Math B Assignments: Introduction to Proofs



Geometry Assignments: Triangle Congruence

|Day |Topics |Homework |HW Grade |Quiz Grade |

|1 |Congruent triangles, SAS |HW TC - 1 | | |

|2 |ASA; CPCTC |HW TC - 2 | | |

|3 |SSS ***QUIZ*** |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*** | | | |

Answers to HW -10

1. [pic] 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.

5. Suppose there are two different lines, [pic] and [pic] 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 [pic] and [pic] were the same line, contradicting the

assumption they are two different lines.

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. [pic] 1. Given

2. Assume (A ( (C (A) 2. Assumption

3. [pic] (S) 3. Given

4. (B ( (B (A) 4. Reflexive

5. (BAE ( (BCD 5. ASA (2, 3, 4)

6. [pic]([pic] 6. CPCTC (5)

7. [pic]([pic] 7. Given

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. [pic] [pic][pic]

14. Statement Reason

1. Scalene (ABC with median [pic] 1. Given

2. [pic] is an altitude 2. Assumption

3. [pic] (S) 3. A median goes to the midpoint of the opp. side; a mdpt. divides a seg. into 2 ( parts

4. [pic] 4. An altitude is ( to the base

5. (AMC ( (BMC (A) 5. ( segs. form rt. (s and all rt. (s are (

6. [pic] (S) 6. Reflexive

7. (AMC ( (BMC 7. SAS (3, 5, 6)

8. [pic] 8. CPCTC

9. [pic] is not an altitude 9. Contradiction in steps 1 and 8

15. Statement Reason

1. [pic]⊥ [pic], [pic]⊥ [pic] 1. Given

2. (ABE and (DCE are rt. (s 2. ( segs. form rt. (s; triangles with rt. (s are rt. (s

3. ΔADE is isosceles with vertex E 3. Given

4. [pic] (H) 4. Legs of an iso. ( are (

5. E is the midpoint of [pic] 5. Given

6. [pic] (L) 6. A mdpt. divides a seg. into 2 ( parts

7. (ABE ( (DCE 7. HL (3, 5, 6)

8. (BEA ” (CED 8. CPCTC

9. (AED ( (AED 9. Reflexive

10. (BED ” (CEA 10. Addition (8, 9)

16a. Statement Reason

1. [pic] ⊥ [pic], [pic] ⊥ [pic] 1. Given

2. (A ( (D (A) 2. ( segs. form rt. (s; all rt. (s are (

3. [pic] || [pic], [pic] 3. Given

4. (AEF ( (CFD (A) 4. When 2 lines are ||, alt. int. s are (

5. [pic] ” [pic] 5. Given

6. [pic] 6. Reflexive

7. [pic] (S) 7. Addition (5, 6)

8. (ABE ( (DCE 8. AAS (2, 4, 7)

9. [pic] ” [pic] 9. CPCTC

10. (B ( (C 10. CPCTC

11. [pic] || [pic] 11. When alt. int. (s are ( (9), lines are ||

b. 16

17a. B. Under a translation along [pic], the image of E, call it E', will still be on that line. Since [pic] and translations preserve distance, we know [pic] so E' must be B.

b. Since angle measure is preserved in rigid motions, (F'AB ( (CAB and after the reflection, ray [pic] will coincide with ray [pic]. Also, since distance is preserved in rigid motions, [pic] 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. 2. 3. 4.

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.

5. 6. 7.

Write complete geometry proofs for the following (diagrams below):

8. Given: [pic] ” [pic], [pic] bisects ∠BAD

Prove: ΔABC ” ΔADC

9. Given: [pic] ⊥ [pic], A is the midpoint of [pic]

Prove: ΔRAS ” ΔTAS

10. Given: [pic] ” [pic], [pic] || [pic], [pic], [pic] ” [pic]

Prove: ΔPQT ” ΔRSU

11. In the diagram at right, [pic],(BAC ( (YAZ and [pic].

a. How do we know (ABC ( (AYZ?

b. A rotation maps ray [pic] onto ray [pic]. The image of B is B'. Explain using the properties of rigid motions how we know point C maps onto point Z.

c. (AB'Z is reflected over [pic]. 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.

