Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles)

Name:______Period:______

Geometry Rules!

Chapter 4 Notes

Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles)

Congruent Figures
Corresponding Sides / Corresponding Angles
Triangle Angle-Sum Theorem
If two ______of one triangle are ______to two ______in another triangle then the third angles in both triangles are ______.

Practice:

1.) If ∆CAT ∆DOG, then complete: (draw a picture first)

_____ _____

TA = _____ _____

2.)

a) Name three pairs of corresponding angles:

b) Name three pairs of corresponding sides:

3.) The two triangles shown are congruent; complete. (It will help to rotate the triangles first, to get them in corresponding positions)

a) b)
c) EV = _____ d)
e) f) /
Isosceles Triangles
Isosceles Triangle Theorem ( )
If two sides of a triangle are congruent, then the angles opposite them are ______.
Converse of the Isosceles Triangle Theorem ( )
If two angles of a triangle are congruent, then the ______opposite them are ______.
Angle bisector in vertex angle of Isosceles Triangle:
Equilateral Triangles

Practice: Solve for x and y

4.)
/ 5.)

6.)
/ 7.)

8.) In equilateral ∆XYZ, and . Find a and b. / 9.) In equiangular ∆ABC, AB = 2x + y,
BC = 6x – 2y, and AC = 10. Solve for
x and y.
10.) What can you conclude from the picture?

11.)
Given:

Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.) Definition of Midpoint
3.)
4.)
12.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.)
3.) Substitution
4.)

Algebra Review: Collecting like terms

Simplfy:

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

Notes #23: Sections 4.2 and 4.3 (Methods of Proving Triangles Congruent)

Q: How can we prove that two triangles are congruent to each other?

A: Four ways: SSS, SAS, ASA, AAS

SSS:
______-______-______Postulate / SAS:
______-______-______Postulate
ASA:
______-______-______Postulate / AAS:
______-______-______Postulate

Are the triangles congruent? If so, write the congruence and name the postulate used.

· Redraw your triangles so they line up
· You need three congruent pairs of sides/angles to follow:
SSS, SAS, ASA, or AAS
· Look for “hidden” pieces in:
- vertical angles
- overlapping sides
- congruent angles formed by parallel lines
- bisected angles
- ITT/Converse of ITT
- midpoints
1.)

/ 2.)

3.)

/ 4.)

5.)

/ 6.)

7.)

/ 8.)

9.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.) / 1.)
2.)
3.)
10.)
Given:
Prove:
Statements / Reasons
1.)
2.)
3.)
4.) / 1.)
2.) Reflexive
3.)
4.)

Factoring Review:

1. Collect like terms

2. Factor out any common terms.

Practice:

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

Notes #24: More Proofs and Section 4.4 (Using Congruent Triangles), CPCTC

***______parts of______triangles are ______***

Are the triangles congruent? If so, write the congruence and name the postulate used.

1.)
/ 2.)

3.)
/ 4.)

5.) Complete:

a) ∆ABC ______because ______
b) AB = ____ because ______
c) AC = EC because ______. Then C is the midpoint of ______by ______. /
d) _____ because ______. Then AB ED because ______.

Complete the proofs: follow these key steps

1. Re-draw and label your picture; mark congruencies
2. Find and list 3 congruencies:
shared sides (reflexive)
vertical angles
alternate interior/corresponding angles (only when lines are )
angle bisectors
midpoints
ITT
3. State ∆∆ by SSS, SAS, ASA, or AAS
4. State part part by CPCTC
6.) Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.)
3.)
4.)
7.) Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.)
5.) / 1.)
2.) ______angles theorem
3.)
4.)
5.)
8.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.)
5.) / 1.)
2.) Definition of Midpoint
3.)
4.)
5.)
9.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.)
5.) / 1.)
2.) Definition of ______
3.)
4.)
5.)
10.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.) Definition of ______
3.)
4.)

Factor Review:

1. Combine like terms

2. Factor out the greatest common factor if possible

3. How many terms?

· 2 terms – Factor using difference of two squares

· 3 terms – Factor using X and box

· 4 terms - Factor using grouping

Two terms:

1. Factor our the greatest common factor :

2. If both terms are perfect squares factor into :

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

Notes #25: Proof Review:

1.) In equilateral , and . Solve for x and y.
/ 2.) Solve for x and y

3.) How can you prove triangles congruent? / 4.) Solve for x and y

7.) What does CPCTC stand for? / 8.) Complete:
a)
b)
c)
d)

Complete each proof by filling in the blanks.

1. Given:

AB || DE

AB @ DE

Prove:

∆ABC @ ∆EDC

1. 1. Given

2. 2. Alt Int Angles

______theorem

3. 3.

2. Given:

AB @ CD

AB || CD

Prove:

∆ADB @ ∆CBD

1. 1.

2. 2. Reflexive

3. 3. Alt Int. Angles

theorem

4. 4.

3. Given:

E is the mdpt

of TP and MR

Prove:

TM @ PR

1. 1. Given

2. TE @ PE 2.

______

3. 3.

4. ∆ @ ∆ 4.

5. 5.

4. Given:

Ð1 @ Ð4; Ð2 @ Ð3

M is the mdpt.

of AB

Prove:

AC @ BD

1. 1.

2. AM @ BM 3.

3. ∆ @ ∆ 3.

4. 4.

5. Given:

AD || ME

MD || BE

M is the mdpt.

of AB

Prove:

MD @ BE

1. 1.

2. Ð2 @ Ð4 2.

3. 3.

4. ∆ @ ∆ 4.

5. 5.

6. Given:

WO @ ZO

XO @ YO

Prove:

ÐW @ ÐZ

1. 1.

2. 2.

3. ∆ @ ∆ 3.

4. 4.

7. Given:

RS @ RT

Prove:

Ð3 @ Ð4

1. 1. Given

2. 3. ITT

3. Ð3 @ Ð1 2.

Ð4 @ Ð2

4. 4.

8. Given:

M is the mdpt

of JK

Ð1 @ Ð2

Prove:

JG @ MK

1. 1.

2. KM @ JM 2.

3. JM @ JG 3.

4. 4.

Algebra Review

Factoring Review:

Three terms; 2x2 – 16x + 30

1. divide all terms by common denominator 2(x2 – 8x +15)

2. Put quadratic in standard form; Coefficient of x2 must be 1. 2(x2 – 8x +15)

3. find factors of last term that will add up to middle coefficient 2(x – 5)(x – 3)

Practice:

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

Notes 27: Section 4.6 (Congruence in Right Triangles) Section 4.7( Using Corresponding Parts of Congruent Triangles)

HL:
______- ______-( )Postulate
/ Hypotenuse: Side opposite the right angle
Leg: Side adjacent to right angle
Which of these triangles are congruent?

Using the HL Postulate:

1.) by ______

2.) by ______

Given: is the perpendicular bisector of .

Prove:

Statements Reasons

1. 1. Given

2. and are rt. 2.

3. 3. Def. of bisector; Def. of midpoint

4. 4. Def. of right triangles

5. 5.

Proving Overlapping Triangles Congruent
For #1-5, complete the following:
a) Separate the overlapping triangles. Mark the side or angle that is/was overlapping.
b) Mark the congruent segments and congruent angles.
c) Are the triangles congruent? If yes, state the postulate used to state the triangle congruence
(SSS, SAS, ASA, AAS, or HL)
1.)

2.)

3.)

4.)

5.)

(the triangles to examine are )

For #6-9, complete the following proofs:
6.) Given:
Prove:
Statements /
Reasons
1.)
2.)
3.) / 1.)
2.)
3.)
7.) Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.)
3.)
4.)
8.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.)
3.)
4.)
9.)
Given:
Prove:
Statements /
Reasons
1.)
2.)
3.)
4.) / 1.)
2.)
3.)
4.)

