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Geometry

Congruent Triangles

2015-10-23

www.njctl.org

http://www.njctl.org

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Table of Contents· Congruent Triangles

· Isosceles Triangle Theorem

· SSS Congruence· SAS Congruence· ASA Congruence · AAS Congruence· HL Congruence

· CPCTC· Triangle Congruence Proofs

click on the topic to go to that section

· Proving Congruence

· PARCC Sample Questions

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Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Congruent Triangles

Return to Tableof Contents

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Similar Triangles

We learned in the Similar Triangles topic (Triangles unit) that if two triangles are similar:

· All their angles are congruent· All their corresponding sides are in proportion

We also learned how to identify the corresponding sides as being opposite to equal angles, or subtended by equal angles

And we learned that the constant of proportionality for the corresponding sides of one triangle to the other was called "k."

If needed, go back to review that topic before proceeding.

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Congruent Triangles

Congruent triangles are a special case of similar triangles.

The constant of proportionality is one, so the corresponding sides are of equal measure.

For congruent triangles, all the angles are congruent AND all the corresponding sides are congruent.

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Naming Congruent Triangles

Just as in the case of similar triangles, the naming of congruent triangles is important: order matters.

The statement:ΔABC is congruent to ΔDEF

indicates that these triangles are congruent.

AND that these angle measures are equal: m∠A = m∠D m∠B = m∠E m∠C = m∠F

AND these lengths are equal:

AB = DEBC = EFCA = FD

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Proving Triangles Congruent

We can prove triangles congruent by proving the measures of all three corresponding angles and the lengths of all three corresponding sides are equal.

Earlier we showed that we need to prove only two angles are congruent to show that triangles are similar, since the third angle must then be congruent.

There are similar shortcuts to proving triangles congruent.

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Third Angle TheoremRecall the proof showing if we know that two pairs of corresponding angles are congruent, then the third pair of corresponding angles are congruent as well.

Statement Reason

1 ∠A ≅ ∠D and ∠B ≅ ∠E Given

2 m∠A = m∠D; m∠B = m∠E Definition of ≅ angles

3 m∠A+ m∠B + m∠C = 180ºm∠D+ m∠E + m∠F = 180º

Triangle Sum Theorem

4 m∠D+ m∠E + m∠C = 180ºm∠D+ m∠E + m∠F = 180º

Substitution Property of Equality

5 m∠D + m∠E + m∠C = m∠D + m∠E + m∠F


6 m∠C = m∠F Subtraction Property of Equality

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Corresponding Parts

Let's review identifying the corresponding parts (angles and sides) of pairs of triangles.

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Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts

A

B

C

D

E

F

Corresponding Parts

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A

B

C

D

E

F

Given that ΔABC is congruent to ΔDEF, the triangles are marked accordingly in this diagram.

Corresponding Parts

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A

B

D

C

E

Part Corresponding PartSegment AB Segment ED

∠A ∠E

Segment AC Segment EC

∠B ∠D

Segment CB Segment CD

∠ACB ∠ECD

ΔABC ≅ ΔEDC

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Example

Corresponding Sides Corresponding Angles

Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and sides. (Draw a diagram)

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1 What is the corresponding part to ∠ J ? A ∠R B ∠K C ∠Q D ∠P

J

K L R Q

P

ΔJKL ≅ ΔPQR

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2 What is the corresponding part to ∠Q?

A ∠R B ∠K C ∠Q D ∠P J

K L R Q

P

ΔJKL ≅ ΔPQR

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3 What is the corresponding part to QP? A JL B LK C KJ D PQ

J

K L R Q

P

ΔJKL ≅ ΔPQR

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Z

X

CV

B

4 The congruence statement for the two triangles is:

A ΔBVC ≅ ΔXCZ

B ΔXCB ≅ ΔBCX

C ΔVBC ≅ ΔZXC

D ΔCBV ≅ ΔCZX

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5 Complete the congruence statement: ΔXYZ ≅ A ΔXWZ B ΔZWX C ΔWXZ D ΔZXW

Y

ZW

X

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Properties of Congruence and Equality

We will be using the three properties of congruence we learned earlier

Reflexive Property of CongruenceSymmetric Property of CongruenceTransitive Property of Congruence

As well as the four properties of equality we learned earlier

Reflexive Property of EqualitySymmetric Property of EqualityTransitive Property of Equality


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Proving Congruence

SSS (Side-Side-Side)


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Congruent triangles have all congruent sides and angles.

However, congruence can be proven by showing less than that.

