Download Geometry Section 4.1 Start Thinking: What does it mean for figures

Geometry Section 4.1
________________________________________________
Start Thinking: What does it mean for figures to be congruent?
Example 1: Give a congruence statement to describe that the figures are
congruent. Then explain what their congruent parts are.
Example 2: Triangle ABC is Congruent to Triangle MNK. Explain what side/ angle is congruent to
the following.

a.) <A is congruent to _______________
b.) <C is congruent to _________________________
c.) AB  ___________________
d.) KM  ____________________________

Example 3: Write a congruence statement for TWO sets of congruent triangles. Be careful with the order
of the points.
USING CONGRUENT TRIANGLES:
EX1: Triangle ABC is congruent to Triangle ABD. If m<DAB=48 and m<c = 60
then what is the m<D?
EX2: Triangle ABD is congruent to Triangle DEA. M<B = 50 m<A = 90.
What is m<ADB?
PROVING TRIANGLES ARE CONGRUENT
To prove that two triangles are congruent you need to make sure that all corresponding
__________________ are congruent and all corresponding ___________________________ are
congruent.
You can then justify that the triangles are congruent by the _____________________________________.
Example 1:
EXAMPLE 1:
EXAMPLE 2:
EXAMPLE 3: (do not use 3rd angle theorem)
Given: Diagram to Right
Prove: Triangle ACB is congruent to Triangle ECD
WRAP UP:
1.) To Prove that two triangles are congruent you need to show that all
__________________________________
__________________________ are congruent and all ______________________________________
are congruent.
2.) If two angles of a triangle are congruent to two angles of another then
________________________________
_________________________________________.
3.)When you write a congruence statement you must be careful with the _______________________ you
list the points.
Geometry Section 4.2
______________________________________
So Far:
In order to prove that two triangles are congruent you must show that all the corresponding
____________________ are congruent and that all corresponding ________________________________ are
congruent.
POSTULATE 4-1: Side – Side – Side (SSS) Postulate
Postulate:
If:
Then:
If the three sides of one triangle
are congruent to three sides of
another triangle then _________
__________________________.
EXAMPLE 1:
EXAMPLE 2:
POSTULATE 4-2: Side – Angle – Side (SAS) Postulate
Postulate:
If:
Then:
If ________________ and the
_______________ of one triangle
are congruent to two sides and
the included angle of another
triangle, then the triangles are
_________________.
EXAMPLES:
1.)
2.)
EX: Determine if there is enough information to determine if the following triangles MUST be congruent.
1.)
4.)
2.)
3.)
5.)
PROOFS:
1.) GIVEN: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷 and ̅̅̅̅
𝐴𝐵 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 ̅̅̅̅
𝐶𝐷
PROVE: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐶𝐵
̅̅̅̅ and 𝐵𝐹
̅̅̅̅
̅̅̅̅ is perpendicular to 𝐶𝐹
2.) GIVEN: 𝐴𝐶
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅ and 𝐶𝐷 ≅ 𝐷𝐹
̅̅̅̅
𝐴𝐶 ≅ 𝐵𝐹
PROVE: ∆𝐷𝐴𝐶 ≅ ∆𝐷𝐵𝐹
3.)GIVEN: ̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐶𝐹 AND C is the midpoint of ̅̅̅̅
𝐵𝐷
PROVE: ∆𝐶𝐴𝐵 ≅ ∆𝐶𝐹𝐷
4.) GIVEN: Figure ABGN is a square
̅̅̅̅̅ ≅ 𝐵𝐹
̅̅̅̅
𝑀𝑁
G is the midpoint of ̅̅̅̅̅
𝑀𝐹
PROVE: ∆𝑀𝑁𝐺 ≅ ∆𝐹𝐵𝐺
WRAP UP
You can prove that two triangles are congruent by showing that all corresponding sides are congruent and all
corresponding angles are congruent OR you can use ______________________ or
_______________________.
If you are going to use SAS you must be sure that the angles is ____________________________ the sides.
Geometry Section 4.4
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Start thinking: You are given the following information about Triangle ABC and Triangle DEF are congruent.
You are also given that 𝑚∠𝐴 = 50 and 𝑚∠𝐷 = 2𝑥. What does x equal? How do you know this?
