Ch 4 Notesheet Key

Ch 4 Note Sheet L1 Key 

Name ___________________________ 

Chapter 4: 

Discovering and Proving 

Triangle Properties 

Note Sheet 

S. Stirling Page 1 of 15


Name ___________________________ 

4.1 Triangles Sum Conjectures 

These notes replace pages 200 – 202 in the book. See step 4 of the Investigation. 

Rigidity is a property that triangles have. They cannot be shifted, like a quadrilateral can. They retain 

their shape. 

Triangulation is a procedure used by surveyors to locate a position by using triangles. 

Triangle Sum Conjecture 

The sum of the measures of the angles in every triangle is 180°. 

Numeric Example: 

Find x. 

A 

56 

B 

101 

101 + 56 + x = 180 

x = 180 – 157 = 23 

x 

C 

Find y. 

x + x + y = 180 

2x + y = 180 

y = 180 – 2x 

E 

x 

F 

x 

y 

D 

Third Angle Conjecture 

If two angles of one triangle are congruent to two angles of another triangle, then the third 

angles of the triangles are congruent. 

Numeric Example: 

If m∠ K = m∠Band m∠ I = m∠ C , find y. Give reasons for your answer! 

K 

101 

B 

A 

y 

56 23 

J 

I 

C 

y = 56 because the third angles are equal also. 

See Ch 4 Worksheet to prove these conjectures. 



Name ___________________________ 

Lesson 4.2 Properties of Isosceles Triangles 

Read top of page 206. This vocabulary is review! Know it!! 

Remember! With If..then.. statements: If a, then b. The converse of the statement is If b, then a. 

Label and define the special vocabulary for the isosceles triangle Δ VBS . 

B 

A leg (of an isosceles triangle) is one of the congruent 

sides. VB and VS 

S 

The base (of an isosceles triangle) is the side that is not a 

leg. BS 

The vertex angle is the angle between the two legs. 

∠ V 

V 

The base angles are the pair of angles whose vertices are 

the endpoints of the base. ∠B 

and ∠ S 

Do group Investigation 1, Base angles of an Isosceles Triangle on page 207. 

Isosceles Triangle Conjecture 

If a triangle is isosceles, then its base angles are congruent. 

Given 

If VB 

ΔVBS 

with VB = VS , then ∠B ≅∠ S or 

≅ VS , then ∠B ≅ ∠ S . 

B 

V 

S 

***Note: Since equilateral triangles are a special case of isosceles triangles, any property 

that applies to isosceles triangles also apply to equilateral triangles. 

Do group Investigation 2, Is the Converse True? on page 208. Use a protractor. 

Converse of the Isosceles Triangle Conjecture 

If a triangle has two congruent angles, then it is an isosceles triangle. 

If ∠B ≅ ∠ S , then ΔVBS 

is isosceles or 

If ∠B ≅ ∠ S , then VB ≅ VS . 

V 

B 

S 



Lesson 4.3 Triangle Inequalities 

Read of page 215. Do group Investigation 1 on page 216. 

Name ___________________________ 

Triangle Inequality Conjecture 

The sum of the lengths of any two sides of a triangle is greater than 

third side. 

Is a triangle measuring 5 cm, 4 cm and 2 cm 

possible? Construct it then explain! 

the length of the 

Is a triangle measuring 5 cm, 3 cm and 2 cm 

possible? Construct it then explain! 

Since 2 + 4 > 5, it makes a triangle. 

Since 2 + 3 = 5, it makes segment NOT 

a triangle. 

Do group Investigation 2, Largest and Smallest Angles in a Triangle? on page 217. 

Side-Angle Inequality Conjecture (Only applies to triangles!) 

In a triangle, if one side is the longest side, then the angle opposite the longest side is the 

largest angle. 

Likewise, if one side is the shortest side, then the angle opposite the shortest side is the 

smallest angle. 

Which is the largest angle? The smallest 

angle? Why? 

K 

J 

2.48 

6.3 

5.34 

I 

Which is the largest side? Why? 

∠D 

largest, so AQ 

largest in Δ ADQ , 

But, m∠ QAU = 100 

D 

A 

90 

54 

23 

Q 

∠ K largest, its opposite the longest 

side, JI . 

∠I 

smallest, its opposite the shortest 

side, JK . 

which makes UQ the 

largest in ΔQAU 

and 

AQ < UQ. 

Overall, UQ is the longest. 

