Name: ____________________________ Period: ________ 4.1 Tricky Triangles

1) Triangle Sum Conjecture: The sum of the measures of the angles in every triangle is ___________.

2) Determine the missing angle in each diagram

a a = ______ b b= ______

570

360

c= ______

c d d= ______

380 120

e e= ______ h= ______

f g 1240 380 f= ______ i= ______

h g= ______ j = ______

i

1380 450

750

3) Snyder Rd

k

m

Prince Rd n

City Designers planned many Tucson streets at right angles. Houghton Rd is perpendicular to both

Snyder Rd and Prince Rd. Melpomene is perpendicular to Prince and Snyder Roads.

a) What can you prove about Snyder Rd and Prince road according to the given information? Explain.

b) The angle Prince Rd makes with Catalina Highway is a 440 angle. Find the unknown angles

k= ______ m= ______ n= ______

line m

4) 2 3

4 1 5

line n

a) Given: m || n

Prove: The sum of the angles of ΔABC is 1800

Name: _______________________________ Period______4.2 Classifying Triangles/Exterior Angle Theorem

1. Use the diagram indicated to prove the exterior angle theorem. Your givens come from the diagram.

The conclusion of your proof should say that 𝑚∠1 = 𝑚∠2 +𝑚∠3

Statements Reasons

Questions 2-5: Classify each triangle by the angles, and sides. Assume that the only given information are

the congruence marks, and angle indicators.

Sketch an example of the type of triangle described. Mark the triangle to indicate what information is

known. If no triangle can be drawn, write “not possible.”

6) acute isosceles 7) right scalene 8) right isosceles

9) right equilateral 10) acute scalene 11) obtuse scalene

12) right obtuse 13) equilateral 12) acute equilateral

Questions 13- 18 Find the measure of each indicated angle (?).

13) 14) 15)

16) 17) 18)

Name: _________________________ Period: ________ 4.3 Overlapping Triangles

For 1 & 2, shade a different triangle in each image.

1)

2)

For 3 & 4, copy the diagram as many times as needed to shade all the different triangles in each image.

3)

4)

5) Draw your own shape that is made up of at least 4 overlapping triangles. Then recopy your design as

many times as needed to shade all the different triangles in your image.

Name: ___________________________ Period: _________ 4.4 Triangle Inequality Problem Set

𝑥 + 𝑦 > 𝑧

Three numbers are given as the side lengths of a triangle. Use the triangle inequality to determine whether

such a triangle can exist

1) 7, 5, 4 2) 3, 6, 2 3) 5, 2, 4

4) 8, 2, 8 5) 9, 6, 5 6) 5, 8, 4

7) 4, 7, 8 8) 11, 12, 9 9) 3, 10, 8

10) 1, 13, 13 11) 2, 15, 16 12) 10, 18, 10

Two side lengths of a triangle are given, determine the range of values that are possible for the 3rd side.

13) 9, 5 14) 5, 8 15) 6, 10

16) 6, 9 17) 11, 8 18) 14, 11

Use your compass and a straight edge to draw a triangle given each set of measurements (label):

19) 7cm, 5cm, 4 cm 20) 8 cm, 8cm, 2cm

21) 3 cm, 6 cm, 2 cm 22) 6 cm, 6 cm, 6 cm

Name: _________________________ Period: ________ 4.6 SSS and SAS

SSS: Three sides of one triangle are congruent to the corresponding sides of another triangle

SAS: Two sides and the included angle of one triangle are congruent to the corresponding

parts of another triangle

Identify which property will prove these triangles congruent, SSS or SAS. If neither method works say

“neither”

1. 2.

3. 4.

5. 6.

7. 8.

Sate what additional information is required in order to know that the triangles are congruent

FOR THE GIVEN REASON. Remember ORDER matters. Write the triangle congruency.

Example: SAS Since one side is marked 𝐻𝐼 ≅ 𝐾𝐼 and from the diagram there

is a pair of vertical angles so ∠𝐻𝐼𝐽 ≅ ∠𝐾𝐼𝑀, so we need

One more piece of needed information: ___________________

To prove ∆𝐻𝐼𝐽 ≅ ∆𝐾𝐼𝑀 𝑏𝑦 𝑆𝐴𝑆

9. prove by: SSS

One more piece of needed information: ___________________

Δ KMH ≅ Δ ________

10. C A

prove by: SAS

T F One more piece of needed information: ___________________

Δ CAT ≅ Δ ________

11. prove by: SAS

One more piece of needed information: ___________________

Δ MKL ≅ Δ ________

12. prove by: SSS

One more piece of needed information: ___________________

Δ XZY ≅ Δ ________

©a O2W0G1j5d aKZu\tcaW cS\oWfHtowXaorHeg LLjLpCU.n e ^ADlYl[ crdikgqhAtlsA arYeqsNeErGvveOdg.w i PMaa`dNeA Kwziytchu jIOnrfCignYivtZeV lGeeBoqmyeytJrSyX.

