a proof using more than one pair of congruent triangles.
• Sometimes two triangles that you want to prove congruent have common parts with two other triangles that you can easily prove congruent.
Median of a TriangleA segment connecting a vertex to the
midpoint of the opposite side.
midpoint
vertex
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.
Altitude of a TriangleA segment drawn from a vertex
perpendicular to the opposite side.
vertex
perpendicular
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.
Special Cases - AltitudesObtuse Triangles: Two of the altitudes are drawn
outside the triangle. Extend the sides of the triangle
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 !
TheoremIf a point lies on the perpendicular bisector of a segment of a
segment, then the point is equidistant from the endpoints.
TheoremIf a point is equidistant from the endpoints of a segment, then
the point lies on the perpendicular bisector of the segment.
TheoremIf a point lies on the bisector of an angle then the
point is equidistant from the sides of the angle.
Construction 10Given a triangle, circumscribe a circle about the triangle.
Given:
Construct:
Steps:
ABC circumscribed about R ABC
Construction 11Given a triangle, inscribe a circle within the triangle.
Given:
Construct:
Steps:
ABC inscribed within R ABC
• An altitude is _____________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know
• A median is ___________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know
• An altitude is ______________ a perpendicular bisector.
A. AlwaysB. SometimesC. NeverD. I don’t know
• An angle bisector is _______________ perpendicular to the opposite side.
A. AlwaysB. SometimesC. NeverD. I don’t know