Ch 4 Worksheet L1 Rev Key - Palisades High School · PDF fileCh 4 Worksheet L1 Rev Key.doc Name _____ S. Stirling Page 1 of ... and the smallest angle ∠S. ... then the angle opposite

Ch 4 Worksheet L1 Rev Key.doc Name ___________________________

S. Stirling Page 1 of 20

4.1 Triangles Sum Conjectures Auxillary line: an extra line or segment that helps you with your proof. Page 202 Paragraph proof explaining why the Triangle Sum Conjecture is true. Conjecture: The sum of the measures of the angles in every triangle is 180°.

Given: ABC∆ with auxiliary line EC AB�� and

angles labeled as shown.

Show: 2 4 5 180m m m∠ + ∠ + ∠ = °

1 2 3 180m m m∠ + ∠ + ∠ = ° Linear pair conjecture AC and CB form transversals between parallel lines EC

�� and AB

��

1 4m m∠ = ∠ and 3 5m m∠ = ∠ because AIA are congruent Substituting into the first equation above 2 4 5 180m m m∠ + ∠ + ∠ = °

Therefore, the sum of the measures of the angles in every triangle is 180°. Page 204 #18 Prove Third Angle Conjecture Conjecture: If two angles of one triangle are congruent to two angles of another triangle, then the third angle in each triangle is congruent to the third angle in the other triangle.

Given: m A m E∠ = ∠ and m B m F∠ = ∠

Show: m C m D∠ = ∠

180m A m B m C∠ + ∠ + ∠ = ° and 180m E m F m D∠ + ∠ + ∠ = ° by the triangle sum conjecture. Since they both equal 180, m A m B m C m E m F m D∠ + ∠ + ∠ = ∠ + ∠ + ∠

Now subtract equal measures m A m E∠ = ∠ and m B m F∠ = ∠ .

m C m D∠ = ∠ Therefore, the third angles are always congruent.

B

A C

DE

F



4.1 Page 203 Exercise #8


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

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

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



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.

A

B

C45

45

90

A

B

C

76

76

28

A

B

C20

20

140



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 ABC∆ is isosceles. Is the converse of the Isosceles Triangle Conjecture true? YES Complete the conjecture in the notes.

70

35

B

C

B C35

6.1 cm 6.1 cm

10 cm

70 8.4 cm

8.4 cm

5.7 cm




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.



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 CAT∆ from

Construct FSH∆ from

Why were you able to construct CAT∆ but not able to constructFSH∆ ? 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.



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



EXERCISES Lesson 4.4 Page 224-225 #3 – 10, 12 – 17 Mark diagrams! If congruence cannot be determined, draw a counterexample.



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.



EXERCISES Lesson 4.5 Page 229-230 #3 – 18 Mark diagrams! If congruence cannot be determined, draw a counterexample.



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.





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



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 BMC∆ ≅ ∆

ASA Congruence

m A m B∠ = ∠

given

AD BC≅

CPCTC or Def. Congruence

bisects DB m ABC∠

given

BD BD≅

Shared side

ABD CBD∆ ≅ ∆

ASA Congruence

90m ADB m BDC∠ = ∠ =

def. of perpendicular

A C∠ ≅ ∠


BD AC⊥

given

m ABD m CBD∠ = ∠

def. of angle bisector



4.8 Proving Special Triangle Conjectures Prove: The bisector of the vertex angle of an isosceles triangle is also the median and altitude to the base.

Given: Isosceles ABC∆ with AB BC= ; bisects BD ABC∠

Prove: BD is a median.

BD is an altitude. Prove: The bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector to the

base.

Given: Isosceles ABC∆ with AB BC= ; bisects BD ABC∠

Prove: BD is a perpendicular bisector.

DA C

B

bisects DB m ABC∠

given

BD BD≅

Shared side

ABD CBD∆ ≅ ∆

ASA Cong.

AD DC=


AB BC=

given



BD is a median

Def. of a Median

DA C

B

bisects DB m ABC∠ given BD BD≅

Shared side

ABD CBD∆ ≅ ∆ ASA Congruence

AD DC=


AB BC=

given



BD is a perpendicular bisector

Def. of a perpendicular bisector



BD is an altitude

Def. of an Altitude


Linear pair supp. and angles equal.

AC BD⊥ Def. Perpendicular





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



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 ZAP∆ ≅ ∆ by AAS Cong.

or

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

T Z∠ ≅ ∠ the lines are parallel. now TOP ZAP∆ ≅ ∆ by ASA Cong.

MSE OSU∆ ≅ ∆ by SSS Cong. or if you state

ESM USO∠ ≅ ∠ because vertical angles = , now

MSE OSU∆ ≅ ∆ by SAS Cong.

TRP APR∆ ≅ ∆ by SAS Cong.

Since GHI HIG∠ ≅ ∠ ,

HG GI≅ because if base angles =, then isosceles.

HGC IGN∠ ≅ ∠ because vertical angles =

CGH NGI∆ ≅ ∆ by SAS Cong.



AMD UMT∆ ≅ ∆ 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 DEC∆ ≅ ∆ 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 OBR∆ ≅ ∆ or ACN RBO∆ ≅ ∆ 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.



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

SLA IRT∆ ≅ ∆ 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 VSE∆ ≅ ∆ 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 AIR∆ ≅ ∆ 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.



4.R Page 253 Exercise #27

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

So f larger than a = g In PXM∆ :

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

So all sides of PXM∆ are equal f = e = d

In AXM∆ :

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

Documents

Ch 4 Worksheet L1 Rev Key - Palisades High School · PDF fileCh 4 Worksheet L1 Rev Key.doc Name _____ S. Stirling Page 1 of ... and the smallest angle ∠S. ... then the angle opposite