57 

U 



Name ___________________________ 

Do group Investigation 3, Exterior Angle of a Triangle on page 217 – 218 . 

Label the drawing with the following terms and define the terms: 

Exterior angle is an angle that forms a linear pair 

with one of the interior angles of a polygon. 

Adjacent interior angle is the angle of a polygon 

that forms a linear pair with a given exterior angle of 

a polygon. 

The remote interior angles (of a triangle) are the 

interior angles of a triangle that do not share a vertex 

with a given exterior angle. 

Triangle Exterior Angle Conjecture 

The measure of an exterior angle of a triangle is equal to the sum of the measures of the 

remote interior angles. 

Given ΔABC 

above. If a = 50 and b = 60, what 

is the measure of ∠ BCD ? Explain. 

50 + 60 = 110 

The exterior angle equals the sum of the 

two remote interior angles. 

Given ΔABC 

above. If x = 80 and b = 30, what 

is the measure of ∠ A ? Explain. 

80 – 30 = 50 

The exterior angle equals the sum of the 

two remote interior angles. 

Before starting Lesson 4.4: 

Vocab for triangles Triangles have 6 “parts”. 

∠A 

is opposite BC and AC is opposite ∠ B 

∠A 

is between BA and AC (angles are between sides) 

BC is between 

∠ B and ∠ C (sides are between angles) 

included angle is an angle formed between two consecutive 

sides of a polygon. 

B 

A 

C 

included side is a side of a polygon between two consecutive angles. 



Name ___________________________ 

Read the top of page 221, then complete the Triangle Congruence Shortcut Investigation, page 1 – 3. 

SSS Congruence Conjecture 

If the three sides of one triangle are congruent to the three sides of another triangle, then 

the triangles are congruent. 

A 

D 

O 

A 

G 

C 

T 

ΔCAT 

≅ Δ _______ 

By SSS Congruence Conjecture. 

C 

T 

O 

ΔCAT 

≅Δ _______ 

By SSS Congruence Conjecture. 

SAS Congruence Conjecture 

If two sides and the included angle of one triangle are congruent to two sides and the 

included angle of another triangle, then the triangles are congruent. 

A 

I 

P 

I 

Z 

C 

T Z P 

ΔPIZ 

≅ Δ _______ 

By SAS Congruence Conjecture. 

E 

D 

ΔZED 

≅Δ _______ 


ASA Congruence Conjecture 

If two angles and the included side of one triangle are congruent to two angles and the 

included side of another triangle, then the triangles are congruent. 

D 

O 

I 

O 

G 

Z 

P 

ΔDOG 

≅ Δ _______ 

By ASA Congruence Conjecture. 

C 

P 

A 

ΔCOP 

≅Δ _______ 

By ASA Congruence Conjecture. 



Name ___________________________ 

SAA or AAS Congruence Conjecture 

If two angles and a non-included side of one triangle are congruent to the corresponding 

angles and side of another triangle, then the triangles are congruent. 

D 

O I 

G 

Z 

P 

ΔPIZ 

≅ Δ _______ 


ΔCAP 

≅Δ _______ 

P 

A 

T 

C 


Special Case: Hypotenuse Leg Congruence Conjecture 

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one 

leg of another right triangle, then the two triangles are congruent. 

D 

O 

I 

O 

G 

Z 

P 

ΔDOG 

≅ Δ _______ 

By HL Congruence Conjecture. 

C 

P 

A 

ΔCOP 

≅Δ _______ 

By HL Congruence Conjecture. 

SSA or ASS Congruence? 

If two sides and the non-included angle of one triangle are congruent to two sides and the 

non-included angle of another triangle, then the triangles are NOT necessarily congruent. 

A 

I 

Draw a counterexample. 

I 

C 

T Z 

P 

ΔZIP 

≅ Δ _______ 

By NOT necessarily congruent. 

Z 

P 



Name ___________________________ 

AAA Congruence Conjecture 

If three angles of one triangle are congruent to the corresponding angles of another 

triangle, then the triangles are NOT necessarily congruent. 

A 

I 

Draw a counterexample. 

I 

C 

T Z 

P 

ΔZIP 

≅ Δ _______ 

By NOT necessarily congruent. 

Z 

P 

How do you get equal parts in order to get congruent triangles? 

To see if you have congruent triangles, you will be checking for a SSS, SAS, ASA or AAS marked on the matching pair of 

triangles that you be given (as shown above). Sometimes the parts will be marked equal in the diagram. That’s the easy stuff. 

Other times, you will be given information that you will “translate” into equal sides and angles in order to get your 

congruence. You will need to deduce this information from definitions or conjectures that you already know to be true. 

Complete the following to help you review these statements. Remember, to mark your diagrams with the equal parts. Also 

never assume things are congruent! You must have a definitions or conjecture to back you up!! 

Ways to Get Equal Segments 

Converse of the Isosceles Triangle Conj. 

If a triangle has two congruent angles, 

then it is an isosceles triangle. V S 

Def. Isosceles Triangle 

If a triangle is isosceles, 

then its legs are congruent. 

V 

S 

If ∠B ≅ ∠ S , 

≅ VS 

then VB 

B 

ΔVBS 

If isosceles 

then VB ≅ VS . 

B 

Def. Midpoint 

If a point is a midpoint, 

then it divides the segment into 

two equal segments. 

If M is the midpoint of SG , 

then SM ≅ MG . 

S 

M 

G 

Def. segment bisector 

If a line (or part of a line) is a bisector, 

then it passes through the 

midpoint of the segment. 

If CM bisects AB , 

≅ MB . 

then AM 

A 

C 

M 

B 

“Same Segment” 

When two triangles share 

the exact same segment, you get 

a pair of equal segments. 

ΔBET 

and ΔWTE 

ET ≅ TE 

share 

B 

E 

O 

T 

W 

Def. Median 

If a segment is a median, 

then it connects the vertex to the 

midpoint of the opposite side. 

If AM is median in Δ ABC , 

then BM ≅ MC . 

A 

B 

M 

C 



Isosceles Triangle Conjecture 

If a triangle is isosceles, 

then its base angles are congruent. 

ΔVBS 

If isosceles 

or VB ≅ VS , 

then ∠B ≅ ∠ S . 

Definition of Angle Bisector 

If you have an angle bisector, 

then the ray cuts the angle 

into two equal angles. 

 

If BD bisects ∠ ABC 

then ∠ABD ≅ ∠ DBC . 

Def. Altitude 

If a segment is an altitude, 

then it goes from a vertex 

perpendicular to the line that 

contains the opposite side. 

Δ TRI 

∠ = ∠ = °. 

If RA is an altitude of 

then m RAT m RAI 90 

“Same Angle” 

When two triangles share 

the exact same angle, you get 

a pair of equal angles. 

ΔTIA 

and ΔNIR 

∠TIA 

≅ ∠ RIN . 

T 

share 

T 

B 

V 

B 

R 

Name ___________________________ 

Ways to Get Equal Angles 

I 

G 

R 

A 

S 

C 

A 

A 

D 

I 

N 

Vertical Angle Conjecture 

If two angles are vertical, 

then they are congruent. 

If ∠VET 

and ∠CER 

then ∠VET ≅∠ CER . 

Def. perpendicular lines 

If two lines are perpendicular, 

then they intersect to form equal 

90° angles. 

If CD 

V 

C 

E 

are vertical, 

⊥ AB , then 

m∠ CMA= m∠ CMB = 90°. 

Corresponding Angles Conjecture 

If two parallel lines are cut by a transversal, 

then corresponding angles are congruent. 

 

 

If AB DC , 

then ∠EFA ≅∠ FGD . 

Alternate Interior Angles Conjecture 

If two parallel lines are cut by a transversal, 

then alternate interior angles are congruent. 

 

If LK NJ , 

then ∠KHM ≅∠ HMN . 

A 

D 

C 

N 

A 

L 

M 

G 

E 

M 

T 

F 

H 

R 

B 

D 

B 

C 

K 

J 

Ways to Get equal Angles and Sides 

Complete the Triangle 

Congruence Shortcut 

Investigation, page 4 – 5. 

Def. perpendicular bisector 

A line (or part of a line) that passes through 

the midpoint of a segment and 

is perpendicular to the segment. 

If CD is the perp. bisector of AB , 

then AM ≅ MB 

∠ CMA =∠ CMB = °. 

and 90 

C 

A 

M 

B 

D 



Name ___________________________ 

4.6 Corresponding Parts of Congruent Triangles 

Note: Once you can prove that two triangles are congruent, by using the conjectures above, then any of 

the corresponding parts will be equal. 

Definition of Congruent Triangles: 

If two triangles are congruent, then all of their corresponding parts (sides and angles) are 

congruent. 

Can also be stated “Corresponding Parts of Congruent Triangles are Congruent” or 

CPCTC. 