We will prove some theorems which you can then use as shortcuts to proving two triangles congruent. It is not necessary to prove that all the angles and sides are congruent.

Proving Congruence

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Side-Side-Side Triangle CongruenceIf two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.

Euclid - Book 1: Proposition 8

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Side-Side-Side Triangle CongruenceEuclid showed that:

Having three equal sides requires having three equal angles.

Therefore, having three pairs of equal sides verifies that two triangles are congruent since all their corresponding sides and angles must be congruent.

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Side-Side-Side Triangle CongruenceEuclid's argument of this (and for some of the following postulates/theorems) was based on transposing one triangle on top of the other.

He confirmed that if all the corresponding sides are equal, once you place one triangle atop the other in the correct orientation, all the sides have to line up and all the angles must as well.

Recall the Fourth Axiom: Things which coincide with one another are equal to one another.

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Side-Side-Side Triangle Congruence

Click here to go to the lab titled, "Triangle Congruence SSS"

This is shown below.

Can you imagine a way that the corresponding angles could be of different measure without changing the length of one of the sides?

This is often called the SSS Triangle Congruence for short.

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sss-lab/

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Example 1

A F

K

BGSolution:

The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other.

By SSS, this proves congruence.

Please note that this requires that all three sides are congruent.

Prove that ΔAFK is congruent to ΔBGK

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F

G HK

J

Example 2

Given: FG = JK, FH = JH, and H is the midpoint of GK

Prove: ΔFGH ≅ ΔJKH

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Statement Reason

1 FG = JK, FH = JH and H is the midpoint of GK Given

2 FG ≅ JK, FH ≅ JHDefinition of congruent segments

3 GH ≅ HK Definition of midpoint

4 ΔFGH ≅ ΔJKH Side-Side-Side Triangle Congruence

G

F

H

J

K

Example 2

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A

B

C H

J

K

6 ΔABC ≅ ΔHJK

TrueFalse

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A

B

C H

J

K

7 ΔCAB ≅ ΔHJK

TrueFalse

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R

S

T U

8 ΔSRT ≅ ΔSUT TrueFalse

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9 Provide the reason for the second step.

Statement Reason

1 RS ≅ US, RT ≅ UT Given

2 ST ≅ ST ?

3 ΔSRT ≅ ΔSUT Side-Side-Side Triangle Congruence

R

S

T U

A Given B Side-Side-Side Triangle CongruenceC Reflexive property of congruence D Substitution property of congruence E Transitive property of congruence

Ans

wer

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10 ΔABC is congruent to

A ΔQRS

B ΔSRQ

C ΔACB

D ΔRSQ

A

B C Q R

S

3

4

5 3

4

5

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Proving Congruence

SAS (Side-Angle-Side)


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Side-Angle-Side Triangle Congruence

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

- Euclid's Elements - Book One: Proposition 4

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As in Side-Side-Side Triangle Congruence, Euclid verifies Side-Angle-Side Triangle Congruence by superposition (transposing one triangle atop the other.) He thereby indicates that if two sides of two triangles, and the angles contained by those sides, are equal, then all of the sides and angles must be equal...showing congruence.

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Given that two triangles have two equal corresponding sides and equal angles contained by those two equal sides, the third sides must also be equal.

Tap below and the third side of each triangle will become visible.


Tap to reveal third side of the triangles

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It is clear that the third side of each triangle is completely defined by the two other sides and their included angle.

So, the third sides must also be congruent.

This is often called the SAS Triangle Congruence for short.


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So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent.


Click here to go to the lab titled, "Triangle Congruence SAS"

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sas-lab/

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1 2

L

M

P

N

O

Example

Given: MP ≅ NP and LP ≅ OP

Prove: ΔMLP ≅ ΔNOP

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11 Provide the reason for line 2.

1 2

L

M

P

N

O

Given: MP ≅ NP and LP ≅ OPProve: ΔMLP ≅ ΔNOP

Statement Reason

1 MP ≅ NP and LP ≅ OP Given

2 ∠1 ≅ ∠2 ?

3 ΔMLP ≅ ΔNOP ?

A Given B Side-Side-Side Triangle CongruenceC Side-Angle-Side Triangle CongruenceD Vertical angles are congruent E Alternate interior angles are congruent

Ans

wer

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12 Provide the reason for line 3.

Statement Reason

1 MP ≅ NP and LP ≅ OP Given

2 ∠1 ≅ ∠2 ?

3 ΔMLP ≅ ΔNOP ?

