Congruent Triangles and Proofs Name_________________________ Pre-AP Geometry 4-2, 4-3, and 4-6 Date________________ Period_____

Mark each of these diagrams with the information given. Then fill in the congruence statement and decide what reason (SSS, SAS, ASA, AAS, or HL) makes the triangles congruent. Last, justify your answer with a two-column proof. If they cannot be proven congruent, write not possible.

Statements Reasons DCAE ≅ Given

∠AEC ≅ ∠DCE Given EX: 2 pairs of sides & 1 pair of

included angles: SAS! Reflexive Prop of ≅ ECEC ≅

∆ACE ≅ ∆DEC SAS

(Hint: This proof has 2 steps that are not "Given", but all the proofs below have 3, except for three proofs that have 4 and one proof that needs only 1. See your warm-up and class notes for the "Reasons" that you will need to use below.)

S: A: S:

DCAE ≅ ∠AEC ≅ ∠DCE ECEC ≅

DEC SAS

12. BE ≅ RA ,

R E

BR ⊥ RE , AE ⊥ RE ΔBRE ≅ Δ__________ by _________

A

D

B

Name: ________________________________________ Date: ________________ Period: _______ Notes - CPCTC C ____________________ P ____________________ of C ____________________ T ____________________ are C ____________________

ΔABC ≅ ΔDEF

F

E

D

C

B

A

**If we START with congruent triangles, THEN we can say we have CORRESPONDING parts. List the corresponding parts: **We can also CONCLUDE from other information that triangles are congruent, THEN we have corresponding parts. The ORDER of the letters is IMPORTANT!!! Use your triangle shortcut properties to prove congruent triangles. Write the triangle congruency statement. Then write the corresponding parts using CPCTC. 1. Given: (see diagram on the right)

U L

B

D E

R Prove: R B∠ ≅ ∠ , , D U∠ ≅ ∠ DE UL≅

Statements Reasons

1. _________________________ 1. _________________________

2. _________________________ 2. _________________________

3. _________________________ 3. _________________________

4. _________________________ 4. _________________________

5. __________, __________, and 5. _________________________

__________

2. Given: CX bisects UZ at A. Z

A

C Prove: (the remaining corresponding parts)

Statements Reasons

1. _________________________ 1. _________________________

2. _________________________ 2. _________________________ X 3. _________________________ 3. _________________________

U 4. _________________________ 4. _________________________

5. _________________________ 5. _________________________

6. __________, __________, and 6. _________________________

__________

Proofs:

41

3 2C

BA

D

3. Given: AD BC , A C∠ ∠ ≅Prove: AB CD≅

Statements Reasons

1. AD BC 1. ________________________________

2. ________________________________ 2. Alternate Interior Angles Theorem

3. 3. ________________________________ A∠ ≅ ∠C

4. DB DB≅ 4. ________________________________

5. ABD CDBΔ ≅ Δ 5. ________________________________

6. AB CD≅ 6. ________________________________

4. Given: AC BD⊥ , C is the midpoint of BD . Prove: AC bisects BAD∠

Statements Reasons DC

A

3 4

1 2B

1. AC BD⊥ 1. ________________________________

2. ∠ 1 and 2 are right angles 2. ________________________________ ∠

3. 3. ________________________________ 1∠ ≅ ∠2

4. C is the midpoint of BD 4. ________________________________

5. BC CD≅ 5. ________________________________

6. AC AC≅ 6. ________________________________

7. BCA DCAΔ ≅ Δ 7. ________________________________

8. 8. ________________________________ 3∠ ≅ ∠4

9. AC bisects BAD∠ 9. ________________________________

5. Given: ΔCAN is an isosceles triangle with vertex ∠N, CA BE .

Prove: BN EN≅

Statements Reasons

1. ________________________________ 1. Given

2. ________________________________ 2. Definition of Isosceles Triangle

A

B

C

E

N

1 2

3 4

3. ________________________________ 3. Isosceles Triangle Theorem (If 2 sides of a ∆ are ≅, then the angles opposite those sides are ≅.)

4. ________________________________ 4. Given

5. ______________ and ______________ 5. Corresponding Angles Postulate

6. ________________________________ 6. Transitive Property of Congruence

7. ________________________________ 7. Converse of the Isosceles Triangle Theorem (If 2 angles of a ∆ are ≅, then the sides opposite those angles are ≅.)

