Download Chapter 4: Triangle Congruence

Chapter 4: Triangle Congruence
Lesson 4.1 Classifying Triangles
Learning Targets
LT-1: Use triangle classification to solve
problems involving angle measure
and side lengths.
Success Criteria
Classify triangles by angle
measure.
Classify triangles by side length.
Use triangle classification to solve
problems.
•
•
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Directions: Work with a partner to find answers to the following questions.
Ex #1: Create a diagram that will help you remember the two ways to classify
triangles. Be sure to include the classification names in each category.
Ex #2: Classify each triangle by its angle measures.
a. △ABD
b. △ACD
c. △BCD
Ex #3: Classify each triangle by its side lengths.
a. △EFH
b.
△FGH
c. △EGH
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Ex #4: Use triangle classification to solve problems.
Find the side lengths of the triangle.
a. JK
b.
KL
c.
JL
Ex #5: Use triangle classification to solve probems.
Draw an example of each type of triangle, or explain why it is not possible.
a. isosceles right
b. equiangular obtuse
c. scalene right
Ex #6: Use triangle classification to solve problems.
An isosceles triangle has a perimeter of 34cm. The congruent sides measure (4x-1) cm.
The length of the third side is x cm. What is the value of x?
Ex #7: Use triangle classification to solve problems.
Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain.
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Chapter 4
Lesson 2: Angle Relationships in Triangles
Learning Targets
LT-2: Use interior and exterior angle
theorems to solve problems
involving triangles.
Success Criteria
•
•
•
•
•
Use the Triangle Sum Theorem to solve problems.
Find angle measures in right triangles.
Apply the Exterior Angle Theorem.
Apply the Third Angle Theorem.
auxiliary line:
Theorem
Triangle Sum Theorem:
C
A
B
B
4
G1.2.1 – Proof of the Triangle Sum Theorem
Statement
∆ABC and line l // to
A
1
3
l
5
2
C
Reason
AC
Alternate Interior Angle Theorem
Definition of Congruent Angles
Angle Addition Postulate and Defn. of Straight
Angle
m∠1 +m∠2+ m∠3 = 180
Ex# 1: Use the Triangle Sum Theorem to Solve Problems.
After an accident, the position of cars are measured by law enforcement to investigate the collision.
Use the diagram drawn from the information collected to find the indicated angle measures.
a. m∠XYZ
b. m∠YWZ
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•
corollary:
Corollary
Hypothesis
Conclusion
The acute angles of a right
triangle are complementary.
The measure of each angle of an
equiangular triangle is 60°.
Ex #2: Find Angle Measures In Right Triangles.
One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
Theorem
Exterior Angle Theorem:
The measures of an exterior angle of a triangle is equal to the
sum of the measures of its remote interior angles.
2
1
3
4
Ex# 3: Apply The Exterior Angle Theorem.
Find m∠B.
Theorem
Hypothesis
Third Angles Theorem:
If two angles of one triangle are
congruent to two angles of
another triangle, then the third
pair of angles are congruent.
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Conclusion
Ex# 4: Apply The Third Angle Theorem.
Find m∠K and m∠J.
Chapter 4 Lesson 3: Congruent Triangles
Learning Targets
Success Criteria
LT-3: Use provided information to prove
triangles congruent and/or to solve
problems involving congruent triangles.
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•
•
•
Name congruent and corresponding parts
of congruent figures.
Use corresponding parts of congruent
figures to solve problems.
Prove triangles congruent.
congruent triangles:
Properties of Congruent Polygons
Diagram
Corresponding Angles
Corresponding Sides
Ex #1: Name Congruent Corresponding Parts.
Given: ΔPQR ≅ ΔSTW. Identify all pairs of congruent corresponding parts.
Ex #2: Using Corresponding Parts of ≅ Triangles To Solve Problems.
Given: ΔABC ≅ ΔDBC.
A. Find the value of x.
B. Find m∠DBC.
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Ex #3: Prove Triangles Congruent
Given: ∠YWX and ∠YWZ are right angles
YW bisects ∠XYZ
W is the midpoint of XZ
XY ≅ YZ
Prove: ΔXYW ≅ ΔZYW
Statement
Reason
∠YWX and ∠YWZ are right angles
YW bisects ∠XYZ
1.
W is the midpoint of XZ
XY ≅ YZ
2.
3.
4.
5.
6.
7.
Ex #4: Prove Triangles Congruent
Given: PR and QT bisect each other.
∠PQS ≅ ∠RTS
QP ≅ RT
Prove: ΔQPS ≅ ΔTRS
Statement
1.
Reason
PR and QT bisect each other
QP ≅ RT
∠PQS ≅ ∠RTS and
2.
3.