A Given B Side-Side-Side Triangle CongruenceC Side-Angle-Side Triangle CongruenceD Vertical angles are congruent E Alternate interior angles are congruent

Given: MP ≅ NP and LP ≅ OPProve: ΔMLP ≅ ΔNOP

1 2

L

M

P

N

O

Ans

wer

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13 What is the included angle of the given sides of the triangle?

A ∠J B ∠KC ∠L

Hint: Draw the triangle!

ΔJKL, sides KL and JK

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P

QR TV4 4

5 5

100° 100°

S

14 List the congruent parts of the triangles below. Is ΔPQR ≅ ΔSTV?

YesNo

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F

GH

X

Y Z46° 46°

1010

77

Why?

15 Is ΔFGH ≅ ΔXYZ by Side-Angle-Side Triangle Congruence?

YesNo

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A B

C D

16 Using SAS Triangle Congruence, what information is needed to show ΔABC ≅ ΔDCB ?

A ∠DBC ≅ ∠ABCB ∠B ≅ ∠CC ∠ABD ≅ ∠DCAD ∠ABC ≅ ∠DCB

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17 What type of congruence exists between the two triangles?

A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information

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18 What type of congruence exists between the two triangles?A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information

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45° 45°12 12

A

B

CD

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Proving Congruence

ASA (Angle-Side-Angle)

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Angle-Side-Angle Triangle CongruenceAnother way to prove two triangles are congruent makes use of Euclid's Fifth Postulate.

This illustration should look familiar from the unit on parallel lines. It shows non-parallel lines intersected by a transversal.

1 2

3 4

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We know from Euclid's Fifth Postulate, that the non-parallel lines will intersect on the side of the transversal on which the sum of the interior angles is less than 180º.

In this case, that's the side of the transversal with angles 2 and 4.

1 2

3 4

Angle-Side-Angle Triangle Congruence

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By extending the lines and decreasing angles 2 and 4, we can see where the non-parallel lines intersect. This forms a triangle in which the transversal is one side and the two non-parallel lines form the other two sides.

1 2

3 4


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You can see that we have formed a triangle on the right side of the transversal, with the transversal providing one side and the two non-parallel lines the other two sides.

Let's examine that triangle.

1 2

3 4

5

A

C

B


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We can see that ∠2 and ∠4 lead to only one possible value for ∠5, since the angles must add to 180º.

That means that two triangles with two corresponding angles which are congruent, must have their third angles equal, so they are similar...but we knew that from earlier.

1 2

3 4

5

A

C

B


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Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size.The two triangles must be congruent. This is called ASA Triangle Congruence for short.

1 2

3 4

5

A

C

B


To see a visual representation of what we discussed for ASA Triangle Congruence, click the link below.

http://www.mathopenref.com/congruentasa.html

Click here to go to the lab titled, "Triangle Congruence ASA"

http://www.mathopenref.com/congruentasa.html

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-asa-lab/

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W

X

Y

23 What is the included side between ∠X and ∠W?

A YX B YW C XW

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W

X

Y

24 What is the included side between ∠X and ∠Y ?

A XW B YX

C YW

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M

N

O

P

25 What information is needed to have ASA Triangle Congruence between the two triangles?

ABCD

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A

C

D B

26 What information is needed to have ASA Triangle Congruence between the two triangles?

ABCD

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E

F GM

H

27 Why is ∠FME ≅ ∠GMH?

A ASA Triangle CongruenceB vertical anglesC included anglesD congruent

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A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information

Q R

U T S

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When you have overlapping figures that share sides and/or angles, marking the diagram with the given information and separating the triangles (when needed) make it easier to understand the problem.

Another strategy that you could use is to look for repeating letters once you separate the two triangles. When 2 letters repeat, then you have a common side shared. When 1 letter repeats, then you have a common angle shared.

Strategy to Prove Congruence

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29 What type of congruence exists between ΔJLM and ΔNLK?


J

M

N

K

L

Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?

Hints:click to reveal

click to reveal

click to reveal

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A

B

C

Q R

30 What type of congruence exists between ΔABQ and ΔCBR?


Mark the diagram with the given information. Be careful you don't always use all information

Hint

click to reveal

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C

B

Q R

A

B

R

31 What type of congruence exists between ΔQAR and ΔRCQ?


Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?

Hints:click to reveal

click to reveal

click to reveal

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ST

ND

A

vertical

32 What type of congruence exists between ΔSAN and ΔDAT?

A SSS Triangle Congruence

B SAS Triangle Congruence

C ASA Triangle Congruence

D Not enough information

At the intersection of two lines you always have _____ angles.HintClick to Reveal Click

Given: SA ≅ DA AN ≅ AT

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33 What type of congruence exists between the two triangles?A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough

information

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D Not enough information C

P M

S

A

Hint:Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal

Given: PA ≅ MA ∠P ≅ ∠M

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Proving Congruence

AAS (Angle-Angle-Side)

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Based on that same logic, if ANY two corresponding angles and one corresponding side of a pair of triangles are congruent, the triangles must also be congruent.