When you write congruent triangles you must match the corresponding vertices. (To show how the match up.) 

So if you know that ΔCAT 

≅ ΔDOG 

then you can say any of the following: 

∠C 

≅ ∠ D CA ≅ DO 

∠A 

≅ ∠ O AT ≅ OG 

∠T 

≅ ∠ G CT ≅ DG 

Examples: How to prove (show deductively) that parts are equal. 

First example, informal paragraph : 

P 

Know: ∠PAC ≅ ∠TAC 

and PA AT 

Is ∠PCA ≅ ∠ TCA ? 

≅ . 

A 

T 

The triangles share a side, so CA 

So… 

= CA of course. 

C 

ΔPAC 

≅ ΔTAC 

by SAS Congruence 

and ∠PCA ≅ ∠ TCA because CPCTC or Corresponding Parts of Congruent Triangles are 

Congruent. 

A bit more formal paragraph: 

Given: Z is midpoint of DI and EP . 

Z 

Prove: PI ≅ DE ? 

Z is midpoint of DI and EP was given, so 

IZ = ZD and PZ = ZE by Definition of Midpoint 

D 

E 

∠PZI 

≅ ∠ EZD by the Vertical Angles Conjecture (or vertical angles are congruent.) 

Therefore, ΔZIP ≅ ΔZDE 

by SAS Congruence and 

PI DE 

≅ because CPCTC or Corresponding Parts of Congruent Triangles are Congruent. 

P 

I 



This one is tricky: 

Know: RE 

Is RE ≅ 

EC 

TC and TR CE . 

? What is congruent? 

Name ___________________________ 

RE TC given so ∠RET ≅ ∠ETC 

and 

TR CE given so ∠RTE ≅ ∠CET 

because parallel lines make equal alternate interior angles. 

TE = TE the triangles share a side, 

ΔTRE 

≅ ΔECT 

by ASA Congruence. So any of the corresponding sides are congruent. 

RE and EC are NOT corresponding sides. So, probably not equal. 

Note: you could say any of the following though: RE 

T 

R 

C 

≅ TC , RT ≅ EC or ∠R ≅ ∠ C . 

E 

This one is trickier! Start by un-overlapping possible pairs of congruent triangles: 

Given: LE = EP and LV = OP 

Prove: ∠ALP ≅ ∠ APL 

O 

A 

V 

E 

L 

P 

LE = EP so ∠ELP ≅ ∠ EPL If isosceles, then base angles are equal. 

LP = PL Same segment 

LV = OP Given information 

∠LOP 

≅ ∠PVL 

by SAS Congruence 

∠ALP 

≅ ∠ APL CPCTC 

Try Example A and Example B on Ch 4 Worksheet. 



4.7 Flowchart Thinking 

4.7 Page 238 Example B Flowchart Proof 

Given: EC AC 

≅ and ER ≅ AR 

Prove: ∠A ≅ ∠ E 

Name ___________________________ 

E 

R 

C 

EC 

Given 

≅ 

AC 

A 

ER ≅ AR 

Given 

RC≅ 

RC 

Same segment 

ΔREC 

≅ ΔRAC 

SSS Cong. Conj. 

∠A 

≅∠E 

CPCTC 

4.7 Page 239 Top. Explain why the angle bisector construction works. Flowchart Proof 

Given: 

∠ABC 

with BA ≅ BC and CD ≅ AD 

ABC 

Prove: BD is the angle bisector of 

∠ . 

BA ≅ 

BC 

Given 

CD ≅ AD 

Given 

BD≅ 

BD 

Same segment 

ΔBAD 

≅ ΔBCD 

SSS Cong. Conj. 

∠1≅∠2 

CPCTC 

BD is the angle bisector of 

∠ ABC . 

Def. of angle bisector 



Name ___________________________ 

What do you need to know to get… 

You’ll need Equal Segments to get… 

Def. Isosceles Triangle 

If a triangle’s legs are congruent, 

then it is isosceles. 

If VB 

then 

≅ VS 

ΔVBS 

isosceles. 

B 

V 

S 

Def. Median 

If a segment connects the vertex to the 

midpoint of the opposite side, 

then it is a median. 

If BM ≅ MC , 

then AM is median in 

A 

Δ ABC . 

B 

M 

C 

Def. Midpoint 

S 

If a point divides the segment into 

two equal segments, 

then it is a midpoint. 

If SM ≅ MG , 

then M is the midpoint of SG . 