1. 2. 3.

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.

4. 5. 6.

Write complete geometry proofs for the following:

7. Given: [pic], [pic] ⊥ [pic], [pic] ⊥ [pic], [pic] ” [pic], [pic] ” [pic]

Prove: a. (“ ” (D

b. [pic]

8. Given: Rhombus* ABCD with sides [pic] extended to Q and [pic] extended to P, [pic] ⊥ [pic], [pic] ⊥ [pic]

Prove: (P ( (Q

*Recall: A rhombus is a quadrilateral with all four sides congruent.

9. Given: [pic],[pic], [pic] bisects [pic], M is the midpoint of [pic], [pic]

a. Prove: [pic]

b. If m(1 = x2 – 4 and m(2 = 6x + 12, find the numerical value of m(1.

7. In the diagram at right, (1 ( (2 and (3 ( (4. Triangle ABC is reflected over [pic].

a. Explain why the image of A is A and the image of B is 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.

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.

1. 2. 3.

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.

4. 5. 6.

Write geometry proofs for the following:

7. Given: Isosceles ΔTYF with vertex Y, Y is the midpoint of [pic], [pic]” [pic].

Prove: (R ” (L

8. Given: [pic], [pic], [pic]

Prove: a. ΔSRU ” ΔSTU

b. [pic] bisects (RST

c. [pic] (You will need a second pair of congruent triangles.)

9. Given: [pic]|| [pic], [pic], [pic]⊥ [pic], [pic]⊥ [pic], [pic]” [pic]

Prove: a. ΔABC ” ΔDEF

b. [pic]|| [pic]

10. In the diagram at right, [pic] and [pic] with [pic] and [pic].

a. Explain using properties of rigid motions why a 180( rotation about point P will prove (APD ( (BPC.

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, [pic], [pic] and (BAE ( (CBD.

a. Tell which triangle congruence theorem proves (ABE ( (BDC.

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.

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:

2. Given: [pic]and [pic], C is the midpoint of [pic],

∠ABF ” ∠EDG

Prove: a. ΔBFC ” ΔDGC

b. (F ( (G

c. [pic]

3. Given: [pic], [pic], [pic], [pic]

[pic]

Prove: [pic]

4. Given: [pic], [pic], [pic]” [pic], [pic]” [pic]

Prove: ΔRSY ” ΔTSX

Geometry HW: Triangle Congruence - 5

Name

1. In ΔABC, if [pic]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.

3. 4. 5.

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

6. 7.

8. In ΔABC, (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 [pic],[pic] bisects ∠ACB

a. Prove (in statement – reason format) that [pic] is a median.

b. Explain (in paragraph format) why [pic] is an altitude.

10. Given: Isosceles (ABC with [pic], M is the midpoint of [pic], (AME ( (BMF

Prove: [pic]

11. The Isosceles Triangle Theorem can be proven using transformations. Given (ABC with [pic] and let D be on [pic] so that [pic] is an angle bisector. Reflect (ACD over [pic].

a. Explain why the image of C is C and the image of D is D.

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

Geometry HW: Triangle Congruence - 6

Name

1. In ΔABC, (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, [pic]. 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 ΔKLM, (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.

4. 5. 6.

7. Given: [pic] ( [pic], [pic], [pic], [pic]

Prove: [pic]

8. Given: Quadrilateral HJKL, (JKM is isosceles with vertex M,

[pic], [pic], and M is the midpoint of [pic]

Prove: (H ( (L

9. Given: [pic], [pic], (GBH ( (THB, [pic]

Prove: [pic]

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.

1. 2. 3.

4. 5. 6.

Write geometry proofs for the following:

7. Given: [pic] bisects (ABC, [pic], [pic]

Prove: [pic]

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: [pic]

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.

9. Given: Isosceles (ABC with [pic] ” [pic], [pic], [pic] ” [pic],

[pic] ⊥ [pic], [pic] ⊥ [pic]

Prove: [pic] ” [pic]

10. Given: Isosceles (ABC with [pic] ” [pic], ∠ADB ” ∠BEA

Prove: [pic] ” [pic]

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.

1. 2. 3.

4. 5. 6.