This follows from the fact that the Triangle Sum Theorem tells us that once we know the measures of two angles, we know the measure of the third, since they must add to 180º.

1 2

3 4

5

A

C

B

Angle-Angle-Side Triangle Congruence

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Another way of looking at these two theorems is that once you show that two corresponding angles in two triangles are congruent, you know that the third angles are congruent and that the two triangles are similar. That means that they have the same shape.

If you can show a side in one of those triangles is congruent to the corresponding side of the other, you know that they are same size. Thus the scale factor, k, is 1.

If they are the same size and shape, they are congruent.

ASA and AAS Triangle Congruence

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It is really just a formality whether you use the term ASA or AAS, since all three angles must be congruent.

However, to note the difference, if the angles are both adjacent to the side which has shown to be congruent, the reason for congruence is ASA (∆ABC ≅ ∆DEF).

If not, it is AAS (∆GHI ≅ ∆JKL).

ASA and AAS Triangle Congruence

A

B C

D

E F

G

H I

J

K L

VS.

ASA AAS

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Example

C

A

H

T

Given: ∠H ≅ ∠C ∠HTA ≅ ∠CTA

Is ΔCTA ≅ ΔHTA ?

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Example

1) Mark the diagram:

C

A

H

T

Given: ∠H ≅ ∠C ∠HTA ≅ ∠CTA

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Example

2) By the reflexive property:

Therefore, ΔHTA ≅ ΔCTA by AAS Triangle Congruence. AT ≅ AT

C

A

H

T

congruence statement?

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Given:

35 What type of congruence exists, if any, between the two triangles?




D AAS Triangle Congruence

E Not enough information

D

E

F

H G

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A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information

A

B C Q

RS

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Q

W

E

R

T



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A

S

D

F

G

H







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C ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information

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AB

C

D



Given: BD bisects ∠ABC,∠A ≅ ∠C

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A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle

CongruenceD AAS Triangle

CongruenceE Not enough

information

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HL Congruence


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The final shortcut to proving congruence is Hypotenuse Leg Triangle Congruence, or the HL Triangle Congruence for short.

This theorem states that if two right triangles have their hypotenuses and one of their legs congruent, then the triangles are congruent.

The HL Triangle Congruence can be considered a corollary of the SSS Triangle Congruence.

Hypotenuse-Leg Triangle Congruence

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In a right triangle, the sum of the squares of the lengths of the two legs must equal the square of the length of the hypotenuse.

c2 = a2 + b2

If we are given that for two right triangles the hypotenuse and one of the legs are equal (c1 = c2 and a1 = a2), then we know that the other leg but also be equal (b1 = b2).

Thus, HL Triangle Congruence can be considered a special case, or corollary, of the Side-Side-Side Triangle Congruence.


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c12 = a1

2 + b12 c2

2 = a22 + b2

2

Solving for b in both equations

c12 - a1

2 = b12 c2

2 - a22 = b2

2

b12 = c1

2 - a12 b2

2 = c22 - a2

2

Substituting c1 = c2 and a1 = a2

b12 = c2

2 - a22 b2

2 = c22 - a2

2

b12 = b2

2

b1 = b2


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A B

C

R S

T

ExampleAre these two triangles congruent?

These are right triangles, so look for HL Triangle Congruence.

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Example

Recall that the side opposite the right angle is the hypotenuse, and the other two sides are called legs.

Hypotenuse: AC ≅ RT

Leg: CB ≅ TS

By the HL Triangle Congruence, ∆ABC ≅ ∆RST

A B

C

R S

T

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Side-Side-Side (SSS): three sides Side-Angle-Side (SAS): two sides and the included angle

Angle-Side-Angle (ASA): two angles and the included side

Angle-Angle-Side (AAS): two angles and one non-included side

Hypotenuse-Leg (HL): hypotenuse and one leg (right triangles)

Postulates/Theorems to Prove Triangles Congruent

To use the congruence postulates/theorems, we need to know or be able to show the following congruences between two triangles:

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Given:

Q

R S

X

Y Z

Mark the given on the diagram. Note that it is a right triangle.

