Module 1 Lesson 22-30

Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Congruence, Proof and Constructions
NAME ____________________________________
DATE
10/20
10/21
10/22
10/23
10/24
Lesson #
22
23
24
25
26
Page(s)
2-3
4
5-6
7-9
10-11
10/27
10/28
10/29
10/30
10/31
11/3
11/4
11/5
11/6
26/27
27
28
11-14
11-14
15-16
29
30
17-18
19-20
Topic
SAS Congruence
Isosceles Triangle Proofs
ASA and SSS Congruence
AAS, HL Congruence
QUIZ
Triangle Congruence Proofs (Part 1)
More Proofs!
Even more proofs!!
Properties of Parallelograms
QUIZ
Special Lines in Triangles (Midsegments)
Special Lines in Triangles (Medians)
Review
Review
Cumulative TEST
1
Homework
Problem set- Lesson 22
No Homework
Problem set- Lesson 24
Problem set- Lesson 25
No Homework
Problem set- Lesson 27
No Homework
Problem Set- Lesson 28
No Homework
Problem Set- Lesson 29
Problem Set- Lesson 30
TBD
TBD- STUDY
No Homework
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 22: Congruence Criteria for Triangles—SAS
If AB =10 and CD=10 we can say : AB and CD have______________________________
 If AB can be mapped onto CD with rigid motion we can say: AB and CD are _____________
o

Therefore: If
can rotate onto
then the triangles would be ___________
It is true that we will not need to show the rigid motion to be able to know that there is one. We are going to show that there
are criteria that refer to a few parts of the two triangles and a correspondence between them that guarantee congruency (i.e.,
existence of rigid motion).
Side-Angle-Side triangle congruence criteria (SAS): Given two triangles
and
so that
triangles are congruent.
(Side),
(Angle),
(Side). Then the
Example 1
What if we had the SAS criteria for two triangles that were not distinct? Consider the following two cases. How would the
transformations needed to demonstrate congruence change?
Case
Diagram
Transformations Needed
A
Shared Side
B
B'''
C
B"
A
Shared Vertex
B
C
C"
2
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Exercises 1–3
Directions: Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of sides or angles are
congruent and why. If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle
onto the other.
1.
Given:
Do
2.
.
and
Given:
Do
meet the SAS criteria?
.
and
meet the SAS criteria?
3
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 23: Base Angles of Isosceles Triangles
Discussion
Given:
, with
.
What type of triangle must this be? ______________________
What other parts in the triangle must be congruent? __________________
Complete the statement: If a triangle is isosceles then the ____________________________________________________.
Write the converse of the statement: ____________________________________________________________________.
Exercises
1.
̅̅̅
Given:
Prove: ̅̅̅
̅̅̅̅
̅̅̅̅.
2.) Given:
Prove: ̅̅̅̅ bisects
.
.
4
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 24: Congruence Criteria for Triangles—ASA and SSS

Angle-Side-Angle triangle congruence criteria (ASA): Given two triangles
and
. If
(Angle),
(Side), and
(Angle), then the triangles are congruent.
More simply we are trying to show _____________________________________________________________
Recall, we should be able to show congruence through rigid motion.
In the diagram to the right, what type of rigid motion would
Map triangle ABC to triangle A’B’C’? ___________________
A’
B’
C’

Side-Side-Side triangle congruence criteria (SSS): Given two triangles ABC and A’B’C’. If
(Side), and
(Side) then the triangles are congruent.
(Side),
More simply we are trying to show ___________________________________________________________________
Recall, we should be able to show congruence through rigid motion.
In the diagram to the right, what type of rigid motion would
Map triangle ABC to triangle A’B’C’? ___________________
A’
C’
Exercises
Based on the information provided, determine whether a congruence exists between triangles. If a congruence between triangles
exists, or if multiple congruencies exist, state the congruencies and the criteria used to determine them.
1.
Given:
is the midpoint of ̅̅̅̅
.
5
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
2.
Given: Rectangle
3.
Given:
with diagonal ̅̅̅̅̅.
,
6
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 25: Congruence Criteria for Triangles—AAS and HL
Discussion: So far we have talked about 3 different congruencies of triangles, they are _____________, _______________ and
_______________. Today we are going to look at a few more ways of proving triangles congruent and determine if they actually
ensure congruence.

Side-Angle-Angle triangle congruence criteria (SAA): Given two triangles
(Angle), and
(Angle), then the triangles are congruent.
and
. If
(Side),
More simply we are trying to show ___________________________________________________________________
Proof: Consider a pair of triangles that meet the AAS criteria. If you knew that two angles of one triangle corresponded to and were
equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each triangle?
__________________________________________________________________________________________________________
Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are
congruent? ____________
Therefore, the AAS criterion is actually an extension of the

