Assignments Triangle Congruence PDF

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Geometry Assignments: Triangle Congruence

Day

Topics

Homework

1

Congruent triangles, SAS

HW TC - 1

2

ASA; CPCTC

HW TC - 2

3

SSS

HW TC - 3

4

Practice

HW TC - 4

5

Isosceles Triangle Theorem

HW TC - 5

6

Practice

HW TC - 6

7

AAS ***QUIZ***

HW TC - 7

8

RHL

HW TC - 8

9

Practice ***QUIZ***

HW TC - 9

10

Indirect proof

HW TC - 10

11

Review

HW TC - Review

12

***TEST***

***QUIZ***

HW Grade

Quiz Grade

Answers to HW -10 1. 2 is a rational number. 2. There is a largest prime number. 3. There is a smallest positive rational number. 4. Suppose a triangle had two right angles. They would sum to 180. This would mean the third “angle” would be 0 which is not possible. So a triangle can’t have more than one right angle.  A B 5. Suppose there are two different lines, PA and PB through P perpendicular to l. Then 1+ APB + 2 = 180 (straight line). But 1 + 2 = 180 (both right angles) so mAPB = 0. This would mean PA and PB were the same line, contradicting the 1 2 assumption they are two different lines. P l 6. Suppose 1 and 2 are supplementary and lines l and m are not parallel. This would mean that those lines intersected somewhere and that they and the transversal form a triangle. But two angles of this triangle already add up to 180 (they’re supplementary) so the third “angle” would be 0 which is not possible. So l and m can’t intersect; they must be parallel. 7. Statement Reason 1. Given 1. ADB, AEC , BE and CD 2. Assume A  C (A) 2. Assumption 3. Given 3. AB BC (S) 4. B  B (A) 4. Reflexive 5. BAE  BCD 5. ASA (2, 3, 4) 6. BD  BE 6. CPCTC (5) 7. Given 7. BD  BE 8. A  C 8. Contradiction in lines 6 and 7 mean assumption in line 2 is wrong. Review Answers 1. BCD by SAS 2. CDE by ASA 3. May not be  4. CDB by SAS 5. CBD by HL 6. CEF by AAS 7. 28 8. 110 9. (4) 10a. 74 b. M 11. (3) 12. (3) 13a. OT ( PT 14. Statement 1. Scalene ABC with median CM 2. CM is an altitude 3. AM MB (S) 4. 5. 6. 7. 8. 9.

CM  AMB AMC  BMC (A) CM CM (S) AMC  BMC AC BC CM is not an altitude

Reason 1. Given 2. Assumption 3. A median goes to the midpoint of the opp. side; a mdpt. divides a seg. into 2  parts 4. An altitude is  to the base 5.  segs. form rt. s and all rt. s are  6. Reflexive 7. SAS (3, 5, 6) 8. CPCTC 9. Contradiction in steps 1 and 8

15. Statement 1. AB  BC , CD  BC 2. ABE and DCE are rt. s 3. ADE is isosceles with vertex E 4. AE DE (H) 5. E is the midpoint of BC 6. BE EC (L) 7. ABE  DCE 8. BEA  CED 9. AED  AED 10. BED  CEA

Reason 1. Given 2.  segs. form rt. s; triangles with rt. s are rt. s 3. Given 4. Legs of an iso.  are  5. Given 6. A mdpt. divides a seg. into 2  parts 7. HL (3, 5, 6) 8. CPCTC 9. Reflexive 10. Addition (8, 9)

16a. Statement 1. AE  AB , CD  DF 2. A  D (A) 3. AE || DF , BFEC 4. AEF  CFD (A) 5. CE  BF 6. EF  EF 7. CF  BE (S) 8. ABE  DCE 9. AE  DF 10. B  C 11. AB || CD

Reason 1. Given 2.  segs. form rt. s; all rt. s are  3. Given 4. When 2 lines are ||, alt. int. s are  5. Given 6. Reflexive 7. Addition (5, 6) 8. AAS (2, 4, 7) 9. CPCTC 10. CPCTC 11. When alt. int. s are  (9), lines are ||

b. 16 17a. B. Under a translation along ABDE , the image of E, call it E', will still be on that line. Since AB DE and translations preserve distance, we know AE '  AB so E' must be B. b. Since angle measure is preserved in rigid motions, F'AB  CAB and after the reflection, ray AF ' will coincide with ray AC . Also, since distance is preserved in rigid motions, AF ' AC so after the reflection, the image of F’ will be C. c. Since ABC is the image of DEF after a rigid motion, the two triangles are congruent.