43 What type of congruence exists, if any, between the two triangles?A SSS Triangle

CongruenceB SAS Triangle

CongruenceC ASA Triangle


CongruenceE HL Triangle CongruenceF Not enough information

HintClick to reveal

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L

M

N O

P

Q

If they are congruent what is the congruence statement?




CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information

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A

B

CD

E

F






CongruenceE HL Triangle CongruenceF Not enough information

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T

U

V

W

X

Y





CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information

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Q

W

EY



A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information

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N

M

O

J

K

L




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E

F

G

H







E HL Triangle CongruenceF Not enough information

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E

F

G

H

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K F

B M

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P O

Y

alternate interior


U

What angles are congruent when parallel lines are cut by a transversal?






E HL Triangle Congruence

F Not enough information

Click to Reveal

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O K

MJ








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A S

XZ




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Triangle Congruence Proofs


page2svg

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First: identify given information

Second: use a diagram that is marked with given information

Third: review congruence postulates/theorems - what information is needed (sides/angles) to use one of these?

SSS SAS ASA AAS HL

Strategy to Prove Congruence

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E

FM

G

90°

90°8

8H

Example

Given: MF = MH = 8 and m∠F = m∠H = 90º

Prove: ΔEFM ≅ ΔGHM

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55 Provide the reason for line 3.A Angle-Side-Angle Triangle

CongruenceB Side-Side-Side Triangle

CongruenceC Side-Angle-Side Triangle

CongruenceD Vertical angles are congruent E Alternate interior angles are congruent

E

FM

G

90°

90°8

8H



Statement Reason

1 MF = MH = 8 and m∠F = m∠H = 90º Given

2 MF ≅ MH and ∠F ≅ ∠H Defn. of congruence

3 ∠FME ≅ ∠HMG ?

4 ΔEFM ≅ ΔGHM ?

Ans

wer

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56 Provide the reason for line 4.A Angle-Side-Angle Triangle

CongruenceB Side-Side-Side Triangle

CongruenceC Side-Angle-Side Triangle

CongruenceD Vertical angles are congruent E Alternate interior angles are congruent

E

FM

G

90°

90°8

8H



Statement Reason

1 MF = MH = 8 and m∠F = m∠H = 90º Given

2 MF ≅ MH and ∠F ≅ ∠H Defn. of congruence

3 ∠FME ≅ ∠HMG ?

4 ΔEFM ≅ ΔGHM ?

Ans

wer

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Congruent Reasons Summary

SSS

SSASAS

AASASA

AAA

HL0

3

1

2

# of congruent angles

postulate/theorem

(Drag those that don't work out of the chart. Then put HL where it would belong.)

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Example

A F

K

GB

Solution (two-column):

1) Given

2) SSS Triangle Congruence

AF ≅ BG, FK ≅ GK KA ≅ KB

1)

2) ΔAFK ≅ ΔBGK

Statements Reasons

Given: AF ≅ BG, FK ≅ GK, and KA ≅ KBProve: ΔAFK ≅ ΔBGK

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3. Reflexive property

2. Definition of ∠bisector

4. SAS Triangle Congruence

Example

A

B

C

DStatements Reasons

1. Given

click ___________

click ___________

click ___________

Given: BC ≅ CD AC bisects ∠BCD

Prove: ∆ABC ≅ ∆ADC

1. BC ≅ CD, AC bisects ∠BCD

2. ∠BCA ≅ ∠DCA

3. AC ≅ AC

4. ∆ABC ≅ ∆ADCclick

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Problem

D

FG

EWrite a two-column proof.Given: DE ‖ FG

DE ≅ FG

Prove: ∆DEG ≅ ∆FGE

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Problem D

FG

E

Statements Reasons

1. DE ‖ FG

2. ∠DEG ≅ ∠FGE

DE ≅ FG

3. GE ≅ EG4. ΔDEG ≅ ΔFGE

1. Given

2. Alternate Interior Angles are ≅3. Reflexive Property of Congruence4. SAS Triangle Congruence

Click

Click

Click

Click

Given: DE ‖ FGDE ≅ FG

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A

C

DT

B

Problem: complete the proof

___________

Given: ∠A and ∠D are right angles; AT ≅ DTProve: ΔATB ≅ ΔDTC

1. ∠A and ∠D are right angles

2. ∠A ≅ ∠D

3. AT ≅ DT

4. ∠ ATB ≅ ∠ DTC

5. ΔATB ≅ ΔDTC Click Click

Click

1. Given

2. right ∠'s are congruent

3. Given

4. Vertical ∠'s are congruent

5. ASA Triangle Congruence

Click

Click

Click

Statements Reasons

Click

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Problem: complete the proofD C

A B

Given: DA ⊥ AB

Prove: ΔDAB ≅ ΔBCD

BC ⊥ CD∠ADB ≅ ∠CBD

Statements Reasons

1. DA ⊥ AB, BC ⊥ CD

2. ∠A and ∠C are right ∠'s

3. ∠A ≅ ∠C

4. ∠ADB ≅ ∠CBD

5. DB ≅ BD

6. ΔDAB ≅ ΔBCD

1. Given

2. Definition of ⊥ lines3. All right angles are congruent4. Given

5. reflexive property of ≅6. AAS Triangle Congruence

Click Click

ClickClick

Click

Click

Click

Click

Click

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Statements Reasons

1)

2)

3)

4)

5)

1)

2)

3)

4)

5)

Given: AC ≅ BD, E is the midpoint of AB and CD

Prove: ΔAEC ≅ ΔBED

A

B

DC

E

Problem

E is the midpoint of AB and CD

SSS Triangle Congruence

AC = BD~

Def. of midpoint

AE = BE~Given

∆AEC ≅ ∆BEDCE = DE~

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CPCTCCorresponding Parts of Congruent Triangles are Congruent

page2svg

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CPCTC states that if two or more triangles are congruent by one of the congruence postulates/theorems - SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent.

Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true.

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Process for proving that two segments or angles are congruent

1. Find two triangles in which the two sides or two angles are corresponding parts

2. Prove that the two triangles are congruent (SSS, SAS, ASA, AAS, HL)

3. State that the two parts are congruent, using as the reason: "Corresponding Parts of Congruent Triangles are Congruent"

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MN

O

E L

57 Which two triangles might you try to prove congruent in order to prove NM ≅ NO ?

ABCD

ΔLOEΔNOLΔLMEΔNME

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58 Which two triangles might you try to prove congruent in order to prove EO ≅ LM ?

A

BCD

ΔEOLΔNOLΔLMEΔNME

MN

O

E L

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59 Which two triangles might you try to prove congruent in order to prove ∠1 ≅ ∠2?

ABCD

ΔLOE

ΔNOLΔLME

ΔNME

MN

O

E L

1 2

Ans

wer

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60 Which two triangles might you try to prove congruent in order to prove EN ≅ LN ?

ABCD

ΔLOEΔNOLΔLMEΔNME

MN

O

E L

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Statements Reasons

Problem: complete the proof

Given: AB ≅ DE, BC ≅ EC, C is the midpoint of AD

Prove: ∠A ≅ ∠DA

B

CD

E

1. AB ≅ DE

2. BC ≅ EC

3. C is the midpoint of AD

4. CA ≅ CD

5. ΔABC ≅ ΔDEC

6. ∠A ≅ ∠D

1. Given

2. Given

3. Given

4. Definition of midpoint

5. SSS Triangle Congruence

6. CPCTCClick

Click

Click

Click

Click

Click

Click

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ProblemAB

C

D E

We are given that ∠BCA ≅ ∠DCE, BC ≅ CD, and ∠B and ∠D are right angles. Since all right angles are congruent, ∠B ≅ ∠D. With the congruent angles and segments, we can conclude that ΔABC ≅ ΔEDC by ASA. Therefore, BA ≅ DE by CPCTC.

Given: ∠BCA ≅ ∠DCE ∠B and ∠D are right angles BC ≅ CDProve: BA ≅ DE

Click Click

Click

Click

Click

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Statements Reasons

Problem: complete the proof W X

P

Z Y

Given: P is the midpoint of WY, P is the midpoint of XZProve: WX ‖ ZY

1. P is the midpoint of WY2. P is the midpoint of XZ3. WP ≅ YP, ZP ≅ XP4. ∠WPX ≅ ∠YPZ 5. ΔWPX ≅ ΔYPZ6. ∠Z ≅ ∠X7. WX ‖ ZY

1. Given2. Given3. Definition of midpoint4. Vertical angles are congruent5. SAS Triangle Congruence6. CPCTC7. If alt. int. angles are congruent, then lines are parallel

Click

Click

Click

Click

Click

Click

Click

Click

Click

Click

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Additional Proof Practice

Website link: Interactive Proofs

http://feromax.com/cgi-bin/ProveIt.pl

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Learning mathematics is like climbing a ladder, one step leads to the next. No step is more difficult than the one before it, as long as you take them one step at a time.

Congruent Triangles are an important step in geometry. They will be used through much of the rest of this course.

For example, the following PARCC-type question looks like it's about parallelograms, but you can answer every part of this question with what you know already, before you even study quadrilaterals.

Try it out.

Using What You've Learned

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A

B C

DGiven: ∠BAC ≅ ∠DCA, BA ≅ DC

Prove: ABCD is a parallelogram

Statements Reasons

1. ∠BAC ≅ ∠DCA, BA ≅ DC

2. AC ≅ CA

3. ΔBAC ≅ ΔDCA4. ∠BCA ≅ ∠DAC

5. BC || AD, AB || DC

6. ABCD is a parallelogram

1. Given2. Reflexive Property of Congruence3. SAS Triangle Congruence4. CPCTC5. If alt. int. angles are congruent, then lines are parallel6. Definition of a parallelogram Click

Click Click

Click

Click

Click

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Isosceles Triangle

Theorems


page2svg

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Isosceles Triangles

leg leg

base

In an isosceles triangle, the base is the side that is not necessarily congruent to the other two sides (legs).

If an isosceles triangle has 3 congruent sides, it is also an equilateral triangle.

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base angles

vertex angle

Isosceles Triangles

The vertex angle is opposite the base, and is the included angle between the legs.

The base angles are the angles opposite the legs, and are included by a leg and the base.

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Base Angles Theorem

The base angles of an isosceles triangle are congruent.

This says that the angles opposite equal sides of a triangle are of equal measure.

In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

Euclid: Book One Proposition 5

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Proof of Base Angles Theorem

Given: In ΔABC, AB ≅ BCProve: ∠A ≅ ∠C

A

B

C

There are several ways to prove this. Euclid's way is pretty complicated.

The link below shows two typical proofs and an alternate third one.

The third proof uses the fact that order DOES matter in making statements of congruence.

It was supposedly generated by a computer.

http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm

http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm

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Statement Reason

1 In ΔABC, AB ≅ BC Given

2 BC ≅ AB Symmetric Property of ≅3 ∠ABC ≅ ∠CBA Reflexive Property of ≅4 ΔABC ≅ ΔCBA SAS Triangle Congruence

5 ∠A ≅ ∠C CPCTC

Proof of Base Angles Theorem

Given: In ΔABC, AB ≅ BC

Prove: ∠A ≅ ∠CA

B

C

Below are the arguments that could be used to explain the third proof from the link on the previous slide (computer generated).

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61 What is the value of x in this triangle? Justify your answer.

x°

y°

44°

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62 What is the value of y in this triangle? Justify your answer.

x°

y°

44°

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x°

y°

72°

63 Solve for x and y. Explain your reasoning.

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64 What is the measure of each base angle?

70°

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65 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?

A 71° B 38° C 83° D 104°

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Converse of Base Angles TheoremIf two angles of a triangle are congruent, then the

sides opposite them are congruent.

The sides opposite equal angles of a triangle are of equal length.

If in a triangle two angles be equal to one another, the sides opposite those angles will also be equal to one another

Euclid: Book One Proposition 6

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D

EF

4

3

66 What is the length of FD?

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D

EF

9

7

67 What is the length of ED?

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Equilateral TrianglesIf three sides of a triangle are equal, each of the

three angles has a measure of 60º .

An equilateral triangle is a special case of an isosceles triangle.

The base and legs are all of equal length.

Since all the angles are opposite sides of equal length, they all have equal measure.

Since the three angles add to 180º and have equal measure, they each have a measure of 60º.

Conversely, if two angles of a triangle are each 60º, the third angle also has a measure of 60º and all the sides are of

equal length.

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Equilateral TrianglesIf three sides of a triangle are equal, each of the

three angles has a measure of 60º .

Conversely, if the angles of a triangle each have a measure of 60º, all the sides are of equal length.

Also, if two angles of a triangle each have a measure of 60º, the third angle must also has a measure of 60º since the Interior Angles Theorem indicates that the angles must add to 180º

Then, all the sides must be of equal length.

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68 Classify the triangle by sides and angles.

A equilateralB isoscelesC scaleneD equiangular

E acuteF obtuseG right

7

40º

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E acuteF obtuseG right

4

4

4

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5

113º

3 3



E acuteF obtuse

G right

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E acuteF obtuse

G right

12

12

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E acuteF obtuse

G right

60º

60º

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Example

Find the value of x and y. Explain your reasoning.

y°

x°

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73 What is the value of y?

A 120°B 70°C 55°D 125°

70°

y°

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74 What is the value of x? Justify your answer.

A 50°B 25°C 30°D 130°

50° x°

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3x - 17

28

75 Solve for x in the diagram.

A 3 2/3B 14C 15D 16

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PARCC Sample Test Questions

The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions.