Theorem 4.4: CPCTC
Thm: 4.4: If two figures are _______________ then
their ___________________ are congruent.
Example:
Example:
Example:
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅
Given: 𝐴𝐵
𝐴𝐶 , M is the midpoint of 𝐵𝐶
Prove: AMB  AMC
Example:
EX:
EX:
EX: Given: ̅̅̅̅
𝐵𝐷 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐴𝐵𝐶
∠𝐵𝐴𝐷 ≅ ∠𝐵𝐶𝐷
̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐶𝐹
̅̅̅̅
̅̅̅̅
𝐸𝐷 ≅ 𝐷𝐹
Prove: <E is congruent to <F
Geometry Section 4.5
________________________________________
Start Thinking: The following are all examples of isosceles triangles.
How would you define an isosceles triangle?
PARTS OF AN ISOSCELES TRIANGLE:
Theorem 4.3: Isosceles Triangle Theorem
Theorem: If two sides of a triangle are congruent, then ___________________________________.
Example:
Prove It:
APPLICATIONS:
1.)
2.)
3.)
4.)
Converse of Isosceles Triangle theorem
If ___________________________________________ then______________________________.
EXAMPLE:
Proof:
Practice: Use the diagram. Tell what theorem or corollary you used.
a. If AE  CE , then  ______   ______
b. If  DAE   DEA, then _______  _______
c. If BDF   DBF   BFD, then ________  _______ _______
d. If AB  BC  AC , then  ______   _______   ______
Find the value of x.
1)
Corollary to the Isosceles Triangle Theorem
IF all sides of a triangle are congruent then _____________________.
Corollary to the CONVERSE OF Isosceles Triangle Theorem
IF all angles of a triangle are congruent then _____________________.
THEOREM 4.5
If the vertex angle of a isosceles triangle is bisected then the bisector is a_________________ of
the base.
Proof:
Practice:
1.)
̅̅̅̅ is an angle bisector of the vertex angle of isosceles Triangle ACD. It
2.) 𝑨𝑩
intersects the base at B. If CD= 40 and CB=2x then what is x? What
m<ABD?
Geometry Section 4.6
_______________________________________________
Start thinking: The current methods that we have to prove that triangles are congruent are ____________,
________________, ___________________, and ___________________.
We know that AAA does not prove congruent but the SSA has always been questionable. See if you can
change the appearance of the triangle below without changing the measure of the angle or the length of
the two sides.
3 KEY CONDITIONS NEEDED TO USE THE _____________________________
Conditions:
Ex1:
Decide whether enough information is given to prove that the triangles are congruent using the
HL Congruence Theorem.
Ex2:
State the third congruence that must be given to prove congruence.
IF:
AC  DF ,  A is a right angle and  A   D, ______  _______ Use HL Thm
IF:
 J is a right angle and  J   D, _______  ________Use HL Thm
Practice: Decide whether enough information is given to prove that the triangles are congruent.
If there is enough information, state the congruence postulate or theorem you would use AND
write a congruence statement.
PROOF ME:
1.) Given: RT  SU , RU  RS
Prove: ∆RUT  ∆RST
2.) Given: AD bisects EB, AB  DE; ECD, ACB
are right angles.
Prove: ∆ACB  ∆DCE
PRACTICE:
1)
GIVEN:
QS  PR , PS  RS , Q R RS

PROVE: PRS  QSR
2)
GIVEN: OM  LN , ML  MN,
PROVE: OML  OMN
R
Statements
Reasons
1.
1. Given
2.
2. If 2 angles are , then they form 4 right s.
3.
3. Right Angle Congruence Theorem
4.
4. Given
5.
OM  OM
6. OML  OMN
5.
6.
Geometry Section 4.7
___________________________________________
Start Thinking: **The methods of proving that triangles are congruent are _____________,
_____________, _____________, __________________, and ______________.
**When proving angles or sides are congruent first you ____________________________
then use _____________________________________.
DEALING WITH INTERLOCKING TRIANGLES:
1st : ______________________________________________________________________
2nd: ______________________________________________________________________
Determining what are shared angles and sides:
EX 1:
YOU TRY:
EX 2:
Example 1:
EX 2:
EX 3:
EX 4:
EX 5
EX6
EX 7
EX 8
EX 9: