Triangle Congruence

5/21/2018 Triangle Congruence

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GUIDE CARD


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Chief Captain, help me! I have

trouble in my airline pathways. As

I drive I noticed that there are

obstacles on my paths. Volcano on

my right side, then a hurricane onmy left and then another airplane

along my way. Would you like to tell

me other paths? I might lost and

crash. Please help me before its

too late!


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Indeed, we see the trouble you

are going through as seen in our

radar. Please double check and

listen to our instructions. You

are heading on a too dangerousmission!


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This is the pictureof what youre

heading. We will

give you ideas on

how to drive the

plane in different

direction.

SHORTEST DISTANCE

SAFE ALTERNATIVE PATH

Note: Due to the gravity,

volcanoes can pull the

airplanes toward to their

crater.

Note: When there is too

much clouds or

hurricanes, pilots doesnt

have clear paths. Huge air

can disrupt airplane

banking.

Just like highways and street paths,

airplanes also have own paths across

the sky. They cant just fly in

whatever direction they want!


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Other

Plane

Volcano

Hurricane

As you can see,

if you will

connect the

paths, you can

form triangles.


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46.92

69.81

How sure you are that we have the same path? You travel at 4.45 miles

69.81 East of North then you turn at 46.92 degree. Do we have the same

distance from the triangles we formed?

4.45 milesYes captain. I remember during my

Geometry class. My triangle in the

screen is just same because we havesame angle (69.81O) then 4.45 miles as

side, then other same angle ( 46.92O).

It is known as ASA! Angle-Side-Angle. If two angles and the included side of

one triangle are congruent to the corresponding two angles and the included

side of another triangle, then the two triangles are congruent!

SAME


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Nice job! Now, proceed to your next path from the last point! Im

sending you the image.

I drove 2.05 miles then turns to my

right at 45.59O degrees. Then at 2.84

miles. Im sure, I cannot meet the

hurricane. It is because, we have thesame triangles?

And I know you are familiar with SAS Postulate! It

means Side- Angle- Side. If two sides and the included

angle of one triangle are congruent to the

corresponding two sides and the included angle ofanother triangle, then the two triangles are congruent.

2.05 miles

2.84 miles45.59O

SAME


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Captain! This is my last trial. Before Ill bump

with other plane. I just want to make sure

that I turn left at 2.84 miles then will drive at

1.92 miles. However captain, I dont know

the angles. Will it still be the same if I

connect the starting point to the endpointand it forms 2.08 miles?

What do you think? Suppose I will give you this

figure. For example, you just drove 1 mile. Will

it have the same angle?

1 mile No its not! Ah, I understand! Same sides of a two

triangles means same angle to the other triangles. That

is SSS or Side-Side-Side! They are just congruent!

SAME

1.92

2.84

2.08

?


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And now Ill be safe to take off my flight!

Imagine if I didnt recall my geometry lessons,

many people will suffer and die! I feel like Im

hero. Good thing that my passengers dont

know the dangers ahead. Thanks muchCaptain. Knowledge in math can save us from

unexpected dangers.


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Side-Angle-Side Postulate (SAS)

If two sides and the included angle of one triangle are

congruent to the corresponding two sides and the included angle of

another triangle, then the two triangles are congruent.

Angle-Side-Angle Postulate (ASA)

If two angles and the included side of one triangle are

congruent to the corresponding two angles and the included side of

another triangle, then the two triangles are congruent.

Side-Side-Side Postulate (SSS)

If three sides of one triangle are congruent to the

corresponding sides of the other triangle, then the two triangles are

congruent

REMEMBER THIS!!!


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a. Materials: Compass, ruler, pencilb. Procedure

Start with three line segments that

will be the three sides of thetriangle ABC.

1. Mark a point A that will be one

vertex of the new triangle.


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2. Set the compass width to the

length of the segment AB. This will

become the base of the newtriangle.

3. With the compass point on A,

make an arc near the future vertex B

of the triangle.

4. Mark a point B on this arc. This

will become the next vertex of the

new triangle.


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7. Use the compass to measure the

length of the segment BC, the

length of the third side of the

triangle.

6. Place the compass point on A

and make an arc in the vicinity of

where the third vertex of the

triangle (C) will be. All points along

this arc are the distance AC from A,but we do not yet quite know

exactly where the vertex C is.


length of the line segment AC.