STUFF YOU SHOULD KNOW: Ways to prove two triangles congruent SAS ASA SSS AAS RHL Corresponding parts of congruent triangles are congruent (CPCTC) If two sides of a triangle are congruent, the angles opposite those sides are also congruent (base angles of an isosceles triangle are congruent). If two angles of a triangle are congruent, the sides opposite those angles are also congruent (if two angles of a triangle are congruent, the triangle is isosceles).

Geometry HW: Triangle Congruence - 1 Name For the following four problems, determine if the information given in the diagram is sufficient to prove the triangles congruent and give a reason for your answer. 1. F B

2.

D

D

C

3. B

A A

E A

4. C

D

C

E

D

B

A

B

C

In the next three problems, name the pair of corresponding sides or angles that would need to be proved congruent, in addition to the ones already shown, in order to prove the triangles are congruent by SAS. C A D A 5. 6. 7. C

C B

E

B A

D

B D B

Write complete geometry proofs for the following (diagrams below): 8. Given: AB  AD , AC bisects BAD Prove: ABC  ADC

E

F

C

A D

R

9. Given: AS  RT , A is the midpoint of RT Prove: RAS  TAS

S

A T

10. Given: PQ  RS , PQ || RS , QUTS , QU  ST Prove: PQT  RSU

P

Q U

T S

R

A

11. In the diagram at right, AB  AY ,BAC  YAZ and AC  AZ . a. How do we know ABC  AYZ? b. A rotation maps ray AC onto ray AZ . The image of B is B'. Explain using the properties of rigid motions how we know point C maps onto point Z.

B

Y C

Z

c. AB'Z is reflected over AZ . Explain using the properties of rigid motions how we know point B' maps onto point Y

d. Explain how these two transformations verify that ABC  AYZ.

Geometry HW: Triangle Congruence - 2 Name In the following three problems, determine if the information given in the diagram is sufficient to prove the triangles congruent. Give a reason for your answer. E

1.

C

2.

D

3.

D

B E

A

B

A

C

B

D

A

C

In the next three problems, two triangles are shown with two pairs of corresponding parts (angles or sides) labeled congruent. For each problem a. name the third pair of corresponding parts that would need to be proved congruent to prove the triangles are congruent by ASA. b. Give a possible reason (other than simply "given") why the parts might be congruent and c. Tell what given(s) (if any) would be needed to support your reason. D

4.

C

5.

B

D

6.

D

C

A

B

E A

B

A

C

B

Write complete geometry proofs for the following: 7. Given: AFCD , FE  AD , BC  AD , AF  CD , BC  EF Prove: a.    D b. AB || ED

A

C F E

D

8. Given: Rhombus* ABCD with sides AD extended to Q and CD extended to P, AP  AD , CQ  CD Prove: P  Q *Recall: A rhombus is a quadrilateral with all four sides congruent.

B A

C D

P

Q

Q 9. Given: ST || PR , PSQ , ST bisects PSQ , M is the midpoint of PR , ST MR a. Prove: QT SM b. If m1 = x2 – 4 and m2 = 6x + 12, find the numerical value of m1.

T

1

P

7. In the diagram at right, 1  2 and 3  4. Triangle ABC is reflected over AB . a. Explain why the image of A is A and the image of B is B.

2

S

A 1 2 C

3 4 B b. Explain using the properties of rigid motions why a reflection over will map point C onto point D.

c. Explain how this proves ABC  ABD.

R

M

D

Geometry HW: Triangle Congruence - 3 Name In the following three problems, determine if the information given in the diagram is sufficient to prove the triangles congruent. Give a reason for your answer. P C

1. D

2.

3.

A B

A

D

E

D B

A

C

C

B

In the next three problems, name the pair of corresponding sides or angles that would need to be proved congruent, in addition to the ones already shown, in order to prove the triangles are congruent by SSS. B A D A 4. 5. 6. A

D

C D

B B

C

E

C

Write geometry proofs for the following:

F

T

F

7. Given: Isosceles TYF with vertex Y, Y is the midpoint of RL , TR  FL . Prove: R  L R

Y

L

S

8. Given: SR ST , RU TU , SVU Prove: a. SRU  STU b. SU bisects RST R c. RV TV (You will need a second pair of congruent triangles.)