Good Luck!

Return to Table of Contents

page2svg

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76 The first step of the construction is to draw an arc centered at point A that intersects both sides of the given angle. What is established by the first step?A AB ≅ BC B AB ≅ AC

C AD ≅ AC D BD ≅ CD

Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)

A

CB

D

Part A

Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

PARCC Released Question (EOY)

http://parcc.pearson.com/practice-tests/math/

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77 The construction creates congruent triangles. ABD and ACD are congruent because of the

_______________ postulate/theorem.

A side-side-sideB angle-side-angle C side-angle-side D angle-angle-side


A

CB

D

Part BComplete the sentence with the choices given below.


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78 It follows that AD must be the angle bisector of ∠BAC because _________________.

A ∠ACD ≅ ∠ABDB ∠BAC ≅ ∠BDCC ∠BAD ≅ ∠CADD ∠BAD ≅ ∠ABD


A

CB

D

Part BComplete the sentence with the choices given below.


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Marcella drew each step of a construction of an angle bisector.

Z

B

AZ Z

B

A Z

B

A

C

Z

B

AC

Step 1 Step 2 Step 3 Step 4 Step 5

Part A

Angle Z is given in Step 1. Describe the instructions for Steps 2 through 5 of the construction.

This is a great problem and draws on a lot of what we've learned.Try it in your groups.Then we'll work on it step by step together by asking questions that break the problem into pieces.

PARCC Released Question (PBA)

http://parcc.pearson.com/practice-tests/math/

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79 Angle Z is given in Step 1. What would be the description used to get from Step 1 to Step 2?

A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.

B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.

C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius

length. Label the points where the arc intersects the angle A and B.



Z

B

AZ Z

B

A Z

B

A

C

Z

B

AC


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80 What would be the description used to get from Step 2 to Step 3?A Construct an arc located in the interior of angle Z using a compass

centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.






Z

B

AZ Z

B

A Z

B

A

C

Z

B

AC


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81 What would be the description used to get from Step 3 to Step 4?A Construct an arc located in the interior of angle Z using a compass

centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.






Z

B

AZ Z

B

A Z

B

A

C

Z

B

AC


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82 What would be the description used to get from Step 4 to Step 5?A Construct an arc located in the interior of angle Z using a

compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.






Z

B

AZ Z

B

A Z

B

A

C

Z

B

AC


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Part BMarcella wants to explain why the construction produces and angle bisector. She makes a new step with line segments AB and BC added to the construction, as shown.

Using the figure, prove that ray ZC bisects angle AZB. Be sure to justify each statement of your proof.

This is a great problem and draws on a lot of what we've learned.Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces.


B

Z

C

A

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83 What have we learned that will help solve this problem?A Construction of an angle bisector w/ a compass

and straightedgeB Ways to prove triangles congruentC The corresponding parts of congruent triangles

are congruent (CPCTC)D All of the above

B

Z

C

A

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84 What should be the first statement in our proof?A ZA ≅ ZBB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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85 Why can we say that these two segments in step #1 are congruent?

A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass

setting, and all radii of a given circle are congruent.

B

Z

C

A

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86 What should be the second statement in our proof?A BC ≅ ACB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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87 Why can we say that these two segments in step #2 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass


B

Z

C

A

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88 What should be the third statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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89 Why can we say that these two segments in step #3 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass


B

Z

C

A

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90 What should be the fourth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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91 Why can we say that these two triangles in step #4 are congruent?A SSS Triangle CongruenceB SAS Triangle CongruenceC AAS Triangle CongruenceD ASA Triangle CongruenceE HL Triangle Congruence

B

Z

C

A

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92 Since we know that the triangles are congruent, what should be the fifth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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93 Why can we say that these two angles in step #5 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass


B

Z

C

A

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94 Since we know that the angles are congruent, what should be the sixth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB

B

Z

C

A

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95 What is the final reason in our proof?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass


B

Z

C

A

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Statements Reasons

1) ZA ≅ ZB 1) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent.

2) AC ≅ BC 2) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent.

3) ZC ≅ ZC 3) Reflexive Property of ≅4) ∆AZC ≅ ∆BZC 4) SSS Triangle ≅5) ∠AZC ≅ ∠BZC 5) CPCTC6) ZC is bisects ∠AZB 6) Definition of an Angle Bisector

Given: The construction of the figure to the right

Prove: ZC bisects ∠AZB

Below is a completed version of the proof that we just wrote.

B

Z

C

A

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