Worksheet by Kuta Software LLC

4.7 Geometry

ASA and AAS Congruence

Name___________________________________

Period____

©l t2_0e1z5z QK[uztjas zSto_fpt[wHaYr^eX KLMLYC\.u [ uAhlTlN orsixgYhztbss PrQelsFewrivjeEdX.

State if the two triangles are congruent. If they are, state how you know.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

11. Use the figure to determine which (if any) triangles

are congruent to one another.

12. Determine if . Explain your reasoning.

13. In the figure, is an equilateral triangle, does that mean

is also equilateral? Explain your reasoning.

Name: ______________________________ Period: _______ 4.8 Congruence Shortcuts that Fail

1. Demonstrate (By construction) that AAA doesn’t produce two congruent triangles.

2. Using an angle of 45 degrees, and side lengths

of 4cm, and 3.5cm, show that SSA will produce

two different triangles which are not congruent.

The diagram on the right shows the same setup

except with an angle of 30 degrees and side

lengths 10 and 6.

Name two short cuts that don’t work: _____________ and _______________

1. Construct one triangle that has lengths 4cm, 5cm, 7cm and another that has lengths 7cm, 7cm, and 4 cm

2. Construct a triangle that has leg lengths 5cm, 6cm, and an included angle of 50 degrees.

3. Construct a triangle that has leg lengths 4cm, 4cm and an included angle of 60 degrees. What kind of

triangle is this?

Name: _________________________ Period: ________ 4.9 Proving Triangles Congruent

For each problem give the correct naming order of the congruent triangles. Write that name in order on the lines for the problem number (see box at bottom). Also, indicate which postulate or theorem is being used.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

___ ___ ___ ___ ___ _O_ ___ ___ _N_ ___ ___ ___ _S_ ___ ___ _E_ ___ _I_ ___ ___ ___ ___ ___ _T_ ___

4 4 4 8 8 8 12 12 12 2 2 2 5 5 5 9 9 9 6

___ ___ ___ _E_ _E_ ___ ___ ___ _O_ ___ ___ _N_ ___ _U_ ___ ___ ___ ___ _T_ ___ _E_ ___ ___ _I_ ___ . 6 6 10 10 10 1 1 1 3 3 3 7 7 7 11 11 11

(When you are done with the puzzle, there are: 3 SAS, 5 AAS, 2 ASA, and 2 SSS instances.)

W

A

B I

C R B

C G

A N B

A S

C E

G

J R

H E C

A E

B Y C

B S

A D

O

M K

N A

H T

A T H

C

B I

A L K

J T

L H D

F S

E N

H A A

B

C G

A K A

B Y

C D

P E S

ABC _______ by _________

ABC _______ by _________

ABC _______ by _________

ABC _______ by _________

GHJ _______ by _________

ABC _______ by _________

DEF _______ by _________

ABC _______ by _________

JKL _______ by _________

ABC _______ by _________

ABC _______ by _________

MNO _______ by _________

13)

A) Translation B) Vertical Reflection

C) Rotation D) Horizontal Reflection

14) There are five different ways to find triangles are congruent: SSS, SAS, ASA, AAS and HL.

For each pair of triangles, select the correct rule. Indicate if there isn’t enough information.

15) a) Mark the diagram with the given information.

b) Look for any other given information that could help show that the two triangles are congruent. Do they

overlap anywhere? Share any side or any angle? Mark it in the diagram.

c) You should have enough information to prove the triangles are congruent. Fill in the proof.

Statements Reasons

1) 1) Given

2) 2) Given

3) 3)

4) 4)

d) What do you think is true about ∠𝐴 𝑎𝑛𝑑 ∠𝐶? Explain:

Name: _________________________ Period: ________ 4.10 Proving Isosceles Conjectures In this space, draw a large isosceles triangle. Use tools appropriately, do not freehand. Mark the two

congruent sides of the triangle. Precision is essential.

1) Label the vertices RED with R begin the vertex angle (included by the two congruent sides).

2) Carefully construct the angle bisector of ∠R. Label point Q on 𝐷𝐸̅̅ ̅̅ where the angle bisector ray

intersects 𝐷𝐸̅̅ ̅̅ . Mark the congruent angles.

3) Complete the proof to show that Δ ERQ ≅ Δ DRQ

Statement Reason

1. ___________________________ 1. Given

2. ∠ERQ ≅ ∠ DRQ 2. ___________________________

3. ___________________________ 3. ___________________________

4. Δ ERQ ≅ Δ DRQ 4. SAS

4) Using CPCTC, what other parts of the triangle are congruent?

________ ≅ ________ and ________ ≅ ________

5) On your diagram, measure 𝐸𝑄 𝑎𝑛𝑑 𝐷𝑄, what can you say about point Q?

6) On your diagram, measure ∠𝐸𝑄𝑅 𝑎𝑛𝑑 ∠𝐷𝑄𝑅, what can you say about these angles?