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8. From point B, draw an arccrossing the first. Where these

intersect is the vertex C of the

triangle

9. Finally, draw the three sides AB,

AC, and BC of the new triangle.


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10. Done. The blue triangle ABC

has each side congruentto the

corresponding line segment.
http://www.mathopenref.com/congruent.htmlhttp://www.mathopenref.com/congruent.html


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Given: QR UT, RS TS, QS = 10,

US = 10

Prove: QRS UTS

QU

R T

Statements Reasons________

1. QR UT, RS TS, 1. _________

QS=10, US=10

2. QS = US 2. Substitution

3. ____________ 3. Def of segs.

4. QRS UTS 4. _________

s


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II. SAS Postulate

Start with two line segments and

the included angle.

1. Mark a point A that will be one

vertex of the new triangle.

2. Draw a ray from point A. This will

become the side AB of the new

triangle, so make it longer than AB.

a. Materials: Compass, ruler, pencil

b. Procedure


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length of the given side AB.

4. Set the compass on A, and mark

a point B on the ray just drawn.

5. With the compass set to any

convenient width, from the point A

on the given angle, draw an arcacross both lines..


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6. Without changing the compass

width, draw a similar sized arc atpoint A on the new triangle.

7. Set the compass to the arc width

at the given angle A. This the

distance between the points where

the arc intersects the sides of the

angle.

8. Make a similar arc on the newtriangle so it crosses the previous

arc.

6. Without changing the compass

width, draw a similar sized arc at

point A on the new triangle.


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9. Draw a ray from A, through

where the arcs intersect and

onwards. This will become side AC

of the triangle so make it longer

than AC.

9. Draw a ray from A, through

where the arcs intersect andonwards. This will become side AC

of the triangle so make it longer

than AC.


distance AC.


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11. With the compass on A, make

an arc across the second ray,

creating point C.

12. Draw the line BC, the third side

of the triangle


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Statements

1. DR AG; AR GR

2. ___________

3.DRG &DRA are rt.s

4.____________

5. DRG DRA

Reasons

1. ____________

2. Reflexive Property

3. lines form 4 rt.s

4. Right s Theorem

5. __________

D

A GR

Given: DR AG and AR GR

Prove: DRA DRG.


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Start with the given line segment

and two angles.

Note: The two given angles are only there to indicate

The measureof the two angles. The lines making up

the given angles have random lengths that have no

significance in the construction.

1. Mark a point A that will be one vertex of the

new triangle.

2. Set the compass width to the length of the

segment AB.

a. Materials: Compass, ruler, pencil

b. Procedure


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3. With the compass point on A, make an arc near the

future vertex B of the triangle.

4. Mark a point B on this arc.

Then draw the line AB. This will be one side of

the new triangle.

5. With the compass at any convenient

width, draw an arc across both lines of the

given angle A.


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6. Without changing the compass width,

draw an arc at point A on the new triangle.

The arc must cross AB and also cross the

future side of the triangle.

7. Set the compass to the arc width at the given

angle A.

This the distance between the points where the arc

intersects the sides of the angle.

8. Near point A draw an arc in a similar

position so it crosses the arc drawn earlier.

This, in effect, 'copies' the measure of theangle at P to the angle at A.


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9. Draw a line from A through the point where

the arcs intersect. This will become the secondside of the triangle. Draw it long.

10. Repeat this process at B. Copying the angle

measure from the given angle B to the new

triangle at B. The point where the lines

intersect is C, the third vertex of the triangle.


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Given: ADEC, BD BCProve: ABDEBC

Plan for proof: Notice that ABD and

EBC are congruent. You are given that

BD BC. Use the fact that AD EC toidentify a pair of congruent angles.

Statements: Reasons:1. Given

2. ___________

3. If || lines, then alt. int.

sare

4. ___________

5. ASA Congruence Postulate

A

B

C

D E

1. BD BC

2. AD EC3. ___________

4. ABD EBC

5. ___________


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What do you call a fruit that

doesnt come from a tree or

plant?

Activity #1

A RIDDLE:

Direction: Verify a set of

conditions thatguarantees triangle

congruence by

definition of triangle

congruence. Write the

letter of the correct

choice in each box that

contains the exercise

number.

E


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F

E

HG

43

75

I

2

86

1

3 4 5 6 7 2 8 4 5 1


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HOW MANY

CHROMOSOMES DOES

A NORMAL HUMAN

BEING HAVE?

