Download Learning Target - Mattawan Consolidated School

Lesson 4.1 Classifying Triangles
Learning Target:
•
Use triangle classification to solve problems involving angle measure
and side lengths.
Directions: Work with a partner to find answers to the following questions.
1. Create a diagram that will help you remember the two ways to classify
triangles. Be sure to include the classifications in each category.
2. Classify each triangle by its angle measures.
a. △ABD
b. △ACD
c. △BCD
3. Classify each triangle by its side lengths.
a. △EFH
b.
△FGH
c. △EGH
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4. Find the side lengths of the triangle.
a.
JK
b.
KL
c.
JL
5. Draw an example of each type of triangle, or explain why it is not possible.
a. isosceles right
b. equiangular obtuse
c. scalene right
6. 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?
7. Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain.
Geometry
Chapter 4 Lesson 2: Angle Relationships in Triangles
Learning Target:
•
Use interior and exterior angle theorems to solve problems involving triangles.
•
auxiliary line:
Theorem
Triangle Sum Theorem:
C
A
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B
B
4
1
A
G1.2.1
Statement
3
l
5
2
C
Reason
∆ABC and line l // to
AC
Alternate Interior Angle Theorem
Definition of Congruent Angles
Angle Addition Postulate and Defn. of Straight
Angle
m∠1 +m∠2+ m∠3 = 180
Ex1: 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.
Y
12°
a. m∠XYZ
b. m∠YWZ
W
•
62°
40°
X
corollary:
Corollary
Hypothesis
The acute angles of a right
triangle are complementary.
The measure of each angle of an
equiangular triangle is 60°.
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Conclusion
Z
Ex2: 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
Ex3: Find m∠B.
Theorem
Hypothesis
Conclusion
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.
Ex4: Find m∠K and m∠J.
( 4y
2
K
J
( 6y 2 - 40)°
)°
Geometry
Chapter 4 Lesson3: Congruent Triangles
Learning Target:
•
Use provided information to prove triangles congruent and/or to solve problems involving
congruent triangles.
•
congruent triangles:
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Properties of Congruent Polygons
Diagram
Corresponding Angles
Naming Congruent Corresponding Parts
Ex1: Given: ΔPQR ≅ ΔSTW. Identify all
pairs of congruent corresponding parts.
Corresponding Sides
Using Corresponding Parts of ≅ Triangles
B
Ex2: Given: ΔABC ≅ ΔDBC.
A. Find the value of x.
B. Find m∠DBC.
D
49.3
C °
(2x - 16)°
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A
Y
Proving Triangles Congruent
Ex3:
Given: ∠YWX and ∠YWZ are right angles.
YW bisects ∠XYZ.
W is the midpoint of XZ .
XY ≅ YZ
Z
X
W
Prove: ΔXYW ≅ ΔZYW
Statement
Reason
1.
2.
3.
4.
5.
6.
7.
Ex4:
The diagonal bars across a gate give it support. Since the angle measures and the lengths of the
corresponding sides are the same, the triangles are congruent.
R
Given: PR and QT bisect each other. Q
∠PQS ≅ ∠RTS
S
QP ≅
Prove:
RT
ΔQPS ≅ ΔTRS
Statement
P
T
Reason
1.
2.
3.
4.
5.
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Geometry
Chapter 4 Lesson 4: Triangle Congruence: SSS and SAS
Learning Target:
•
Prove triangles are congruent using SSS and SAS theorems.
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
Conclusion
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.
Ex1: Use SSS to explain why ΔABC ≅ ΔDBC.
B
A
C
Ex2: The diagram shows part X
of the support structure for a
tower. Use SAS to explain why
ΔXYZ ≅ ΔVWZ.
D
W
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Y
Z
V
Ex3: 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.
6
M
N
Q
5
B: ΔSTU ≅ ΔVWX, when y = 4.
U
3x - 9
x
P
x+2
7
O
R
X
(20y + 12)°
y+3
2y + 3
T
S
92° 11
7
W
V
B
Ex4: Proving Triangles Congruent
Given: BC // AD , BC ≅ AD
A
Prove: ΔABD ≅ ΔCDB
Statement
C
D
Reason
1.