Complete each proof by filling in the blanks.

1. Given:

DB @ DN

OD bisects ÐBDN

Prove:

Ð3 @ Ð4

1. DB @ DN 1. Given

2. OD bisects ÐBDN 3. Given

3. 2.

4. 4.

5. DOBD @ DOND 5.

6. 6.

7. 7.

8. 8.

9. DOBS @ DONS 9.

10. 10.

2.Given:

ÐNGI @ ÐNAI

Ð1 @ Ð2

Prove:

GT @ AT

1. ÐNGI @ ÐNAI 1. Given

2. 2.

3. Ð1 @ Ð2 3. Given

4. DGIN @ DAIN 4.

5. 5.

6. 6.

7. 7.

8. DGTN @ DATN 8.

9. 9.

3. Given:

BU @ CH

UC @ HB

Ð1 @ Ð2

Prove:

UE @ HL

1. BU @ CH 1. Given

2. UC @ HB 2. Given

3. 3.

4. DBUC @ DCHB 4.

5. ÐBCU @ ÐCBH 5.

6. Ð1 @ Ð2 6. Given

7. DUCE @ DHBL 7.

8. 8.

4. Given:

UC || HB

UB || HC

BE @ CL

Prove:

Ð1 @ Ð2

1. UC || HB 1. Given

2. 2.

3. 3.

4. UB || HC 4. Given

5. 5.

6. DBUC @ DCHB 6.

7. 7.

8. BE @ CL 8. Given

9. DBUE @ DCHL 9.

10. 10.

Algebra Review: Factoring quadratics with an x2 coefficient not equal to 1.

1. Put in standard form

2. Divide by greatest common factor if possible

3. Use X and box to factor

Practice:

1.) 2.)

3.) 4.)

5.) 6.)

Notes 28: Chapter 4 Review:

Proof Review

1.) Given: X is the mdpt. of

Prove: @

1. ____ 1. Given

2. ______2.

3. _____ 3.

4. 4.

5. 5.

2. Given: @

Prove: @

1. LM @ JK; LM || JK 1. Given

2. 2. Alt. int. angle thm.

3. 3.

4. ______4.

5. 5.

3.) Given:
Prove: /
1.______
2.______
3.______
4.______
5.______ / 1.______
2.______
3.______
4.______
5.______
4.) Given:
Prove: /
1.______
2.______
3.______
4.______ / 1.______
2.______
3.______
4.______

Are the triangles congruent? If so, write the congruence and name the postulate used.

5.)

/ 6.)

7.)

/ 8.)

9.)

/ 10.)

Algebra Review

1.) Factor: / 2.) Factor:
3.) Factor: / 4.) Factor:
5.) Factor: / 6.) Factor:
7.) Factor: / 8.) Factor:
9.) Factor: / 10.) Factor:

Chapter 4 Study Guide:

1. Given:
Prove: /
Statements Reasons
1. ______1. ______
2. ______2. ______
______
3. ______3. ______
4. ______4. ______
5. ______5. ______
2. Given:
Prove: /
Statements Reasons
1. ______1. ______
2. ______2. ______
3. ______3. ______
4. ______4. ______CPCTC______
5. ______5. ______
3. Given:
Prove: /
Statements Reasons
1. ______1. ______
2. ______2. ______
3. ______3. ______
4. ______4. ______
5. ______5. ______
6. ______6. ______
4. Given:
Prove: /
Statements Reasons
1. ______1. ______
2. ______2. ______
3. ______3. ______
4. ______4. ______
5. is equilateral. If and , solve for x and y.
6. In . If solve for
7. Are the pairs of triangles congruent? If so, name the congruence and the postulate used.
a) / b)
c) / d)

8. a) Solve for x: / b) Solve for y:

9. Factor:(Show work on separate sheet of paper)

a) b)

c) d)

e) f)

g) h)