USE OF TRIANGLES

transcript

& CONGRUENT TRIANGLES

NCSCOS: 2.02; 2.03

U.E.Q: How do we prove the congruence of

triangles, and how do we use the congruence of triangles solving real-life

problems?

The triangle is the first geometric shape you will study. The use of this shape has a long history. The

triangle played a practical role in the lives of ancient Egyptians and Chinese as an aid to surveying land.

The shape of a triangle also played an important role in triangles to represent art forms. Native

Americans often used inverted triangles to represent the torso of human beings in paintings or

carvings. Many Native Americans rock carving called petroglyphs. Today, triangles are frequently

used in architecture.

Temple of Diana at EphesusTemple of Diana at Ephesus

Pyramids of GizaPyramids of GizaStatue of ZeusStatue of Zeus

On a cable stayed bridge the cables attached to each tower transfer the weight of the roadway to the tower.

You can see from the smaller diagram that the cables balance the weight of the roadway on both sides of each tower.

In the diagrams what type of angles are formed by each individual cable with the tower and roadway?

What do you notice about the triangles on opposite sides of the towers?

Why is that so important?

We can find triangles everywhere:

In nature In man-made structures

ReplayReplaySlideSlide

Classifying Triangles

Equilateral Equilateral

3 congruent sides3 congruent sides

Isosceles Isosceles

At least 2 congruent sidesAt least 2 congruent sides

Scalene Scalene

No congruent sidesNo congruent sides

Equilangular Equilangular

3 congruent angles3 congruent angles

Acute Acute

3 acute angles3 acute angles

Obtuse Obtuse

1 obtuse angle1 obtuse angle

Right Right

1 right angle1 right angle

Vertex: the point where two sides of a triangle meet

Adjacent Sides: two sides of a triangle sharing Adjacent Sides: two sides of a triangle sharing a common vertexa common vertexHypotenuse: side of the triangle across from Hypotenuse: side of the triangle across from the right anglethe right angleLegs: sides of the right triangle that form Legs: sides of the right triangle that form the right anglethe right angleBase: the non-congruent sides of an Base: the non-congruent sides of an isosceles triangleisosceles triangle

Label the following on the right

triangle:VerticesHypotenuseLegs

VertexVertex

VertexVertexVertexVertex

HypotenuseHypotenuse

LegLeg

Label the following on the isosceles triangle:

BaseCongruent adjacent

sidesLegs

m<1 = m<A + m<B

Adjacent Adjacent sideside

BaseBase

Adjacent Adjacent SideSide

LegLeg LegLeg

Interior Angles: angles inside the triangle

(angles A, B, and C)

Exterior Angles: Exterior Angles: angles adjacent to the angles adjacent to the interior anglesinterior angles (angles 1, 2, and 3)(angles 1, 2, and 3)

The sum of the measures of the interior angles of a triangle is 180o.

<A + <B + <C = 180<A + <B + <C = 180oo

The measure of an exterior angle of a triangle is equal to the sum of the measures of two nonadjacent interior angles.

m<1 = m <A + m <B m<1 = m <A + m <B

The acute angles of a right triangle are complementary. BB

m<A + m<B = 90m<A + m<B = 90oo

NCSCOS: 2.02; 2.03NCSCOS: 2.02; 2.03

2 figures are 2 figures are congruent if they congruent if they have the exact same have the exact same size and shape.size and shape.

When 2 figures are When 2 figures are congruent the congruent the corresponding parts corresponding parts are congruent. are congruent. (angles and sides)(angles and sides)

Quad ABDC is Quad ABDC is congruent to Quad congruent to Quad EFHGEFHG

______

)))))))))))))) ((

(())))))))

))))))((((

If If ΔΔ ABC is ABC is to to ΔΔ XYZ, which XYZ, which angle is angle is to to C?C?

If 2 If 2 s of one s of one ΔΔ are are to 2 to 2 s of another s of another ΔΔ, , then the 3rd then the 3rd s are s are also also ..

2222oo

)))) 8

))))(4x+15)(4x+15)oo

22+87+4x+15=18022+87+4x+15=1804x+15=714x+15=71

4x=564x=56x=14x=14

9191oo

8686oo

9cm9cm

(5y-12)(5y-12)oo

4x-3cm4x-3cm

113113oo

4x-3=9 5y-12=1134x-3=9 5y-12=113

4x=12 5y=1254x=12 5y=125

x=3 y=25x=3 y=25

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.