Congruent Triangles Recording Document Label the drawing with the information given by your partner. Name the triangle that is congruent to the given triangle and state why the triangles are congruent. Use all eight theorems (SSS, SAS, ASA, AAS, HA, HL, LL, LA)

1. ΔPOR ≅ Δ_________ by ________

2. ΔLFE ≅ Δ_________ by ________

3. ΔPNO ≅ Δ_________ by ________

4. ΔHOM ≅ Δ_________ by ________

5. ΔMIT ≅ Δ_________ by ________

6. ΔRUJ ≅ Δ_________ by ________

7. ΔXTE ≅ Δ_________ by ________

8. ΔNEW ≅ Δ_________ by ________

L

E T F

P

E O N

H

M

H

O E

Y R

J

U

T

E

X A

S W S

N E

P

R T

O

T

I M

E

9. ΔPTC ≅ Δ_________ by ________

10. ΔTUH ≅ Δ_________ by ________

11. ΔMAO ≅ Δ_________ by ________

12. ΔNOC ≅ Δ_________ by ________

13. ΔULP ≅ Δ_________ by ________

14. ΔHRO ≅ Δ_________ by ________

15. ΔDGI ≅ Δ_________ by ________

16. ΔANY ≅ Δ_________ by ________

T C

I P

U

H

O T

A

M

R

O

N E

O C

A P

L U

I

G

D

E

R N S

K N

Y

A

H

R O

M

E

Picture: Description:

Draw the median from A to

BC : (include all necessary markings)

CB

A

Significance: Point of Concurrency:

Think about the Median of a road, which goes down the middle of the road

Picture: Description: Draw the altitude from A

to BC : (include all necessary markings)

CB

A

Significance: Point of Concurrency:

The pilot announces, “Passengers, we are flying

at an A ltitude of 20,000 ft – this is our height”

Picture: Description: Draw the perpendicular

bisector of

Picture: Description: Draw the angle bisector of

BC : (include all necessary markings)

CB

A

Significance: Point of Concurrency:

Perpendicular Bisector says: “Give me a midpoint and make me perpendicular, if I go through a vertex too, then that’s just a bonus” ☺

A∠ : (include all necessary markings)

A

CB

Significance: Point of Concurrency:

Remember, “bi” means two and “sect” means cut – so Angle Bisector cuts an angle into two equal parts

Special Segments

Special Segments of Triangles Name:___________________________ Coordinate Plane Date:_________________ Period:_____ Graph ΔABC with vertices A(-6, 1), B(2, 6), and C(6, -7). Then find the following (show work on a separate sheet): 1) What are the coordinates of M if BM is a median of ΔABC? M_______ 2) Slope of BM = _______ 3) Equation of BM ________________ 4) If BH is an altitude of ΔABC which 7) What are the coordinates of D if vertex is a point on BH ? ____(____,____) D is the midpoint of AC ? D_______ 5) Slope of AC = _______ ⊥ slope=______ 8) Slope of AC = _______ ⊥ slope= ______ (altitude slope) 6) Equation of BH ________________ 9) Equation of perpendicular bisector of AC

________________

10) What do the median BM and altitude BH have in common? ________________________

What do the median BM and ⊥ bisector of AC have in common? ____________________

What do the altitude BH and ⊥ bisector of AC have in common? ____________________

Graph ΔABC with vertices A(1,3), B(5,-4) and C(-3,-3). Then find the following (show work on a separate sheet):

11) Equation of AB _______________

12) Equation of BC _______________

13) Equation of AC _______________

14) AD is a median of ∆ABC. Find the coordinates of D_______ Show your work.

Find and draw the perpendicular bisector of AC . Follow these steps:

15) Find the coordinates of E where the ⊥ bisector crosses AC . E_______ Show your work. Graph this point.

16) What is the slope of the ⊥ bisector of AC ? _______ Explain how you know. 17) Use the slope of the ⊥ bisector to “count” from E to find another point on the ⊥ bisector.

Label this point F. What are its coordinates? F_______ 18) What is the equation of the line containing the ⊥ bisector? _______________

19) Draw EF . What other special segments (besides the perpendicular bisector) does this line contain? Justify your answer.

20) Based on the sides, what kind of triangle do you think this is? Without measuring, justify your answer.

Show your work.

