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Triangles and Angles. Standard/Objectives:. Standard 3: Students will learn and apply geometric concepts. Objectives: Classify triangles by their sides and angles. Find angle measures in triangles

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Standard objectives
Standard/Objectives:

Standard 3: Students will learn and apply geometric concepts.

Objectives:

  • Classify triangles by their sides and angles.

  • Find angle measures in triangles

    DEFINITION: A triangle is a figure formed by three segments joining three non-collinear points.


Names of triangles
Names of triangles

Triangles can be classified by the sides or by the angle

Equilateral—3 congruent sides

Isosceles Triangle—2 congruent sides

Scalene—no congruent sides


Acute triangle
Acute Triangle

3 acute angles


Equiangular triangle
Equiangular triangle

  • 3 congruent angles. An equiangular triangle is also acute.


Right triangle

1 right angle

Right Triangle

Obtuse Triangle


Parts of a triangle

Each of the three points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices.

Two sides sharing a common vertext are adjacent sides.

The third is the side opposite an angle

Parts of a triangle

adjacent

Side opposite A

adjacent


Right triangle1

Red represents the hypotenuse of a right triangle. The sides that form the right angle are the legs.

Right Triangle

hypotenuse

leg

leg


An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is thebase.

Isosceles Triangles

leg

base

leg


Identifying the parts of an isosceles triangle

Explain why case it is ∆ABC is an isosceles right triangle.

In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle. Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle.

Identifying the parts of an isosceles triangle

About 7 ft.

5 ft

5 ft


Identifying the parts of an isosceles triangle1

Identify the legs and the hypotenuse of case it is ∆ABC. Which side is the base of the triangle?

Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse. Because AC BC, side AB is also the base.

Identifying the parts of an isosceles triangle

Hypotenuse & Base

About 7 ft.

5 ft

5 ft

leg

leg


Using angle measures of triangles
Using Angle Measures of Triangles case it is

Smiley faces are interior angles and hearts represent the exterior angles

Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.


Ex 3 finding an angle measure
Ex. 3 Finding an Angle Measure. case it is

Exterior Angle theorem: m1 = m A +m 1

x + 65 = (2x + 10)

65 = x +10

55 = x

65

(2x+10)

x


Finding angle measures

Corollary to the triangle sum theorem case it is

The acute angles of a right triangle are complementary.

m A + m B = 90

Finding angle measures

2x

x


Finding angle measures1

X + 2x = 90 case it is

3x = 90

X = 30

So m A = 30 and the m B=60

Finding angle measures

B

2x

x

A

C



Standards objectives
Standards/Objectives: case it is

Standard 2: Students will learn and apply geometric concepts

Objectives:

  • Identify congruent figures and corresponding parts

  • Prove that two triangles are congruent


Identifying congruent figures
Identifying congruent figures case it is

  • Two geometric figures are congruent if they have exactly the same size and shape.

NOT CONGRUENT

CONGRUENT


Triangles

Corresponding angles case it is

A ≅ P

B ≅ Q

C ≅ R

Corresponding Sides

AB ≅ PQ

BC ≅ QR

CA ≅ RP

Triangles

B

Q

R

A

C

P


Z case it is

  • If Δ ABC is  to Δ XYZ, which angle is  to C?


Thm 4 3 3 rd angles thm
Thm 4.3 case it is 3rd angles thm

  • If 2 s of one Δ are  to 2 s of another Δ, then the 3rd s are also .


Ex find x
Ex: find x case it is

)

)

22o

))

87o

))

(4x+15)o


Ex continued
Ex: continued case it is

22+87+4x+15=180

4x+15=71

4x=56

x=14


Ex abcd is to hgfe find x and y

9 cm case it is

91°

86°

113°

Ex: ABCD is  to HGFE, find x and y.

F

E

(5y-12)°

G

4x – 3 cm

H

4x-3=9 5y-12=113

4x=12 5y=125

x=3 y=25


Thm 4 4 props of s
Thm 4.4 case it is Props. of Δs

A

B

  • Reflexive prop of Δ - Every Δ is  to itself (ΔABC ΔABC).

  • Symmetric prop of Δ- If ΔABC ΔPQR, then ΔPQR ΔABC.

  • Transitive prop of Δ - If ΔABC ΔPQR & ΔPQR ΔXYZ, then ΔABC  ΔXYZ.

C

P

Q

R

X

Y

Z


Proving s are sss and sas

Proving case it is Δs are  : SSS and SAS


Standards benchmarks
Standards/Benchmarks case it is

Standard 2: Students will learn and apply geometric concepts

Objectives:

  • Prove that triangles are congruent using the SSS and SAS Congruence Postulates.

  • Use congruence postulates in real life problems such as bracing a structure.


Remember
Remember? case it is

  • As of yesterday, Δs could only be  if ALL sides AND angles were 

  • NOT ANY MORE!!!!

  • There are two short cuts to add.


Post 19 side side side sss post
Post. 19 case it is Side-Side-Side (SSS)  post

  • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .


Meaning

A case it is

Meaning:

___

___

___

___

If seg AB  seg ED, seg AC  seg EF & seg BC  seg DF, then ΔABC ΔEDF.

B

C

___

___

E

___

___

___

___

___

___

D

F


Given seg qr seg ut rs ts qs 10 us 10 prove qrs uts
Given case it is : seg QR  seg UT, RS  TS, QS=10, US=10Prove: ΔQRS ΔUTS

U

Q

10

10

R

S

T


Proof
Proof case it is

Statements Reasons

1. 1. given

2. QS=US 2. subst. prop. =

3. Seg QS  seg US 3. Def of  segs.

4. Δ QRS Δ UTS 4. SSS post


Post 20 side angle side post sas
Post. 20 case it is Side-Angle-Side post. (SAS)

  • If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .


  • If seg BC case it is  seg YX, seg AC  seg ZX, and C X, then ΔABC  ΔZXY.

B

Y

)

(

C

A

X

Z


Given seg wx seg xy seg vx seg zx prove vxw zxy
Given: seg WX case it is  seg. XY, seg VX  seg ZX, Prove: Δ VXW Δ ZXY

W

Z

X

1

2

Y

V


Proof1
Proof case it is

Statements Reasons

1. seg WX  seg. XY 1. given seg. VX  seg ZX

2. 1 2 2. vert s thm

3. Δ VXW Δ ZXY 3. SAS post


Given seg rs seg rq and seg st seg qt prove qrt srt
Given: seg RS case it is  seg RQ and seg ST  seg QTProve: Δ QRT  Δ SRT.

S

Q

R

T


Proof2
Proof case it is

Statements Reasons

1. Seg RS  seg RQ 1. Given seg ST  seg QT

2. Seg RT  seg RT 2. Reflex prop 

3. Δ QRT Δ SRT 3. SSS post


Given seg dr seg ag and seg ar seg gr prove dra drg
Given: seg DR case it is  seg AG and seg AR  seg GRProve: Δ DRA  Δ DRG.

D

R

A

G


Proof3

Statements case it is

seg DR  seg AG

Seg AR  seg GR

2. seg DR  Seg DR

3.DRG & DRA are rt. s

4.DRG   DRA

5. Δ DRG  Δ DRA

Reasons

Given

reflex. Prop of 

 lines form 4 rt. s

4. Rt. s thm

5. SAS post.

Proof



Objectives
Objectives: case it is

  • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem

  • Use congruence postulates and theorems in real-life problems.


Postulate 21 angle side angle asa congruence postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate


Theorem 4 5 angle angle side aas congruence theorem

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Theorem 4 5 angle angle side aas congruence theorem1

Given: congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.A  D, C  F, BC  EF

Prove: ∆ABC  ∆DEF

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Theorem 4 5 angle angle side aas congruence theorem2

You are given that two angles of congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Ex 1 developing proof

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof


Ex 1 developing proof1

A. In addition to the angles and segments that are marked, state the postulate or theorem you would use. Explain your reasoning.EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Ex. 1 Developing Proof


Ex 1 developing proof2

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof


Ex 1 developing proof3

B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Ex. 1 Developing Proof


Ex 1 developing proof4

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

UZ ║WX AND UW

║WX.

Ex. 1 Developing Proof

1

2

3

4


Ex 1 developing proof5

The two pairs of parallel sides can be used to show state the postulate or theorem you would use. Explain your reasoning.1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate.

Ex. 1 Developing Proof

1

2

3

4


Ex 2 proving triangles are congruent

Given: AD state the postulate or theorem you would use. Explain your reasoning.║EC, BD  BC

Prove: ∆ABD  ∆EBC

Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC

. Use the fact that AD ║EC to identify a pair of congruent angles.

Ex. 2 Proving Triangles are Congruent


Proof4

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1.

Proof:


Proof5

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1. Given

Proof:


Proof6

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Proof:


Proof7

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:


Proof8

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:


Proof9

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:


Note: state the postulate or theorem you would use. Explain your reasoning.

  • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.


Using congruent triangles

Using Congruent Triangles state the postulate or theorem you would use. Explain your reasoning.


Objectives1
Objectives: state the postulate or theorem you would use. Explain your reasoning.

  • Use congruent triangles to plan and write proofs.

  • Use congruent triangles to prove constructions are valid.


Planning a proof
Planning a proof state the postulate or theorem you would use. Explain your reasoning.

  • Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.


Planning a proof1

For example, suppose you want to prove that state the postulate or theorem you would use. Explain your reasoning.PQS ≅ RQS in the diagram shown at the right. One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate. Then you can use the fact that corresponding parts of congruent triangles are congruent to conclude that PQS ≅ RQS.

Planning a proof


Ex 1 planning writing a proof

Given: AB state the postulate or theorem you would use. Explain your reasoning.║ CD, BC ║ DA

Prove: AB≅CD

Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.

Ex. 1: Planning & Writing a Proof


Ex 1 planning writing a proof1

Solution: First copy the diagram and mark it with the given information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

Ex. 1: Planning & Writing a Proof


Ex 1 paragraph proof

Because AD information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.║CD, it follows from the Alternate Interior Angles Theorem that ABD ≅CDB. For the same reason, ADB ≅CBD because BC║DA. By the Reflexive property of Congruence, BD ≅ BD. You can use the ASA Congruence Postulate to conclude that ∆ABD ≅ ∆CDB. Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.

Ex. 1: Paragraph Proof


Ex 2 planning writing a proof

Given: A is the midpoint of MT, A is the midpoint of SR. information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

Prove: MS ║TR.

Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that M ≅ T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.

Ex. 2: Planning & Writing a Proof


Given a is the midpoint of mt a is the midpoint of sr prove ms tr

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Given a is the midpoint of mt a is the midpoint of sr prove ms tr1

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Definition of a midpoint

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Given a is the midpoint of mt a is the midpoint of sr prove ms tr2

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Definition of a midpoint

Vertical Angles Theorem

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Given a is the midpoint of mt a is the midpoint of sr prove ms tr3

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Definition of a midpoint

Vertical Angles Theorem

SAS Congruence Postulate

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Given a is the midpoint of mt a is the midpoint of sr prove ms tr4

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Definition of a midpoint

Vertical Angles Theorem

SAS Congruence Postulate

Corres. parts of ≅ ∆’s are ≅

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Given a is the midpoint of mt a is the midpoint of sr prove ms tr5

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A is the midpoint of MT, A is the midpoint of SR.

MA ≅ TA, SA ≅ RA

MAS ≅ TAR

∆MAS ≅ ∆TAR

M ≅ T

MS ║ TR

Reasons:

Given

Definition of a midpoint

Vertical Angles Theorem

SAS Congruence Postulate

Corres. parts of ≅ ∆’s are ≅

Alternate Interior Angles Converse.

Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.


Ex 3 using more than one pair of triangles

Given: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.1≅2, 3≅4.

Prove ∆BCE≅∆DCE

Plan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE.

Ex. 3: Using more than one pair of triangles.

2

4

3

1


Given 1 2 3 4 prove bce dce

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Given 1 2 3 4 prove bce dce1

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Reflexive property of Congruence

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Given 1 2 3 4 prove bce dce2

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Reflexive property of Congruence

ASA Congruence Postulate

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Given 1 2 3 4 prove bce dce3

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Reflexive property of Congruence

ASA Congruence Postulate

Corres. parts of ≅ ∆’s are ≅

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Given 1 2 3 4 prove bce dce4

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Reflexive property of Congruence

ASA Congruence Postulate

Corres. parts of ≅ ∆’s are ≅

Reflexive Property of Congruence

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Given 1 2 3 4 prove bce dce5

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

1≅2, 3≅4

AC ≅ AC

∆ABC ≅ ∆ADC

BC ≅ DC

CE ≅ CE

∆BCE≅∆DCE

Reasons:

Given

Reflexive property of Congruence

ASA Congruence Postulate

Corres. parts of ≅ ∆’s are ≅

Reflexive Property of Congruence

SAS Congruence Postulate

Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCE

2

4

3

1


Ex 4 proving constructions are valid
Ex. 4: Proving constructions are valid information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

  • In Lesson 3.5 – you learned to copy an angle using a compass and a straight edge. The construction is summarized on pg. 159 and on pg. 231.

  • Using the construction summarized above, you can copy CAB to form FDE. Write a proof to verify the construction is valid.


Plan for proof

Show that information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.∆CAB ≅ ∆FDE. Then use the fact that corresponding parts of congruent triangles are congruent to conclude that CAB ≅ FDE. By construction, you can assume the following statements:

AB ≅ DE Same compass setting is used

AC ≅ DF Same compass setting is used

BC ≅ EF Same compass setting is used

Plan for proof


Given ab de ac df bc ef prove cab fde

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

AB ≅ DE

AC ≅ DF

BC ≅ EF

∆CAB ≅ ∆FDE

CAB ≅ FDE

Reasons:

Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE

2

4

3

1


Given ab de ac df bc ef prove cab fde1

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

AB ≅ DE

AC ≅ DF

BC ≅ EF

∆CAB ≅ ∆FDE

CAB ≅ FDE

Reasons:

Given

Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE

2

4

3

1


Given ab de ac df bc ef prove cab fde2

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

AB ≅ DE

AC ≅ DF

BC ≅ EF

∆CAB ≅ ∆FDE

CAB ≅ FDE

Reasons:

Given

Given

Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE

2

4

3

1


Given ab de ac df bc ef prove cab fde3

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

AB ≅ DE

AC ≅ DF

BC ≅ EF

∆CAB ≅ ∆FDE

CAB ≅ FDE

Reasons:

Given

Given

Given

SSS Congruence Post

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE

2

4

3

1


Given ab de ac df bc ef prove cab fde4

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

AB ≅ DE

AC ≅ DF

BC ≅ EF

∆CAB ≅ ∆FDE

CAB ≅ FDE

Reasons:

Given

Given

Given

SSS Congruence Post

Corres. parts of ≅ ∆’s are ≅.

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅FDE

2

4

3

1


Given qs rp pt rt prove ps rs

Statements: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

QS  RP

PT ≅ RT

Reasons:

Given

Given

Given: QSRP, PT≅RTProve PS≅ RS

2

4

3

1


Isosceles equilateral and right s

Isosceles, Equilateral and Right information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.s

Pg 236


Standards objectives1
Standards/Objectives: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

Standard 2: Students will learn and apply geometric concepts

Objectives:

  • Use properties of Isosceles and equilateral triangles.

  • Use properties of right triangles.


Isosceles triangle s special parts
Isosceles triangle’s special parts information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

A

A is the vertex angle (opposite the base)

 B and C are base angles (adjacent to the base)

Leg

Leg

C

B

Base


Thm 4 6 base s thm
Thm 4.6 information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.Base s thm

  • If 2 sides of a  are @, the the s opposite them are @.( the base s of an isosceles  are )

A

If seg AB @ seg AC, then  B @  C

)

(

B

C


Thm 4 7 converse of base s thm
Thm 4.7 information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.Converse of Base s thm

  • If 2 s of a  are @, the sides opposite them are @.

A

If  B @ C, then seg AB @ seg AC

)

(

C

B


Corollary to the base s thm
Corollary to the base information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.s thm

  • If a triangle is equilateral, then it is equiangular.

A

If seg AB @ seg BC @ seg CA, then A @ B @C

B

C


Corollary to converse of the base angles thm
Corollary to converse of the base angles thm information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

  • If a triangle is equiangular, then it is also equilateral.

A

)

If A @B @C, then seg AB @ seg BC @ seg CA

)

B

(

C


Example find x and y
Example: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.find x and y

  • X=60

  • Y=30

Y

X

120


Thm 4 8 hypotenuse leg hl @ thm
Thm 4.8 information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.Hypotenuse-Leg (HL) @ thm

A

  • If the hypotenuse and a leg of one right  are @ to the hypotenuse and leg of another right , then the s are @.

_

B

C

_

Y

_

X

_

If seg AC @ seg XZ and seg BC @ seg YZ, then  ABC @ XYZ

Z


Given d is the midpt of seg ce bcd and fed are rt s and seg bd @ seg fd prove bcd @ fed
Given: D is the midpt of seg CE, information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.BCD and FED are rt s and seg BD @ seg FD.Prove:  BCD @ FED

B

F

D

C

E


Proof10

Statements information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

D is the midpt of seg CE,  BCD and <FED are rt  s and seg BD @ to seg FD

Seg CD @ seg ED

 BCD  FED

Reasons

Given

Def of a midpt

HL thm

Proof


Are the 2 triangles @
Are the 2 triangles information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are [email protected] ?

(

Yes, ASA or AAS

)

)

(

(

(


Find x and y
Find x and y. information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

y

x

60

75

90

y

x

x

x=60

2x + 75=180

2x=105

x=52.5

y=30

y=75


Find x
Find x. information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

)

56ft

(

8xft

)

))

56=8x

7=x

((


Triangles and coordinate proof

Triangles and Coordinate Proof information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.


Objectives2
Objectives: information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

  • Place geometric figures in a coordinate plane.

  • Write a coordinate proof.


Placing figures in a coordinate plane
Placing Figures in a Coordinate Plane information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

  • So far, you have studied two-column proofs, paragraph proofs, and flow proofs. A COORDINATE PROOF involves placing geometric figures in a coordinate plane. Then you can use the Distance Formula (no, you never get away from using this) and the Midpoint Formula, as well as postulate and theorems to prove statements about figures.


Ex 1 placing a rectangle in a coordinate plane
Ex. 1: Placing a Rectangle in a Coordinate Plane information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

  • Place a 2-unit by 6-unit rectangle in a coordinate plane.

  • SOLUTION: Choose a placement that makes finding distance easy (along the origin) as seen to the right.


Ex 1 placing a rectangle in a coordinate plane1

One vertex is at the origin, and three of the vertices have at least one coordinate that is 0.

Ex. 1: Placing a Rectangle in a Coordinate Plane


Ex 1 placing a rectangle in a coordinate plane2

One side is centered at the origin, and the x-coordinates are opposites.

Ex. 1: Placing a Rectangle in a Coordinate Plane


Note: are opposites.

  • Once a figure has been placed in a coordinate plane, you can use the Distance Formula or the Midpoint Formula to measure distances or locate points


Ex 2 using the distance formula

A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse.

Ex. 2: Using the Distance Formula


Ex 2 using the distance formula1

One possible placement is shown. Notice that one leg is vertical and the other leg is horizontal, which assures that the legs meet as right angles. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate.

Ex. 2: Using the Distance Formula


Ex 2 using the distance formula2

You can use the Distance Formula to find the length of the hypotenuse.

d = √(x2 – x1)2 + (y2 – y1)2

= √(12-0)2 + (5-0)2

= √169

= 13

Ex. 2: Using the Distance Formula


Ex 3 using the midpoint formula

In the diagram, hypotenuse. ∆MLN ≅ ∆KLN). Find the coordinates of point L.

Solution: Because the triangles are congruent, it follows that ML ≅ KL. So, point L must be the midpoint of MK. This means you can use the Midpoint Formula to find the coordinates of point L.

Ex. 3 Using the Midpoint Formula


Ex 3 using the midpoint formula1

L (x, y) = x hypotenuse. 1 + x2, y1 +y2

2 2

Midpoint Formula

=160+0 , 0+160

2 2

Substitute values

= (80, 80)

Simplify.

Ex. 3 Using the Midpoint Formula


Writing coordinate proofs
Writing Coordinate Proofs hypotenuse.

  • Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure.


Ex 4 writing a plan for a coordinate proof
Ex. 4: Writing a Plan for a Coordinate Proof hypotenuse.

  • Write a plan to prove that SQ bisects PSR.

  • Given: Coordinates of vertices of ∆PQS and ∆RQS.

  • Prove SQ bisects PSR.

  • Plan for proof: Use the Distance Formula to find the side lengths of ∆PQS and ∆RQS. Then use the SSS Congruence Postulate to show that ∆PQS ≅ ∆RQS. Finally, use the fact that corresponding parts of congruent triangles are congruent (CPCTC) to conclude that PSQ ≅RSQ, which implies that SQ bisects PSR.


Ex 4 writing a plan for a coordinate proof1

Given: Coordinates of vertices of hypotenuse. ∆PQS and ∆RQS.

Prove SQ bisects PSR.

Ex. 4: Writing a Plan for a Coordinate Proof


NOTE: hypotenuse.

  • The coordinate proof in Example 4 applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates.

  • For instance, you can use variable coordinates to duplicate the proof in Example 4. Once this is done, you can conclude that SQ bisects PSR for any triangle whose coordinates fit the given pattern.



Ex 5 using variables as coordinates

Right hypotenuse. ∆QBC has leg lengths of h units and k units. You can find the coordinates of points B and C by considering how the triangle is placed in a coordinate plane.

Point B is h units horizontally from the origin (0, 0), so its coordinates are (h, 0). Point C is h units horizontally from the origin and k units vertically from the origin, so its coordinates are (h, k). You can use the Distance Formula to find the length of the hypotenuse QC.

Ex. 5: Using Variables as Coordinates

C (h, k)

hypotenuse

k units

Q (0, 0)

B (h, 0)

h units


Ex 5 using variables as coordinates1

OC = hypotenuse. √(x2 – x1)2 + (y2 – y1)2

= √(h-0)2 + (k - 0)2

= √h2 + k2

Ex. 5: Using Variables as Coordinates

C (h, k)

hypotenuse

k units

Q (0, 0)

B (h, 0)

h units


Ex 5 writing a coordinate proof

Given: Coordinates of figure OTUV hypotenuse.

Prove ∆OUT  ∆UVO

Coordinate proof: Segments OV and UT have the same length.

OV = √(h-0)2 + (0 - 0)2=h

UT = √(m+h-m)2 + (k - k)2=h

Ex. 5 Writing a Coordinate Proof


Ex 5 writing a coordinate proof1

Horizontal segments UT and OV each have a slope of 0, which implies they are parallel. Segment OU intersects UT and OV to form congruent alternate interior angles TUO and VOU. Because OU  OU, you can apply the SAS Congruence Postulate to conclude that ∆OUT  ∆UVO.

Ex. 5 Writing a Coordinate Proof


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