4.1 triangles sum conjectures - palisades school … 4 worksheet l2 key 3 post name _____ s....

Ch 4 Worksheet L2 Key 3 post Name ___________________________

S. Stirling of 18

4.1 Triangles Sum Conjectures 4.1 Page 203 Exercise #8 Look for overlapping triangles!

4.1 Page 203 Exercise #9

Hint: look for large overlapping triangles (ie. The one with the 40°, 71° and a.) a = 69, b = 47, c = 116, d = 93, e = 86

m = 30, n = 50, p = 82, q = 28, r = 32, s = 78, t = 118, u = 50

Hint: Fill in angles that do not have a variable and l ook for large overlapping triangles! There are many!!


S. Stirling of 18

4.2 Group Investigation 1, Base Angles of an Isosceles Triangle Each of the triangles below is isosceles. Carefully measure the angles of each triangle. (Make sure the triangles’ angles sum is 180° right?) If you disregard measurement error, are there any patterns for all isosceles triangles? Finish the following conjecture using the vocabulary you learned about isosceles triangles. Isosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. Complete the conjecture in the notes and the example problem.

A

B

C45

45

90

A

B

C76

76

28

A

B

C20

20

140


S. Stirling of 18

4.2 Group Investigation 2, Is the Converse True? Write the converse of the Isosceles Triangle Conjecture below. Converse of the Isosceles Triangle Conjecture If a triangle has two congruent angles, then it is an isosceles triangle. Is this converse true? In this investigation, you are going to make congruent angles and then measure the sides to see if the triangle is isosceles. For each of the following, make B C∠ ≅ ∠ . Extend the sides to form A∠ . Then measure the sides to see if ABCΔ is isosceles. Is the converse of the Isosceles Triangle Conjecture true? YES Complete the conjecture in the notes and the example problem.

70

35

B

C

B C 35

6.1 cm 6.1 cm

10 cm

70 8.4 cm

8.4 cm

5.7 cm


S. Stirling of 18


4.2 Page 209 Exercise #11 In the problem, they state that the angles around the center are congruent. Note: In order for the pattern of tiles to look symmetric, all of the triangles of the same size must be congruent! How many of the tiles are isosceles triangles?

a = 124, b = 56, c = 56, d = 38, e = 38, f = 76, g = 66, h = 104, k = 76, n = 86, p = 38

Hint: Look for the overlapping triangle involving e, d and 66°. Do you see 3 equal angles?

a = 36, b = 36, c = 72, d = 108, e = 36 All of the triangles are isosceles.


S. Stirling of 18

4.3 Group Investigation 1, Lengths of the Sides of a Triangle For each of the following, construct the triangle given the three sides. Compare your results with your group members. When is it possible to construct a triangle from 3 sides and when is it not possible? Measure the three sides in centimeters. How do the numbers compare? Construct CATΔ from Construct FSHΔ from Why were you able to construct CATΔ but not able to construct FSHΔ ? Give more examples of three side lengths that will NOT make a triangle. Will sides of 4 cm, 6 cm and 10 cm make a triangle? State your observations in the conjecture. Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Complete the conjecture in the notes and the example problems.

C A

TA

TC

F S

S H

HF

TC

F S

Various examples: 2, 5, 9 because 2 + 5 < 9 4, 6 and 10? No because 4 + 6 = 10 NOT a triangle, it’s a segment.


S. Stirling of 18

4.3 Group Investigation 2, Largest and Smallest Angles in a Triangle For each of the following triangles, carefully measure the angles. Label the angle with the greatest measure L∠ , the angle with the second largest measure M∠ , and the smallest angle S∠ . Now measure the sides in centimeters. . Label the side with the greatest measure l, the side with the second largest measure m, and the shortest side s. Which side is opposite L∠ ? M∠ ? S∠ ? Write a conjecture that states where the largest and smallest angles are in a triangle, in relation to the longest and shortest sides. Side-Angle Inequality Conjecture In a triangle, if one side is the longest side, then the angle opposite the longest side is

the largest angle. (And visa versa.) Likewise, if one side is the shortest side, then the angle opposite the shortest side is

the smallest angle. (And visa versa.) Does this property apply to other types of polygons? Test it out! Would you really need to measure these? Complete the conjecture in the notes and the example problems.

LS

M

19 128

33

m

s l

MS

L75

35 70

l

m s

E∠ is the largest angle but it is opposite the shortest side

AT .

Can’t be true for polygons with an even number of sides because angles are opposite angles and sides are opposite sides.

P

E

N

TA

Q

U A

D


S. Stirling of 18

EXERCISES Lesson 4.4 Page 224-225 #3 – 10, 12 – 17

For Exercises 4 – 9, decide whether the triangles are congruent, and name the congruence shortcut you used (SSS or SAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.

FD FD≅ shared side SSS Cong.

SSS Cong. Not congruent, sides are not matched. But, COT NAPΔ ≅ Δby SAS Cong.

CV CV≅ shared side SSS Cong.

KA KA≅ shared side SAS Cong. AY YR≅ Def. midpoint

BY YN≅ If base angles =, BNYΔ isosceles. SSS Cong.

Perimeter ABCΔ = 180 11 11 180

3 18060

x x xxx

+ + + − ===

60AC = , 60 11 71AB = + = m DAE m BAC∠ = ∠ Vertical angles = So ABC ADEΔ ≅ Δ by SAS Cong.

I

Z P Z P

I


S. Stirling of 18

For Exercises 12 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS or SAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.

ANT FLEΔ ≅ Δ SSS Cong.

Can’t be determined since SSA is not a congruence guarantee.

GIT AINΔ ≅ Δ SSS Cong. Or if you state GIT AIN∠ ≅ ∠ because vertical angles = then GIT AINΔ ≅ Δ by SAS Cong.

Can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS.

SAT SAOΔ ≅ Δ SAS Cong. Since AS AS≅ a shared side.

Even with WO WO≅ a shared side. Can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS.


S. Stirling of 18

EXERCISES Lesson 4.5 Page 229-230 #3 – 18

For Exercises 4 – 9, determine whether the triangles are congruent, and name the congruence shortcut you used (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.

I

Z PI

Z P

AMD RMC∠ ≅ ∠ because vertical angles = then AMD RMCΔ ≅ Δ by ASA Cong.

HOW FEW∠ ≅ ∠ because vertical angles = Even with this there is not enough information to determine congruence.

Even with FS FS≅ a shared side. Can’t be determined. This would be a SSA which does not guarantee congruence.

GAS IOLΔ ≅ Δ AAS Cong. Or if you use the third angle conjecture: state S L∠ ≅ ∠ , then GAS IOLΔ ≅ Δ by ASA Cong.

BOX CARΔ ≅ Δ ASA Cong. INT TLA∠ ≅ ∠ and

NIT TAL∠ ≅ ∠ because lines ||, so alternate interior angles =.

NTI LTA∠ ≅ ∠ because vertical angles = Even with all of this the triangles are not necessarily congruent because AAA does not guarantee congruence.


S. Stirling of 18

For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.

FAD FEDΔ ≅ Δ SSS Cong. Since FD FD≅ a shared side.

H T∠ ≅ ∠ and O A∠ ≅ ∠ because lines ||, so

alternate interior angles =. WHO WTAΔ ≅ Δ by AAS Cong.

or… HWO YWA∠ ≅ ∠ because

vertical angles = H T∠ ≅ ∠ because lines ||, so

alternate interior angles =. WHO WTAΔ ≅ Δ by ASA Cong.

LAT SAT∠ ≅ ∠ def. of angle bisector Since TA TA≅ a shared side,

LAT SATΔ ≅ Δ by SAS Cong.

Overlapping Triangles POE PRNΔ ≅ Δ by ASA

Cong. and

OSN RSE∠ ≅ ∠ because vertical angles =

SON SREΔ ≅ Δ by ASA Cong.

DMA RMC∠ ≅ ∠ because vertical angles = Still can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS.

Overlapping Triangles: MR MR≅ a shared side.

RMF MRAΔ ≅ Δ by SAS Cong.


S. Stirling of 18

B K∠ ≅ ∠ and L C∠ ≅ ∠ because lines ||, so alternate interior angles =.

BAL KAC∠ ≅ ∠ because vertical angles = Even with all of this the triangles are not necessarily congruent because AAA does not guarantee congruence.

WL WL≅ a shared side. LAW WKLΔ ≅ Δ by ASA

Cong.

Perimeter ABCΔ = 138 4 4 138

3 13846

x x xxx

+ + + − ===

46AC = m DAE m BAC∠ = ∠ Vertical angles =

E C∠ ≅ ∠ and D B∠ ≅ ∠ because lines ||, so alternate interior angles =. So ABC ADEΔ ≅ Δ by AAS Cong. or by ASA Cong.