T V U

F 9. Given: AB || DE , ADCF , BC  AF , EF  AF , AD  CF Prove: a. ABC  DEF b. CB || FE

C E

D

A

10. In the diagram at right, APB and CPD with AP  BP and CP DP . a. Explain using properties of rigid motions why a 180 rotation about point P will prove APD  BPC.

B

A C P D B

b. Explain why this does this does not prove that “side-side” is sufficient to prove two triangles congruent.

Geometry HW: Triangle Congruence - 4

Name 1. In the diagram at right, A, B and C are collinear, AB BC , AE BD and BAE  CBD. a. Tell which triangle congruence theorem proves ABE  BDC.

C D

B b. A translation maps B to C. Let the image of E be E'. What is the image of A? Justify your answer using the properties of rigid motions. A E c. Name a second rigid motion that will map BE'C onto BDC. Explain using the properties of rigid motions why E' must map onto D.

Write geometry proofs for the following: F 2. Given: ABCDE and FCG , C is the midpoint of BD , ABF  EDG Prove: a. BFC  DGC b. F  G c. BF || DG

D A

B

E

C G

3. Given: AB || CD , BFED , AE  BD , CF  BD DE  FB Prove: AE FC

A

B F E

D

C

S 4. Given: SXR , SYT , SX  SY , XR  YT Prove: RSY  TSX

X R

Y T

Geometry HW: Triangle Congruence - 5 Name 1. In ABC, if AB BC and mB = 80 find mC.

2. Each of the congruent angles of an isosceles triangle measures 9 less than 4 times the vertex angle. Find the measures of all three angles of the triangle.

Find the value of x in each of the diagrams below. A 3.

B

B

4.

x

130

D

A

C

50

x = mABD

D C

55

A

x AD  DE , AB  CB , ACD and BCE

AD BD , BDC mADC = 130

B

5.

50

C

E

AB  BC CD

Find the values of x and y in each of the diagrams below. 6.

(2x)

(5x)

l1

7. y

l1 || l1 y

(2x) l2

(3x)

D

8. In ABC, A  C, AB = 5x + 6, BC = 3x + 14, and AC = 6x – 1. Find the lengths of all three sides of the triangle.

9. Prove the following: In an isosceles triangle, the median from the vertex, the altitude from the vertex and the angle bisector of the vertex are all the same. Given:

Isosceles triangle ABC with AC BC , CP bisects ACB

a. Prove (in statement – reason format) that CP is a median.

b. Explain (in paragraph format) why CP is an altitude.

10. Given: Isosceles ABC with AC BC , M is the midpoint of AB , AME  BMF Prove: CE CF

C

E A

11. The Isosceles Triangle Theorem can be proven using transformations. Given ABC with AC BC and let D be on AB so that CD is an angle bisector. Reflect ACD over CD . a. Explain why the image of C is C and the image of D is D.

M

B

C

1 2

A

b. Explain using the properties of rigid motions how we know the image of A is B.

F

D

B

Geometry HW: Triangle Congruence - 6 Name 1. In ABC, A  B, AB = 2x + 3, BC = 3x – 2, and AC = x + 6. Determine if the vertex angle of the triangle is larger or smaller than the base angles and justify your answer.

2. In EFG, EF FG . If mE = 2x + 10, mF = 2y – x, and mG = y. Find the numerical values of the measures of all three angles of the triangle.

3. In KLM, K  M, KL is 10 less than twice KM, and the perimeter is 70. Find the lengths of the sides.

Find the values of x and y in the diagrams below. B

4.

5. D

B x

60

y 50

A

B

6.

x

C

In ABC, AC BC and AD is an altitude.

A

110 y

x D

C AB BC , AD BD

A

D y

C

In ABC, AC BC and AD is an  bisector.