Write complete geometry proofs for the following:

7. Given: Isosceles (ABC with vertex B,

[pic], [pic].

Prove: a. (ABD ( (CBD

b. [pic] bisects (ABC

8. Given: [pic], [pic], [pic], ∠ABE ” ∠CDF,

[pic] ” [pic]

Prove: a. ΔABE ” ΔCDF

b. [pic]

9. Given: M is the midpoint of [pic], [pic] ⊥ [pic], [pic] ⊥ [pic], [pic] ” [pic]

Prove: a. ∠CAB ” ∠CBA

b. (ABC is isosceles

10. Given: In (PRQ, (P ( (Q; [pic], [pic], [pic], [pic]

a. Prove: (SRV ( (TRV

b. If SV = x – y, VT = y + 1, SR = 3x – 4y and RT = [pic], find the value of RV.

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.

1. 2. 3.

4. 5. 6.

7. 8.

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, [pic] is the angle bisector of (BAC, and P is a point on [pic].

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 [pic] and [pic].

Label the points of intersection F and G.

d. Prove: [pic]

10. Given: (BAC with point P in the interior, F is on [pic] and G is on [pic] such that [pic] and [pic], [pic].

Prove: [pic] bisects (BAC

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 [pic]

Prove: a. [pic] and [pic]

b. (A ( (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 [pic] (choose one) must/may/cannot be congruent to [pic].

b. If (COW ( (PIG, then [pic] (choose one) must/may/cannot be congruent to [pic].

Geometry HW: Triangle Congruence - 10

Name

State the assumption that would be made to prove each of the following indirectly.

1. [pic] is an irrational number.

2. There is no largest prime number.

3. There is no smallest positive rational number.

Write indirect proofs of the following. The proofs may be done in paragraph form but must contain enough justification to be convincing. A labeled diagram can be helpful.

4. A triangle can not have more than one right angle.

5. Given a line l and a point P on l, there can be only one line through P and perpendicular to l.

6. If (1 and (2 are supplementary, then lines l and m are parallel.

Write an indirect proof in statement-reason format for the following.

7. Given: [pic], [pic]; [pic]([pic]

Prove: (A ( (C

Geometry HW: Triangle Congruence - Review

Name

For each question #1 - 6,

a) use the givens and your knowledge of geometry to mark corresponding parts of the

diagram as congruent, and then

b) either identify the triangle that is congruent to the given triangle and give a reason why

they are congruent or circle the statement "The triangles may not be congruent."

1. Given: [pic] is the perpendicular bisector of [pic].

ΔACD ” Δ by .

or

The triangles may not be congruent.

2. Given: [pic] ⊥ [pic], [pic] ⊥ [pic], [pic] bisects [pic].

ΔABE ” Δ by .

or

The triangles may not be congruent.

3. Given: [pic] ” [pic], ∠BCA ” ∠DAC.

ΔABC ” Δ by .

or

The triangles may not be congruent.

4. Given: [pic] ” [pic], [pic] bisects ∠ABC.

ΔADB ” Δ by .

or

The triangles may not be congruent.

5. Given: [pic] ⊥ [pic]. [pic] ⊥ [pic], [pic] ” [pic].

ΔABD ” Δ by .

or

The triangles may not be congruent.

6. Given: [pic], [pic], ∠A ” ∠C, [pic] ” [pic].

ΔADF ” Δ by .

or

The triangles may not be congruent.

7. In ΔPQR, (P ” (R, PQ = 3x + 2, QR = x + 8, and PR = 2x. Find the perimeter of the triangle.

8. In the diagram at right, [pic], and [pic] and [pic] are angle bisectors of (ABC. If m(A = 40(, find m(D.

9. Given ΔABC and ΔDEF with [pic]” [pic], (BCA ” (EFD, and [pic]” [pic].  Which is true?

(1) ΔABC ” ΔDEF by SAS (2) ΔABC ” ΔDEF by SSS

(3) ΔABC ” ΔDEF by SSA. (4) ΔABC and ΔDEF may not be congruent.

10. In the diagram, ΔELK ( ΔRAM.

a. Find the perimeter of (RAM.

b. Determine which is the largest angle of (RAM.