S. Stirling of 18

4.4 Page 226 Exercise #23 4.6 Page 234 Exercise #18

a = 37, b = 143, c = 37, d = 58 e = 37, f = 53, g = 48, h = 84, k = 96, m = 26, p = 69, r = 111, s = 69

a = 112, b = 68, c = 44, d = 44, e = 136, f = 68, g = 68, h = 56, k = 68, l = 56, m = 124


S. Stirling of 18

4.6 Corresponding Parts of Congruent Triangles 4.6 Page 232 Example A

Given: AM MB≅ and m A m B∠ = ∠

Prove: AD BC≅ Example B

Given: BD AC⊥ and bisects DB m ABC∠ Prove: A C∠ ≅ ∠

2

1M

C B

A D

DA C

B

AM MB≅

given

1 2m m∠ = ∠

Vertical angles = AMD BMCΔ ≅ Δ

ASA Congruence

m A m B∠ = ∠

given

AD BC≅

CPCTC or Def. Congruence

bisects DB m ABC∠

given

BD BD≅

Shared side

ABD CBDΔ ≅ Δ

ASA Congruence

90m ADB m BDC∠ = ∠ =

def. of perpendicular

A C∠ ≅ ∠

CPCTC or Def. Congruence

BD AC⊥

given

m ABD m CBD∠ = ∠

def. of angle bisector


S. Stirling of 18



a = 72, b = 36, c = 144, d = 36 e = 144, f = 18, g = 162, h = 144, j = 36, k = 54, m = 126

a = 128, b = 128, c = 52, d = 76 e = 104, f = 104, g = 76, h = 52, j = 70, k = 70, l = 40, m = 110, n = 58


S. Stirling of 18

EXERCISES Ch 4 Review Page 252 #7 – 24 For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.

The triangles are not necessarily congruent because SSA does not guarantee congruence.

Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence.

OPT APZ∠ ≅ ∠ vertical angles = TOP ZAPΔ ≅ Δ by AAS Cong.

or if you state TO AZ because alternate interior angles =, then

T Z∠ ≅ ∠ the lines are parallel. now TOP ZAPΔ ≅ Δ by ASA Cong.

MSE OSUΔ ≅ Δ by SSS Cong. or if you state

ESM USO∠ ≅ ∠ because vertical angles = , now

MSE OSUΔ ≅ Δ by SAS Cong.

TRP APRΔ ≅ Δ by SAS Cong.

Since GHI HIG∠ ≅ ∠ , HG GI≅ because if base angles =, then isosceles.

HGC IGN∠ ≅ ∠ because vertical angles =

CGH NGIΔ ≅ Δ by SAS Cong.


S. Stirling of 18

AMD UMTΔ ≅ Δ by SAS Cong. AD UT≅ Def. of

Congruent Triangles or CPCTC

If isosceles, then base angles =. So O T∠ ≅ ∠ . WH WH≅ a shared side Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence.

Since AB CD A D∠ ≅ ∠ and B C∠ ≅ ∠

because lines ||, so alternate interior angles =. Also BEA CED∠ ≅ ∠ because vertical angles =

ABE DECΔ ≅ Δ by AAS Cong. or ASA Cong.

Since it is a regular polygon all sides and angles are =:

C B∠ ≅ ∠ and CN CA OB BR= = = So

ACN OBRΔ ≅ Δ or ACN RBOΔ ≅ Δ by SAS

Cong.

Can’t be determined because you can not get more equal sides nor angles and AAA does not guarantee congruence.

Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence.


S. Stirling of 18

Since LA TR , A T∠ ≅ ∠ because lines ||, so alternate interior angles =.

SLA IRTΔ ≅ Δ by AAS Cong. TR LA≅ Def. of Congruent Triangles or CPCTC

Parts do not match. Both triangles are AAS but the angles do not match.

INK VSEΔ ≅ Δ by SSS Cong. But not needed because EV IK= and EV VI IK VI

EI VK+ = +

=

by addition.

Since MN CT , MNT NTC∠ ≅ ∠ because

lines ||, so alternate interior angles =. NT NT≅ a shared side Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence.

Overlapping triangles: ALZ AIRΔ ≅ Δ by ASA

Cong. because A A∠ ≅ ∠ same angle.

Since SPT PTO∠ ≅ ∠ , the alternate interior angles = and lines ||. SP TO Since OPT PTS∠ ≅ ∠ , the alternate interior angles = and lines ||. OP TS . Since the opposite sides are parallel, STOP is a parallelogram.


S. Stirling of 18

4.R Page 253 Exercise #27

In PCXΔ : 30m CPX∠ = triangle sum 180 – 30 – 120 = 30

So f larger than a = g In PXMΔ :

60m PXM∠ = straight angle 180 – 30 – 90 = 60 60m PMX∠ = triangle sum 180 – 60 – 60 = 60

So all sides of PXMΔ are equal f = e = d In AXMΔ :

45m XMA∠ = triangle sum 180 – 90 – 45 = 45 Since base angles =, triangle is isosceles. So So c larger than d = b So c is the largest overall!

X

4.1 triangles sum conjectures - palisades school … 4 worksheet l2 key 3 post name _____ s....

Documents