G

7. Given: GOT  GAS , GO GA , GOT  OM , GAS  AM Prove: OM  AM

A

O T

S

M

J

K

8. Given: Quadrilateral HJKL, JKM is isosceles with vertex M, HJ JK , JK KL , and M is the midpoint of HL Prove: H  L H

M

L

T

9. Given: BUG , HUT , GBH  THB, TU GU Prove: BT GH

G U

B

H

Geometry HW: Triangle Congruence - 7 Name In the following six problems, determine if the information given in the diagram is sufficient to prove the triangles congruent. Give a reason for your answer. B

1.

2.

B

A

3.

B

C

C

A

A

E C

D

E

D

D

E

4. A

5.

B

D

6.

D

C

C

B C

D A

E

F

A

B

Write geometry proofs for the following: 7. Given: BD bisects ABC, AB  AD , CB  CD Prove: AD CD

B

C

A

D

8. We want to prove the converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are also congruent. Given: ABC with A  B Prove: AC BC Note: Remember, we cannot use a theorem to prove itself. Start off the same way we did in the notes when we proved the Isosceles Triangle Theorem.

A 9. Given: Isosceles ABC with AB  AC , BDFC , BF  DC , ED  BC , GF  BC Prove: ED  FG

E

B

D

G

F

C

10. Given: Isosceles ABC with AC  BC , ADB  BEA Prove: AE  BD

C

D A a

E B

Geometry HW: Triangle Congruence - 8 Name In the following three problems, determine if the information given in the diagram is sufficient to prove the triangles congruent. Give a reason for your answer. C E B 1. 2. 3. A A A

4. A

B C

E

D

B

F

B

5.

C

A

B

6. B

D C

D

C

D

E

C

F

A

D E

B Write complete geometry proofs for the following: 7. Given: Isosceles ABC with vertex B, AB  AD , CB  CD . Prove: a. ABD  CBD b. BD bisects ABC

A

C

D

8. Given: AFEC , BE  AC , DF  AC , ABE  CDF, AF  CE Prove: a. ABE  CDF b. AB || CD

D

C E F

A

B

C 9. Given: M is the midpoint of AB , ME  AC , MF  CB , ME  MF Prove: a. CAB  CBA b. ABC is isosceles

E A

F M

B

10. Given: In PRQ, P  Q; PSR , QTR , PS QT , RV  SVT a. Prove: SRV  TRV 2 b. If SV = x – y, VT = y + 1, SR = 3x – 4y and RT = x  y , find the value of RV. 3 S P

R

V

T Q

Geometry HW: Triangle Congruence - 9 Name In the following eight problems, determine if the information given in the diagram is sufficient to prove the triangles congruent. Give a reason for your answer. A A 1. 2. A 3. C B D E B C b D B D C E

C

4.

A

7.

A

B

D

C

5.

B

A

C

8.

D

D

C

6.

A

B

D

C

A

B

D

B

We want to prove that the points equidistant from the sides (rays) of an angle are exactly those points on the angle bisector. In logic terms, we need to prove: A point is equidistant from the sides of an angle if and only if it is on the angle bisector. This is a biconditional; we need prove each part separately. 9. Given: BAC, AZ is the angle bisector of BAC, and P is a point on AZ . a. In the space at right, draw a diagram. (Do not draw a tiny diagram. Do not make P too close to A.)

b. Recall (from coordinate geometry unit): How do we measure the distance form a point to a line? c. Draw perpendiculars from P to AB and AC . Label the points of intersection F and G. d. Prove: PF PG

10. Given: BAC with point P in the interior, F is on AB and G is on AC such that PF  AB and PG  AC , PF PG . Prove: AP bisects BAC

B F P

A

G

11. The definition of a parallelogram is a quadrilateral with both pairs of opposite sides parallel. You should already know a bunch of additional facts about parallelograms. We want to prove the following: In a parallelogram, a. opposite sides are congruent AND b. opposite angles are congruent. Given: Parallelogram ABCD with diagonal BD Prove: a. AB CD and AD BC b. A  C

C

12. Triangle ABC is congruent to DEF. If BC is represented by 3x + 2, EF is represented by x + 10, and AB is represented by x + 2, then find a) The value of x

b) The numerical value of AB

c) The numerical value of DE.

13. a. If CAT  DOG, then CT (choose one) must/may/cannot be congruent to DG . b. If COW  PIG, then CW (choose one) must/may/cannot be congruent to PG .

Geometry HW: Triangle Congruence - 10 Name State the assumption that would be made to prove each of the following indirectly. 1.

2 is an i...