4.
5.
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Chapter 4 Lesson 4: Triangle Congruence: SSS and SAS
Learning Targets
Success Criteria
LT-4: Prove triangles are congruent using SSS
and SAS Theorems.
•
•
Use SSS to Prove Triangles Congruent.
Use SAS to Prove Triangles Congruent.
Triangles can be proved congruent without using all six pairs of corresponding and congruent parts
(three pairs of sides and three pairs of angles). This lesson will show how to prove triangles congruent
using just three pairs of congruent corresponding parts.
Side-Side-Side (SSS) Congruence
Postulate
Hypothesis
Conclusion
If three sides of one triangle are
congruent to three sides of
another triangle, then the
triangles are congruent.
Side-Angle-Side (SAS) Congruence
Postulate
Hypothesis
If two sides and the included
angle of one triangle are
congruent to two sides and the
included angle of another
triangle, then the triangles are
congruent.
Ex #1: Use SSS To Prove Triangles Congruent.
Use SSS to explain why ΔABC ≅ ΔDBC.
Ex #2: Use SAS To Prove Triangles Congruent.
The diagram shows part
of the support structure for a
tower. Use SAS to explain why
ΔXYZ ≅ ΔVWZ.
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Conclusion
Ex #3: Use SSS and SAS To Prove Triangles Congruent.
Use the given information to show that the triangles are congruent for the given values of the variable.
Give the name of the theorem that supports your work.
A: ΔMNO ≅ ΔPQR, when x = 5.
B: ΔSTU ≅ ΔVWX, when y = 4.
Ex #4: Prove Triangles Congruent
Given: BC // AD , BC ≅ AD
Prove: ΔABD ≅ ΔCDB
Statement
BC
1.
Reason
//
AD ,
BC
≅
AD
2.
3.
4.
Chapter 4 Lesson 5: Triangle Congruence: ASA, AAS, and HL
Learning Targets
Success Criteria
LT-5: Prove triangles congruent by using ASA,
AAS, and HL Theorems.
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•
•
•
included side:
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Use ASA to prove triangles congruent.
Use AAS to prove triangles congruent.
Use HL to prove triangles congruent.
Angle-Side-Angle (ASA) Congruence
Postulate
Hypothesis
Conclusion
If two angles and the included
side of one triangle are
congruent to two angles and the
included side of another triangle,
then the triangles are congruent.
Ex #1: Use ASA to Prove Triangles Congruent.
Determine if you can use ASA to prove
the triangles congruent. Explain.
Angle-Angle-Side (AAS) Congruence
Theorem
Hypothesis
If two angles and a nonincluded
side of one triangle are
congruent to the corresponding
angles and nonincluded side of
another triangle, then the
triangles are congruent.
Proof of Angle-Angle-Side Congruence
Given: ∠G ≅ ∠K, ∠J ≅ ∠M, HJ ≅ LM
Prove: ΔGHJ ≅ ΔKLM
Statement
1.
∠G ≅ ∠K, ∠J ≅ ∠M,
Reason
HJ
≅
LM
2.
3.
4.
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Conclusion
Ex #2: Use AAS to prove the triangles congruent.
Given: AB // ED , BC
≅ DC
Prove: ΔABC ≅ ΔEDC
Statement
1.
AB
Reason
//
ED ,
BC
≅
DC
2.
3.
4.
Hypotenuse-Leg (HL) Congruence
Theorem
Hypothesis
Conclusion
If the hypotenuse and a leg of
one triangle are congruent to the
hypotenuse and a leg of another
right triangle, then the triagles
are congruent.
Ex #3 Use HL to Prove Triangles Congruent.
Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell
what else you need to know.
A.
B.
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Chapter 4 Lesson 6: Triangle Congruence: CPCTC
Learning Targets
Success Criteria
LT-6: Use Corresponding Parts of Congruent
Triangles are Congruent(CPCTC) to prove
statements about triangles and/or solve
problems involving triangles.
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Use CPCTC to find angle measures and
side lengths.
Use CPCTC in a proof.
Ex #1: Use CPCTC to Find Angle Measures and Side Lengths.
A and B are on the edges of a ravine.
What is AB? Explain.
Ex2: Use CPCTC in a Proof.
Given: YW bisects XZ and
XY
≅
YZ
Prove: ∠XYW ≅ ∠ZYW
Statement
1.
Reason
YW bisects XZ
XY
≅ YZ
2.
3.
4.
5.
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Ex #3: Using CPCTC in a Proof.
Given: NO // MP , ∠N ≅ ∠P
Prove: MN // OP
Statement
1.
NO //
Reason
MP , ∠N ≅ ∠P
2.
3.
4.
5.
6.