M 

G 

Def. segment bisector 

C 

If a line (or part of a line) passes through the 

midpoint of the segment, 

then it is a bisector. 

M 

If AM ≅ MB, 

A 

then CM bisects AB . 

B 

You’ll need Equal Angles to get… 

Converse of the Isosceles Triangle Conj. 

If a triangle has two congruent angles, 

then it is an isosceles triangle. V S 

If ∠B ≅ ∠ S , 

ΔVBS 

then 

isosceles. 

B 

Def. perpendicular lines 

If two lines intersect to form equal 

adjacent angles, 

then they are perpendicular. 

If m∠ CMA= m∠ CMB , 

⊥ AB . 

then CD 

C 

A 

M 

B 

D 

Definition of Angle Bisector 

If a ray cuts the angle into two 

equal angles, then you have 

an angle bisector. 

If ∠ABD ≅ ∠ DBC , 

ABC 

then BD bisects ∠ . 

B 

C 

A 

D 

Corresponding Angles Conjecture 

If two lines are cut by a transversal and 

the corresponding angles are congruent, 

then the lines are parallel. 

If ∠EFA ≅∠ FGD , 

 

DC 

then AB . 

A 

D 

G 

E 

F 

B 

C 

Def. Altitude 

If a segment goes from a vertex 

perpendicular to the line that 

contains the opposite side, 

then it is an altitude. 

If m∠ RAT = m∠ RAI = 90°, 

TRI 

then RA is an altitude of 

T 

R 

A 

Δ . 

I 

Alternate Interior Angles Conjecture 

If two lines are cut by a transversal and 

the alternate interior angles are congruent, 

then the lines are parallel. 

If ∠KHM ≅∠ HMN , 

 

NJ 

then LK 

. 

L 

N 

M 

H 

K 

J 



Name ___________________________ 

You’ll need Equal Angles and Sides to get… 

Def. perpendicular bisector 

If a line (or part of a line) that passes through the midpoint of 

a segment and is perpendicular to the segment, 

then it is the perpendicular bisector. 

C 

If AM 

≅ MB and ∠ CMA = 90°, 

then CD is the perp. bisector of AB . 

A 

M 

B 

D 

Summary 

Basic Procedure for Proofs 

“parts” refers to sides and/or angles. 

1. Get equal parts by using given info. and known definitions and conjectures. 

2. State the triangles are congruent by SSS, SAS, ASA or AAS. 

3. Use CPCTC to get more equal parts. 

4. Connect that info. to what you were trying to prove . 

Hints [if you get stuck]: 

Mark the diagram with what you have stated as congruent in your proof. 

( If given M is the midpoint of AB , convert it to AM = MB by def. of 

midpoint before marking the diagram!) 

Look at the diagram to find equal parts. 

Brainstorm and then apply previous conjectures and definitions. 

Work (or think) backwards! 

Draw overlapping triangles separately. 

Re-draw figures without all of the “extra segments” in there. 

Draw an auxillary line. 

Break a problem into smaller parts. 



4.8 Proving Special Triangle Conjectures 

Name ___________________________ 

Equilateral/Equiangular Triangle Conjecture 

Every equilateral triangle is equiangular. 

Conversely, every equiangular triangle is equilateral. 

Vertex Angle Bisector Conjecture 

In an isosceles triangle, the bisector of the vertex angle is also the altitude and the median 

to the base and the perpendicular bisector of the base. 

Isosceles Triangle with vertex angle A, AB = AC. 

Median BM 

A 

Altitude BL 

L 

Angle Bisector BS 

AB = 3.36 in. 

AC = 3.36 in. 

CB = 4.34 in. 

C 

M 

S 

B 

Isosceles Triangle with Vertex Angle B 

B 

A 

EMD 

C 

Median BM 

AM = 1.64 in. 

MC = 1.64 in. 

Altitude BE 

m∠AEB = 90° 

Angle Bistector BD 

m∠ABD = 27° 

m∠DBC = 27° 

Medians to the Congruent Sides Theorem 

In an isosceles triangle, the medians to the congruent sides are congruent. 

Angle Bisectors to the Congruent Sides Theorem 

In an isosceles triangle, the angle bisectors to the congruent sides are congruent. 

Altitudes to the Congruent Sides Theorem 

In an isosceles triangle, the altitudes to the congruent sides are congruent. 

A 

A 

A 

N 

M 

S 

P 

T 

L 

C 

C 

C 

B 

B 

B 

Medians MC = BN Angle Bisectors PC = BS Altitudes BT = LC

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