triangle congruence criterion.
Hypotenuse-Leg triangle congruence criteria (HL): Given two right triangles
and
If
(Leg) and
(Hypotenuse), then the triangles are congruent.
with right angles
and
More simply we are trying to show ___________________________________________________________________
Label the parts in the right triangle below:
Write a congruence statement:
7
.
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Criteria that do not determine two triangles as congruent: SSA and AAA
Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well
as the non-included angle of 23˚. Yet, the triangles are not congruent.
Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations.
The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be congruent, but
they might not be; therefore we cannot categorize SSA as congruence criterion.
Angle-Angle-Angle (AAA): Given two triangles ABC and A’B’C’ you can draw them in such a way that, even though the
angle measures may be the same, the sides can be proportionally larger, draw two such triangles below:
Therefore, you can have ______________________ triangles in addition to a congruent triangle.
QUICK REVIEW
List all the triangle congruence criteria here: ______________________________________________________________
List the criteria that do not determine congruence here: ____________________________________________________
8
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Examples
1.
Given:
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
Prove:
2.
Given:
Prove:
9
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 26: Triangle Congruency Proofs – Part I
Classwork
Exercises 1–6
1. Given: ̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
.
Prove:
2.
.
Given:
Prove:
.
10
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
̅̅̅̅
3. Given:
Prove:
4.
̅̅̅̅
.
Given:
Prove:
̅̅̅̅
̅̅̅̅ .
11
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 27: Triangle Congruency Proofs – Part II
Classwork
Exercises 1–6
1.
Given:
Prove:
2.
Given:
Square
⃡
,⃡
Square
.
Prove:
12
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
3.
Given:
Prove:
4.
Given:
.
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ ,
Prove:
13
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
5.
Given:
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
Prove:
6.
Given:
Prove:
14
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 28: Properties of Parallelograms
Vocabulary:
Parallelogram- A 4-sided flat shape with straight sides where opposite sides are parallel.
Use each picture along with a compass and the definition of a parallelogram above to justify why the given statement
must be true. Then write the converse of each statement.
1.) If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure.
Converse: ___________________________________________________________________________________
___________________________________________________________________________________________
2.) If a quadrilateral is a parallelogram, then the diagonals bisect each other.
Converse: ___________________________________________________________________________________
___________________________________________________________________________________________
15
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
3.) If the parallelogram is a rectangle, then the diagonals are equal in length.
Converse: ___________________________________________________________________________________
___________________________________________________________________________________________
EXAMPLES:
1.) Given: QDU
UD
Q
Prove: QUAD is a parallelogram
Q
U
D
A
2.) Given: T is the midpoint of NS
NE || OS
E
Prove: ONES is a parallelogram
S
T
N
O
16
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 29: Special Lines in Triangles
Construct the mid-segment of the triangle below. A mid-segment is a line segment that joins the midpoints of two sides of a triangle
or trapezoid. For the moment, we will work with a triangle.
1.
2.
Use your compass and straightedge to determine the midpoints of ̅̅̅̅ and ̅̅̅̅ as
Draw mid-segment ̅̅̅̅
Compare
and
; compare
and
and , respectively.
.
WHEN YOU DRAW A MIDSEGMENT, TWO THINGS HAPPEN:
1.) The mid-segment of the triangle is _____________________________________ to the third side of the triangle.
2.) The mid-segment is ___________________________ the length of the third side of the triangle
17
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Exercises 1–6
Apply what you know about the properties of mid-segments to solve the following examples.
1.
_______
Perimeter of
3.
In


4.
2.
_______
_______
_______
, the midpoints of each side have been marked by points
and .
Mark the halves of each side divided by the midpoint with a congruency mark. Remember to distinguish congruency
marks for each side.
Draw mid-segments ̅̅̅̅, ̅̅̅̅ , and ̅̅̅̅. Mark each mid-segment with the appropriate congruency mark from the sides of
the triangle.
What conclusion can you draw about the four triangles within
? Explain Why.
___________________________________________________________________________________________
5.
G, H, and I are midpoints of AC, AB, and BC respectively.
18
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 30: Special Lines in Triangles
Opening Exercise
In
below:
at the right,
is the midpoint of ̅̅̅̅ ,
is the midpoint of ̅̅̅̅ , and
is the midpoint of ̅̅̅̅ . Complete each statement
̅̅̅̅ is parallel to _______ and measures _______ the length of _______.
̅̅̅̅ is parallel to _______ and measures _______ the length of _______.
̅̅̅̅ is parallel to _______ and measures _______ the length of _______.
DEFINE:
Median- ___________________________________________________________________________________________
Sketch the medians in the triangle below:
The three medians of a triangle are concurrent at the
, or the center of gravity. This point of concurrency
divides the length of each median in a ratio of
; the length from the vertex to the centroid is ___________
the length from the centroid to the midpoint of the side.
19
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Example 1
In the figure at the right,
,
a.
___________
b.
___________
c.
___________
d.
___________
e.
___________
f.
___________
,
. Find each of the following measures.
Example 2
. Points E, F and G are the midpoints of ̅̅̅̅, ̅̅̅̅ , and
In the figure below,
is reflected over l to create
̅̅̅̅, respectively. If AK=24, prove that LE=12.
l
K
L
20
Lesson 22-30 Module 1
NYS COMMON CORE MATHEMATICS CURRICULUM
GEOMETRY
Lesson 31: Construct a Square
Classwork
With a partner, use your construction tools and what you learned in Lessons 1–5 to attempt the construction of a square. Once you
are satisfied with your construction, write the instructions to perform the construction.
21

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