9292oo

9292o o

Statements ReasonsStatements Reasons1. 1. given1. 1. given2. m2. mP=mP=mN 2. subst. N 2. subst. prop =prop =

3. 3. P P N N 3. def of 3. def of s s

4. 4. RQP RQP MQN 4. vert MQN 4. vert s s thmthm

5. 5. R R M 5. 3M 5. 3rdrd s thms thm6. 6. ΔΔRQP RQP ΔΔ MQN 6. def of MQN 6. def of ΔΔss

In Lesson 4.2, you learned that if all six In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and pairs of corresponding parts (sides and angles) are congruent, then the angles) are congruent, then the triangles are congruent.triangles are congruent.

Corresponding PartsCorresponding Parts

ABC ABC DEFDEF

1.1. AB AB DEDE2.2. BC BC EFEF3.3. AC AC DFDF4.4. A A D D5.5. B B E E6.6. C C F F

If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)

Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.

AC = BC =

AB = MO = NO =

2 25 7 74

ABC MNOV V

K is the angle between JK and KL. It is called the included angle of sides JK and KL.

What is the included angle for sides KL and JL?

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS)

JKL PQRV V by SAS

Given: N is the midpoint of LW N is the midpoint of SK

Prove: LNS WNKV V

N is the midpoint of LWN is the midpoint of SK

,LN NW SN NK Definition of Midpoint

LNS WNK Vertical Angles are congruent

LNS WNKV V SAS Postulate

JK is the side between J and K. It is called the included side of angles J and K.

What is the included side for angles K and L?

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. (ASA)

JKL ZXYV V by ASA

Given: HA || KS

Prove: HAW SKWV V

HA || KS, Given

HAW SKW Alt. Int. Angles are congruent

HWA SWK Vertical Angles are congruent

HAW SKWV V ASA Postulate

METEORITESMETEORITESWhen a meteoroid (a piece of rocky or When a meteoroid (a piece of rocky or metallic matter from space) enters Earth’s metallic matter from space) enters Earth’s atmosphere, it heatsup, leaving a trail of atmosphere, it heatsup, leaving a trail of burning gases called a meteor. Meteoroid burning gases called a meteor. Meteoroid fragments that reach Earth without fragments that reach Earth without burningup are called meteorites.burningup are called meteorites.

On December 9, 1997, an extremely bright meteor lit up the sky

above Greenland. Scientists attempted to find meteorite fragments by collecting data from eyewitnesses who had seen the meteor pass through the sky. As shown, the scientists were able to describe sightlines from observers in different towns. One sightline was from observers in Paamiut (Town P) and another was from observers in Narsarsuaq (Town N). Assuming the sightlines were accurate, did the scientists have enough information to locate any meteorite fragments? Explain. ( this example is taken from your text book pg. 222

Note: is not Note: is not SSS, SAS, or ASA.SSS, SAS, or ASA.

M L P N

Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.

ABC STRV V by SSSby SSSPNO VUWV V by SASby SAS

Given:

Prove: MH HT

MATV is isosceles with vertex bisected by AH.MAT

• Sides MA and AT are congruent by the definition of an isosceles triangle.

• Angle MAH is congruent to angle TAH by the definition of an angle bisector.

• Side AH is congruent to side AH by the reflexive property.

• Triangle MAH is congruent to triangle TAH by SAS. • Side MH is congruent to side HT by CPCTC.

A line to one of two || lines is to the other line.A line to one of two || lines is to the other line.

|| ,QM PO QM MOQM PO

Given:

Prove: QN PN

QM PO QM MOQM PO MO

GivenGiven

has midpoint Nhas midpoint N

om QMNom PON

Perpendicular lines intersect at 4 right Perpendicular lines intersect at 4 right angles.angles.

QMN PON Substitution, Def of Congruent AnglesSubstitution, Def of Congruent Angles

Definition of MidpointDefinition of Midpoint

QMN PONV V SASSAS

QN PN CPCTCCPCTC

Triangles may be proved congruent by Side – Side – Side (SSS) PostulateSide – Angle – Side (SAS) Postulate, and Angle – Side – Angle (ASA) Postulate.

Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).

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

Do you need Do you need all six ?all six ?

NO !NO !

SSSSSSSASSASASAASAAASAAS

Solve a real-world problemSolve a real-world problem

Structural SupportStructural Support

Explain why the bench with the diagonal support is Explain why the bench with the diagonal support is stable, while the one without the support can collapse.stable, while the one without the support can collapse.

Solve a real-world problemSolve a real-world problem

The bench with a diagonal support forms triangles with The bench with a diagonal support forms triangles with fixed side lengths. By the fixed side lengths. By the SSSSSS Congruence Postulate, Congruence Postulate, these triangles cannot change shape, so the bench is these triangles cannot change shape, so the bench is stable. The bench without a diagonal support is not stable. The bench without a diagonal support is not stable because there are many possible quadrilaterals stable because there are many possible quadrilaterals with the given side lengths.with the given side lengths.

SOLUTIONSOLUTION

Warning:Warning: No SSA Postulate No SSA Postulate

NOT CONGRUENTNOT CONGRUENT

There is no such There is no such thing as an SSA thing as an SSA

postulate! postulate!

Warning:Warning: No AAA Postulate No AAA Postulate

There is no such There is no such thing as an AAA thing as an AAA

postulate! postulate!

NOT CONGRUENTNOT CONGRUENT

Tell whether you can use the Tell whether you can use the given information at determine given information at determine whether whether ABC ABC DEF DEF

A A D, ABD, ABDE, ACDE, ACDF DF

AB AB EF, BC EF, BC FD, AC FD, AC DE DE

The Congruence Postulates & The Congruence Postulates & TheoremTheorem

SSS SSS correspondencecorrespondence

ASA ASA correspondencecorrespondence

SAS SAS correspondencecorrespondence

AAS AAS correspondencecorrespondence

SSA SSA correspondencecorrespondence

AAA AAA correspondencecorrespondence

Name That PostulateName That Postulate

SASSAS ASAASA

SSSSSSSSASSA

(when possible)(when possible)

Name That PostulateName That Postulate(when possible)(when possible)

ASAASA

SASSAS

AAAAAA

SSASSA

Name That PostulateName That Postulate(when possible)(when possible)

SASSAS

SASSASReflexive Reflexive PropertyProperty

Vertical Vertical AnglesAngles

Reflexive Reflexive PropertyProperty SSASSA

HW: Name That PostulateHW: Name That Postulate(when possible)(when possible)

ClosureClosureIndicate the additional information needed to Indicate the additional information needed to enable us to apply the specified congruence enable us to apply the specified congruence postulate.postulate.

For ASA: For ASA:

For SAS:For SAS:

For AAS:For AAS:

Let’s PracticeLet’s PracticeIndicate the additional information needed to Indicate the additional information needed to enable us to apply the specified congruence enable us to apply the specified congruence postulate.postulate.

For ASA: For ASA:

For SAS:For SAS:

B B DD

For AAS:For AAS: A A FF AC AC FEFE

Now For The Fun Part…

Write a two column ProofGiven: BC bisects AD and A D

Prove: AB DC

A A C C

BB D D

The two angles in an isosceles triangle adjacent to the base of the triangle are called base angles.

The angle opposite the base is called the vertex angle.

Base AngleBase Angle Base AngleBase Angle

Vertex AngleVertex Angle

If two sides of a triangle are congruent, then the angles opposite them are congruent.

CBthenACABIf ,

If two angles of a triangle are congruent, then the sides opposite them are congruent.

CAABthenCBIf ,

If a triangle is equilateral, then it is equiangular.

If a triangle is equiangular, then it is equilateral.

YesYes YesYesNoNo

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

DEFABCthenDFACandEFBCIf ,

Find the measure of the missing angles and tell which theorems you used.

50°50°AA

m B=80° m B=80°

(Base Angle Theorem)(Base Angle Theorem)

m C=50°m C=50°

(Triangle Sum (Triangle Sum Theorem)Theorem)

m A=60°m A=60°

m B=60°m B=60°

m C=60°m C=60°

(Corollary to the Base (Corollary to the Base Angles Theorem)Angles Theorem)

Is there enough information to prove the triangles are congruent?

YesYesNoNo NoNo

USE OF TRIANGLES

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