Direction:

To answer this questions,

follow the instructions. Two

congruent triangles are indicated in

each exercise. Assuming you know

only that the marked parts are

congruent, blacken the circle of the

theorem that proves that the two

triangles are congruent. Write the

letter of the correct choice in the box

that contains the exercise number.

Activity #2

TRIVIA:


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1. 5.

2. 6.

3. 7.

4. 8.

SSS

SAS

ASA

SSS

SAS

ASA

ASA

SSS

SAS

SSS

SAS

ASA

SAS

ASA

SSS

ASA

SSS

SAS

SSS

ASA

SAS

ASA

SSS

SAS

4 7 5 1 3 8 2 6


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SELF CHECK:

Put a check mark on the blankcorresponding to your answer.

After reading the lesson:

__ I now understand the lesson

__ I still do not understand

The activities have been:

__ easy

__ difficult

My score is:

__ 18 (very good)__ 12-17 (good)

__ below 12 (needs improvement)


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Enrichment Card


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Tell whether the following pairs of triangles are

congruent. If yes, state the congruence postulate.


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Internet http://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-

3%20SSS%20and%20SAS.pdf

http://www.worldofteaching.com/powerpoints/maths/geometrycongruence.ppt http://www.mathopenref.com/consttriangleasa.html http://www.google.com.ph/search?hl=tl&biw=1024&bih=435&tbm=isch&sa=1&q=asa+congruence+postu

late&oq=asa+congruence+postulate&aq=f&aqi=&aql=&gs_sm=e&gs_upl=436308l436680l0l437473l3l2l0l0l0l0l0l0ll0

http://www.google.com.ph/search?hl=tl&biw=1024&bih=406&tbm=isch&sa=1&q=sas+congruence+postulate&oq=sas+congruence+postulate&aq=f&aqi=&aql=&gs_sm=e&gs_upl=397585l398087l0l398851l2l2l0l0l0l0l0l0ll0

http://www.google.com.ph/search?hl=tl&biw=1024&bih=406&tbm=isch&sa=1&q=sss+congruence+postulate&oq=sss+congruence+postulate&aq=f&aqi=&aql=&gs_sm=e&gs_upl=415281l415849l0l416618l2l2l0l0l0l0l706l706l6-1l1

Books Geometry by Isidore Dresser pp 110 Geometry by Dilao, Soledad and Julieta Bernabe (Revised Edition) pp. 89 -100 Geometry by Ferdinand Pa
http://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-3%20SSS%20and%20SAS.pdfhttp://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-3%20SSS%20and%20SAS.pdfhttp://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-3%20SSS%20and%20SAS.pdfhttp://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-3%20SSS%20and%20SAS.pdfhttp://www.nexuslearning.net/books/ML-Geometry/Chapter4/ML%20Geometry%204-3%20SSS%20and%20SAS.pdf


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Given: QR UT, RS TS, QS = 10,

US = 10

Prove: QRS UTS

Q U

R T

S

Statements Reasons________

1. QR UT, RS TS, 1. Given

QS=10, US=10

2. QS = US 2. Substitution

3. QS US 3. Def of segs.

4. QRS UTS 4. SSS Postulate


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Statements1. DR AG; AR GR

2. DR DR

3.DRG & DRA are rt. s

4.DRG DRA

5. DRG DRA

Reasons

1. Given

2. Reflexive Property

3. lines form 4 rt.

s

4. Right s Theorem

5. SAS Postulate

D

A GR

Given: DR AG and AR GR

Prove: DRA DRG.


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Given: ADEC, BD BCProve: ABDEBC

Plan for proof: Notice that ABD and

EBC are congruent. You are given that

BD BC. Use the fact that AD EC toidentify a pair of congruent angles.

Statements:Reasons:1. Given

2. Given

3. If || lines, then alt. int. s

are

4. Vertical Angles Theorem

5. ASA Congruence Postulate

A

B

C

D E

1. BD BC

2. AD EC3. D C

4. ABD EBC

5. ABD EBC


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3 4 5 6 7 2 8 4 5 1

J U I C Y F R U I T

4 7 5 1 3 8 2 6

F O R T Y S I X


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SSS Postulate

SAS Postulate

SAS Postulate

Not Possible

ASA PostulateNot Possible

Not Possible


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Documents

Triangle Congruence