Special Segments of Triangles Name:_____________________ More Practice Date:____________ Period:____ Draw and label a figure to illustrate each situation. Be sure to mark your drawing appropriately (i.e. congruent angles, congruent segments, etc.) 1) OQ is a median of Δ POM. 2) KT is an altitude of Δ KLM

and L is between T and M. 3) HS is an angle bisector of Δ GHI. 4) Δ KHS is a right triangle where KH and KS are

altitudes, and BH is a perpendicular bisector. Use the drawing and the given information to determine if each statement is true or false. ____5) If BD ⊥ AC, then BD is a median. ____6) If CF is a median of Δ ABC, then AF = FB. ____7) If AE is an angle bisector of Δ ABC and

m∠ BAE = 30 , then m ∠ EAC = 60. ____8) If AE is an angle bisector of Δ ABC, then BE = EC. ____9) If AB is an altitude of Δ ABC, then ∠ ABC is a right angle.

A

B

CD

EF

____10) If BD is a perpendicular bisector of Δ ABC, then AD = DC. ____11) If CF is an angle bisector of Δ ABC and ∠ CAE is an acute angle, then ∠ BAE is

an acute angle. Determine if each statement is true or false. ____12) An angle bisector of a triangle can be on the exterior of the triangle. ____13) A perpendicular bisector of a triangle must pass through the midpoint of one

side of the triangle. ____14) A median of a triangle must pass through a vertex of the triangle. ____15) An altitude of a triangle must be perpendicular to one side of the triangle.

16. Identify each of the following as a circumcenter, incenter, orthocenter, or centroid.

Demos are available at http://www.keymath.com/x23078.xml and http://www.keymath.com/x19398.xml Theorem 5-1 Triangle Midsegment Theorem: If a segment joins the __________ of two sides of a triangle, then the segment is __________ to the third side and is __________ its length. 17. In ΔLOB, the points A, R, and T are midpoints. LB = 19 cm, LO = 35 cm, and OB = 29 cm.

a. Find RT. b. Find AT. c. Find AR.

Use the given measures to identify three pairs of parallel segments in each diagram. 18. 19. 20. A triangle has vertices at A (-4, 6), B(1, 7), and C(4,2).

a) Find the equation of the line segment formed by the median from point B.

b) Find the equation of the line formed by the perpendicular bisector of AC.

c) Find the equation of the line segment formed by the altitude from point B.

d) Find the equation of the line segment formed by the altitude from point A.

e) Find the location of the orthocenter of the triangle by solving the system of equations from c) and d) for both x and y.

Review for Pre-AP Test 2.3: Congruence & Special Segments Name: ___________________________ in ∆s (chapters 4 & 5, except 5-4 and 5-5) and previous units! Date: ________________ Period: _____ In addition to this review, study your homework and quizzes for this unit! 1. Given: ΔTHK is isosceles with base KH , ΔTAN is isosceles with base AN , ∠1 ≅ ∠2

K

N

A

H

T

1 2

Prove: C

M

2. Given: In ∆ABC, CM is the median to AB . AB bisects CP . Prove: ∠CBM ≅ ∠PAM

3. Given: ∠3 ≅ ∠4, ∠1 ≅ ∠2.

Prove: ΔABC is isosceles with base AC 4. Given: BA ≅ DA , EA bisects ∠BAD, BE ⊥ CB , and ED is the altitude to CD .

Prove: CA bisects ∠BCD 5. Given: AB ⎪⎪ DE , AB DE≅ , BC FE≅

Prove: ∆ABF ∆DEC ≅

F

C D

E

B

A

A

B

C

E1 2

3 4

B

C

D

A E 2 1

4 3

5

6

A B

P

6. Luis began walking up a hill at a spot where the 7. elevation in 0.9 km. After he walked 3 km, he saw a sign indicating the elevation as 0.95 km. At this rate, how far will he have walked when he reaches an elevation of 1.1 km.? (Hint: draw a diagram, and then redraw the overlapping triangles separately.) 8

12

x + 3

2x - 4

8. The point (3, 2) is reflected about a certain straight line. If the image is located at (5, 6), what is the equation of the line of reflection?

For each transformation below, give the mapping notation and find the image of the point (–3, 8). 9. reflection over the x-axis 13. 180◦ rotation 10. 90◦ rotation 14. 270◦ rotation 11. translation to the right 4 and down 8 15. translation to the left 4 and up 5 12. reflection over the y-axis 16. dilation of 1/4 17. For a regular nonagon, find the measure of each exterior angle _____ and interior angle _____. 18. Find x, the measures of the angles in the triangle, and classify the triangle by angles and by sides.

(8x + 6)°

12x ° 30°

19. In the following pentagon, ST TR⊥ and VR TR⊥ . S V∠ ≅ ∠ . m∠M = 40º. Find the m∠S. M

VS

RT

20. Three regular polygons meet at point A. How many sides does the largest polygon have? 

  21. Review special segments: writing the linear equation of a median, altitude, and perpendicular bisector.