2.
3.
4.
Geometry
Chapter 4 Lesson 5: Triangle Congruence: ASA, AAS, and HL
Learning Target:
•
Prove triangles congruent by using ASA,
AAS, and HL theorems.
•
included side:
Angle-Side-Angle (ASA) Congruence
Postulate
Hypothesis
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.
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Conclusion
Ex1: A mailman has to collect mail from mailboxes at A and B and drop it off at the post office at C.
Does the table give enough information to determine the location of the mailboxes and the post office?
Bearing
Distance
A to B
N 65° E
8 mi
B to C
N 24° W
C to A
S 20° W
Ex2: Determine if you can use ASA to prove
the triangles congruent. Explain.
1
2
3 4
m∠1 = m∠2
m∠3 = m∠4
Angle-Angle-Side (AAS) Congruence
Theorem
Hypothesis
Conclusion
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 ≅
Prove: ΔGHJ ≅ ΔKLM
Statement
L
H
LM
J
G
Reason
1.
2.
3.
4.
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K
M
Using AAS to Prove Triangles Congruent
Ex3: Use AAS to prove the triangles congruent
Given: AB // ED , BC
≅ DC
Prove: ΔABC ≅ ΔEDC
Statement
Reason
1.
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.
Ex4: 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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Geometry
Chapter 4 Lesson 6: Triangle Congruence: CPCTC
Learning Target:
Use corresponding parts of congruent triangles to prove statements about triangles and/or solve
problems involving triangles.
You can use congruent triangles to estimate distances.
Ex1: A and B are on the edges of a ravine.
What is AB?
Y
Ex2: Proving Corresponding Parts Congruent
Given: YW bisects XZ and XY
≅ YZ
Prove: ∠XYZ ≅ ∠ZYW
X
Statement
Reason
1.
2.
3.
4.
5.
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W
Z
Ex3: Using CPCTC in a Proof
Given: NO // MP , ∠N ≅ ∠P
Prove: MN // OP
M
Statement
O
N
P
Reason
1.
2.
3.
4.
5.
6.
Ex4: Using CPCTC in the Coordinate Plane
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
•
Appropriately place geometric figures in the coordinate plane to prove statements and/or solve
problems about these figures.
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.
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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
Ex1: Position a square with
a side length of 6 units
in the coordinate plane.
Ex2: Writing 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.
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.
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Assigning Coordinates to Vertices
Ex3: Position each figure in the coordinate plane and give the coordinates of each vertex.
A. rectangle with width m and length twice the
width
B. right triangle with legs of lengths s and r
Ex4: Writing a Coordinate Proof
Given: Rectangle PQRS with P(0, b), Q(a, b)
R(a, 0), and S(0, 0).
Prove: The diagonals are congruent.
Geometry
Chapter 4 Lesson 8: Isosceles and Equilateral Triangles
Learning Target:
•
Use the properties of isosceles and equilateral triangles to prove statements and/or solve
problems about them.
•
isosceles triangle
•
leg
•
vertex angle
•
base
•
base angle
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Isosceles Triangle
Theorem
Isosceles Triangle Theorem
If two sides of a triangle are
congruent, then the angles
opposite those sides are
congruent.
Hypothesis
Conclusion
A
B
Converse of Isosceles Triangle
Theorem If two angles of a
triangle are congruent, then the
sides opposite those angles are
congruent.
C
A
B
C
B
Proof of Isosceles Triangle Theorem
Given: AB
≅ AC
Prove: ∠B ≅ ∠C
A
X
C
Statement
Reason
1
2
3
4
5
6
7
Ex1: The length of YX is 20 ft. Explain
why the length of YZ is the same.
Ex2: Find each angle measure.
a. m∠F
b. m∠G
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Equilateral Triangle
Corollary
Hypothesis
Conclusion
If a triangle is equilateral, then it
is equiangular.
If a triangle is equiangular, then
it is equilateral.
Ex3: Find each value.
a. x
b. y
Ex4: 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 AB , and y is the midpoint of AC .
1
AC
Prove: XY =
2
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