7) Describe a series of transformations that would map Δ ERQ onto Δ DRQ. Be specific.

The isosceles triangle theorems are frequently abbreviated as

1. ΔXYZ is an isosceles triangle with perimeter 48 cm

If XY = 18

Find XW _______

2. M Δ MPQ is an isosceles triangle. 𝑚∠𝑃𝑄𝑀 = 45° 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚∠𝑄𝑃𝑁

N

P Q

3. 2x 5

2𝑥 − 5 Find AB

2x

4. (3y – 5)0 Solve for y

400

5. (5x + 15)0 solve for x

Name: _________________________ Period: ________ 4.11 CPCTC worksheet

MARK THE DIAGRAMS WITH THE GIVEN INFORMATION!

#1: HEY is congruent to MAN by ______.

What other parts of the triangles are congruent by CPCTC?

______ ______

______ ______

______ ______

#2:

CAT ______, by _____

THEREFORE:

______ ______, by CPCTC

______ ______, by CPCTC

______ ______, by CPCTC

#3:

Given: ARAC and 21

Prove: 43

Proof:

1. ARAC

2. ____________

3. RASCAL

4. LCA SRA

5. 43

1. _______________

2. Given

3. ________________

4. ________________

5. ________________

M

A

N

Y

E

H

L

C

S

R

4 3

2 1

C

T P

A

R

A

MARK THE DIAGRAMS WITH THE GIVEN INFORMATION!

#4:

Given: LNONLM and MNLOLN

Prove: OM

Proof:

1. LNONLM

2. _________________

3. _________________

4. LMN ______

5. _________________

1. _________________

2. Given

3. Reflexive Property of

4. _________________

5. _________________

#5

Given: BCAC and BXAX

Prove: 1 2

Proof:

1. __________________________ 1. Given

2. __________________________ 2. Reflexive Prop. of Congruence

3. AXC _______ 3. ____________

4. ________________ 4. ____________

#6

Given: 1 2 and 3 4

Prove: ZWXY

Proof:

1. __________________________ 1. Given

2. XZXZ 2. ________________

3. XWZ _______ 3. ________________

4. ________________ 4. ________________

M

N O

L

C

X B A

1 2

4 3

W

X Y

Z

1

2 3

4

Name: ______________________________Period: __________ 4.12 Using Proof Blocks

2

3.

4.

5.

6.

7.

8.

Name: ______________________________ Period: __________ 4.13Proof blocks & CPCTC

1. Given: 𝐵𝐶 ≅ 𝐷𝐸 & ∠𝐵 ≅ ∠𝐸

Prove: 𝐴𝐶 ≅ 𝐴𝐷

2. Given: ∠𝐷 ≅ ∠𝑃, ∠𝐸 ≅ ∠𝑄, 𝐸𝐷 ≅ 𝑃𝑄

Prove: 𝐷𝐹 ≅ 𝑃𝑅

3. Given that ∠𝐺 ≅ ∠𝐾, and the information in the diagram,

prove 𝐻𝐼 ≅ 𝐽𝐿

4. Using the information in the diagram prove that ∠𝑀 ≅ ∠𝑂.

5. Given that 𝐺𝐻 ∥ 𝐽𝐼, I is the midpoint of 𝐻𝐾 and 𝐺𝐻̅̅ ̅̅ ≅ 𝐽�̅� Prove: ∠𝐺 ≅ ∠𝐽

6. Given that 𝑀𝑁 ⊥ 𝑂𝑃, and the information in the diagram to prove

that 𝑂𝑃 is the angle bisector of ∠𝑀𝑃𝑁

Name: _________________________________Period: _________ 4.14 Quiz 4 Review

1) Classify ∆ABC by its angles and its side lengths ___________ _________________

2) Classify each triangle by its side length

∆ABD ______________ and ∆ADC ___________________

3) While surveying a triangular plot of land, a surveyor finds 𝑚∠𝑆 = 430.

The measure of ∠𝑅𝑇𝑃 is twice that of ∠𝑅𝑇𝑆. What is the 𝑚∠𝑅?

Given ∆XYZ ≅ ∆JKL, identify the congruent corresponding parts

4) 𝐽�̅� ≅ _____ 5) ∠𝑌 ≅ ____ 6) ∠𝐿 ≅ ____ 7) 𝑌𝑍̅̅̅̅ ≅ ___

8) Given: T is the midpoint of both 𝑃𝑅̅̅ ̅̅ 𝑎𝑛𝑑 𝑆𝑄̅̅̅̅

Prove: ∆PTS ≅ ∆RTQ Statements Reasons

9 & 10: Find the Measures of each angle

11. Find the measure of ∠𝐴𝑈𝑇

12. Find the measure of ∠CBD

13. 14.

18.

15. Look at the diagrams below and determine which (if any) triangle congruence theorems you can use to

prove two triangles are congruent. Circle the ones that work and label them with SSS, SAS, AAS, ASA or HL.