11. In ΔABC and ΔDEF, [pic]” [pic], ∠C ” ∠F. Which of the following additional pieces of information could be used to prove ΔABC ” ΔDEF?

I. ∠A ” ∠D II. [pic]” [pic] III. [pic]” [pic]

(1) I only (2) II only (3) Either I or II (4) Any of the above.

12. In the figure at right, ∠BAC ” ∠DCA and

∠BCA ” ∠DAC, which must also be true?

(1) [pic]” [pic] (2) [pic]” [pic]

(3) [pic]” [pic] (4) [pic]” [pic]

13. Gomer needs to prove the following theorem: A tangent line* to a circle is perpendicular to the radius it intersects. He has drawn the diagram shown with tangent line [pic] and radius [pic]. (*A tangent line is a line that intersects a circle in exactly one point. We will cover this later in the course.) What should be the assumption Gomer makes if he plans to do an indirect proof?

14. Write an indirect proof of the following: A median of a scalene triangle is not an altitude of the triangle.

Given: Scalene (ABC with median [pic]

Prove: [pic] is not an altitude of (ABC

15. Given: [pic]⊥ [pic], [pic]⊥ [pic], ΔADE is isosceles with vertex E,

E is the midpoint of [pic]

Prove: a. (BEA ” (CED

b. (BED ” (CEA

16. Given: [pic] || [pic], [pic] ⊥ [pic], [pic] ⊥ [pic],[pic] ” [pic], [pic].

a. Prove: 1) [pic] ” [pic]

2) [pic] || [pic]

b. If AB = x + y, CD = 3(x – y), BE = 3x – 4 and CF = [pic], find

the length of AE.

17. In the diagram, [pic], [pic], (CAB ( (EDF, and [pic].

a. After a translation, the image of D is A and the image of F is F'. What is the image of E? Justify your answer using the properties of rigid motions.

b. (AF'B is reflected over [pic]. Explain using the properties of rigid motions why the image of F' is C.

c. Explain how the transformations above prove that (DEF ( (ABC.

Geometry: Triangle Congruence Answers

HW – 1

1. Yes; SAS 2. No; not included (s 3. Yes; SAS 4. No; ( in (ABD not included

5. (A ( (E 6. [pic] 7. [pic]

HW – 2

1. Yes; ASA 2. Yes; ASA 3. Yes; SAS

4a. (DCA ( (BAC b. Alt. int. (s ( c. [pic]

5a. (EAD ( (BEC b. Vert. (s ( c. [pic]

6a. [pic] b. Reflexive c. Reflexive does not need a given

10b. 60

HW – 3

1. Yes; SSS 2. No; only SS 3. Yes; SAS

4. [pic] 5. [pic] 6. [pic]

HW – 5

1. 50 2. 79(, 79(, 22( 3. 65 4. 40 5. 135 6. (15, 105) 8. 26, 26, 23

11. (18, 40, 5)

HW – 6

1. Larger 2. 50(, 50(, 80( 3. 18, 26, 26 4. (40, 20) 5. (30, 40) 6. (40, 20)

HW – 7

1. Yes; AAS 2. Yes; SAS 3. No; SS only 4. No; ASS does not work

5. Yes; AAS 6. Yes; SSS

HW – 8

1. Yes; HL 2. Yes; SAS 3. No; ASS 4. Yes; HL 5. Yes; HL

6. Yes: AAS 11a. 4 b. 6 c. 6

HW – 9

1. Yes; SAS 2. Yes; AAS 3. No; AAA 4. Yes; AAS 5. Yes; HL

6. Yes; SAS 7. No; ASS 8. Yes; SSS

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A

E

Bb

D

C

A

B

E

C

D

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

A

D

B

A

B

C

D

A

B

F

P

C

G

1

2

l

m

D

B

E

A

C

C

D

A

B

A

E

B

D

C

A

C

D

B

A

D

C

B

A

B

C

D

E

Aa

C

D

B

F

A

B

C

D

E

L

K

R

A

M

x2

2x + 8

3x + 10

5x + 6

A

C

D

B

O

P

T

E

A

D

C

B

A

F

E

D

B

C

C

A

F

D

B

E

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