Ex #4: Use CPCTC in a Coordinate Proof.
Given: D(-5,-5), E(-3, -1), F(-2, -3), G(-2, 1), H(0, 5),
and I(1, 3)
Prove: ∠DEF ≅ ∠GHI
Geometry
Chapter 4 Lesson 7: Introduction to Coordinate Proofs
Learning Target (LT-7)
•
Appropriately place geometric figures in the coordinate plane to prove statements and/or solve
problems about these figures.
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Learning Targets
Success Criteria
LT-7: Appropriately place geometric figures in
the coordinate plane to prove statements
and/or solve problems about these figures.
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Position a figure in the coordinate plane.
Write a proof using coordinate geometry.
Assign coordinates to vertices.
You have used coordinate geometry to find the midpoint of a line segment and to find the distance
between two points. Coordinate geometry can also be used to prove conjectures.
A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step is to
position the given figure in the plane. You can use any position, but some strategies can make the steps
of the proof simpler.
Strategies for Positioning Figures in the Coordinate Plane
●
●
●
●
Use the origin as a vertex, keeping the figure in Quadrant 1
Center the figure at the origin
Center a side of the figure at the origin
Use one or both axes as a side
Ex #1: Position a Figure in the Coordinate Plane.
A. Position a square with a side length
of 6 units in the coordinate plane.
Ex #2: Write a Proof Using Coordinate Geometry.
Given: Rectangle ABCD with A(0, 0),
B(4, 0), C(4, 10), and D(0, 10).
Prove: The diagonals bisect each other.
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B. Position a right triangle with legs of length
3 and 6 in the coordinate plane.
A coordinate proof can also be used to prove that a certain relationship is always true. You can prove
that a statement is true for all right triangles without knowing the side lengths. To do this, assign
variables as the coordinates of the vertices.
Ex #3: Assign Coordinates to Vertices.
Position each figure in the coordinate plane and give the coordinates of each vertex
A. A rectangle with width m and length twice the B. A right triangle with legs of lengths s and r.
width.
Ex #4: Write a Proof Using Coordinate Geometry.
Given: Rectangle PQRS with P(0, b), Q(a, b)
R(a, 0), and S(0, 0).
Prove: The diagonals are congruent.
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Chapter 4 Lesson 8: Isosceles and Equilateral Triangles
Learning Targets
Success Criteria
LT-8: Use the properties of isosceles and
equilateral triangles to prove statements
and/or solve problems about them.
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isosceles triangle
•
leg
•
vertex angle
•
base
•
base angle
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Apply properties of isosceles and
equilateral triangles to solve problems.
Write proofs using properties of isosceles
and equilateral triangles.
Isosceles Triangles
Theorem
Hypothesis
Isosceles Triangle Theorem
If two sides of a triangle are
congruent, then the angles
opposite those sides are
congruent.
Converse of Isosceles Triangle
Theorem If two angles of a
triangle are congruent, then the
sides opposite those angles are
congruent.
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Conclusion
Proof of Isosceles Triangle Theorem
Given: AB
≅ AC
Prove: ∠B ≅ ∠C
Statement
1
AB
≅
Reason
AC
2
3
4
5
6
7
Ex #1: Use Properties of Isosceles and Equilateral Triangles to Solve Problems.
The length of YX is 20 ft. Explain
why the length of YZ is the same.
Ex #2: Use Properties of Isosceles and Equilateral Triangles to Solve Problems.
Find each angle measure.
A. m∠F
B. m∠G
Equilateral Triangle
Corollary
Hypothesis
If a triangle is equilateral, then it
is equiangular.
If a triangle is equiangular, then
it is equilateral.
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Conclusion
Ex #3: Use Properties of Isosceles and Equilateral Triangles to Solve Problems.
Find each value.
A. x
B. y
Ex #4: Write proofs using properties of isosceles and equilateral triangles.
Prove that the segment joining the midpoint of two sides of an isosceles triangle is half the base.
Given: In isosceles ΔABC, X is the midpoint of
1
AC
Prove: XY =
2
AB , and y is the midpoint of
Lesson
Problems
4.1 p. 219
#12-19, 21-37, 39, 41-43, 52, 56
4.2 p. 228
#15-24, 26, 28-33, 37, 41-43, 46
4.3 p. 234
#11, 13-25, 28, 30, 31-34
4.4 p. 246
#8-22, 24, 27-31, 42-44
4.5 p. 257
#11-17, 20, 22, 24, 26-29, 30
4.6 p. 263
#7-9, 11-12, 17, 18, 22, 24-28, 37
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AC .
4.7 p. 270
#8-13, 15-20, 22, 25, 27-30
4.8 p. 277
#12-26, 28, 29, 33, 44
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