Triangle Congruence and Similarity - Mr. Montrella · Name _ Date _ Example 2 Example 3 Determine...

Name ~ __Date _~------

California Standards

Geometry 5.0

Students prove that triangles are congruent or similar, and they are able

to use the concept of corresponding parts of congruent triangles.

Triangle Congruence and Similarity

In two congruent figures, all the parts of one figure are congruent to thecorresponding parts ofthe other figure.

Corresponding angles:LA :=LF,LB:= LE,LC:= LD

Corresponding sides:AB :=FE, BC:=ED,AC:=FD

B E

~~A C F 0

6ABC:= 6FEDWhen you write a congruence statement for two polygons,always list the corresponding vertices in the same order.

Side-Side-Side (SSS) Congruence

Postulate

If three sides of one triangle are congruent tothree sides of a second triangle, then the twotriangles are congruent. 6ABC:= 6PQR

Side-Angle-Side (SAS) Congruence

Postulate

If two sides and the included angle of one.triangle are congruent to two sides and theincluded angle of a second triangle, then thetwo triangles are congruent. 6DEF:= 6STUAngle-Side-Angle (ASA) Congruence

Postulate

If two angles and the included side of onetriangle are congruent to two angles and theincluded side of a second triangle, then the twotriangles are congruent. 6DEF:= 6MNO

B Q

66A CPR

L1~o F S U

E. N

D~ M~F o

Hypotenuse-Leg (HL) Congruence

Theorem

If the hypotenuse and a leg of a right triangleare congruent to the hypotenuse and a leg of asecond right triangle, then the two triangles arecongruent. 6JKL:= 6XYZ I K

J X

~~L y z

Angle-Angle-Side (AAS) Congruence

Theorem

H W

If two angles and a non-included side of one I /\ /\triangle are congruent to two angles and a non- ~ ~included side of a second triangle, then t4e two I Gtriangles are congruent. 6 GHI:= 6 VWX

v X

California Standards Review and Practice

14 Geometry Standards .

r

>-c:coCl.EaUc:

>-croCl.Eouc

~~c

.8..c:OJ

=>.0

:::r::'+-o

c

.<:::l'":~"0

ro

--

r

Name _

Example 1

Date _

!'T t

Triangle Similarity., Example !l!ii" -- -- - ,--

Angie-Angie (AA) Similarity K

Postulate

PL QIf two angles of one triangle are congruent to yRtwo angles of another triangle, then the two

triangles are similar. l1JKL ~ £"PQR

J

P

Side-Side-Side (SSS) Similarity B K

Theorem

~~

If the corresponding side lengths of two

triangles are proportional, then the trianglesA C

are similar.J L

AB = BC = AC. l1ABC ~ l1JKLJK KL 4'

Side-Angle-Side (SAS) Similarity M

Theorem

s~~If an angie of one triangle is congruent to

an angle of a second triangle and the lengths

of the sides including these angles are

proportional, then the triangles are similar.U T

SU = ST . l1STU ~ l1MNOMO MN'

Prove Triangles Are Congruent

Write a proof.

GIVEN ~ M is the midpoint of AC.

M is the midpoint of BD.

A B

PROVE ~ l1AMB =: l1CMD

o c

Solution

Statements

3. Vertical Angles Congruence Theorem

4. SAS Congruence Postulate

Reasons

1.M is the midpoint ~f AC.

M is the midpoint of BD.-- --

2.AM=:MC

BM=:MD

3. LAMB =: L CMD

4. l1AMB =: l1CMD

1. Given

2. Definition of Segment Midpoint

California Standards Review and Practice

Geometry Standards 15

Name _ Date _

Example 2

Example 3

Determine Information to Show CongruenceState the third congruence that must be given to prove that LABC == LPQR using theindicated postulate or theorem. B Q

a. LB == LQ, LC == LRUse the ASA Congruence Postulate.

b.BC == QR, LB == LQUse the SAS Congruence Postulate.

Solution c R

a. Two angles in the first triangle are congruent to two angles in the second triangle. Touse the ASA Congruence Postulate, we need to know that the included side in thefirst triangle is congruent to the included side in the second triangle, or BC == QR.

c. One side and one angle in the first triangle are congruent to one side and one anglein the second triangle. To use the SAS Congruence Postulate, we need to know thatanother side of the first triangle is congruent to the corresponding side of the secondtriangle, such that the congruent angles are the included angles. So, AB == PQ.

Determine Whether Triangles Are CongruentDecide whether the congruence statement is true. Explain your reasoning.

a. LWYZ== LYWX wA .:;> X

. b. L VXY == LzxwzV

y

X

zc. LJKL == LMNO K N

J M

L .0

Solutiona.Yes, by the HL Congruence Theorem. L WYZ is a right angle by the Corresponding. Angles Postulate. WY == WYby the Reflexive Property of Congrµent Segments, andZW == XY is given.

b. Yes, by the AAS Congruence Theorem. LX== LXby the Reflexive Property ofCongruent Angles and L V == L Z is given. ZW = ZT + TWand VY = VT + TY bythe Segment Addition Postulate. ZW = VYby the Transitive Property of Equality,and ZW == VY by the Definition of Congruent Segments.

c. No; SSA is not one of the triangle congruence postulates or theorems.

California Standards Review and PracticeGeometry Standards'

I 16

r

c.8.s:::Ol

:::l

o::r:4-oCo.",

:~-0co--

>-ccoD.-

Eou.c=~~c

.8..cOJ

:::J

o:r:

aco'0:;:.~"0

co

Name ~ ~ __

Example 4

Example 5

Example.6

Date _

Use Corresponding Parts

Write a congruence statement for the triangles. Identify an pairs of congruent

corresponding parts.

p

Solution

The diagram indicates that 6.ABC == 6.RPQ.

Corresponding angles LA == LR, LB == LP, L C == L Q

Corresponding sides AB == RP, BC == PQ, CA == QR

ShowTriangles Are Similar

Show that the triangles are similar and write a similarity statement. Explain your

reasomng.

P 3

~RT 4 S 16

SQlution

Since we know the lengths of the sides, calculate the ratios of corresponding sides.

QS 8 4 QR 12 12 4 SR 16 16 4

PT - 10 - '5 PR 3 + 12 - 15 - '5 TR - 4 + 16 - 20 - '5

;: = ;; = ~~. = ~, thus the triangles are similar by the SSS Similarity Theorem.

Answer 6. TPR ~ 6.SQR by the SSS Similarity Theorem.

Prove Triangles Are Similar

GIVEN ~ KP==LP,JL = 21,KM= 14,

LQ = 24,NK= 16J K L M

< <; J

PROVE ~ 6.JQL ~ 6.MNK

Solution

L QLJ == L NKM by the Base Angles Theorem.

JL _ ~ -1. QKM 14 2 .

LQ 24_1.

NK 16 2

The measures of the corresponding sides that include angles QLJ and NKM are

proportional, so the triangles are similar by the SAS Similarity Theorem.

CaliforniaStandardsReview andPractice

Geometry Standards 17

Name __

Exercises

Date _

1. /:"IKL and /:"PQR are two triangles such that

L K := L Q. Which of the following is sufficient

to prove the triangles are similar?

@ JK=PQ

© LJ is right.

2. In the figure below, wz II XY.

V

I ". Z

. X y

Which theorem or postulate can be used to prove

/:,.VWZ~ /:"VXY?

@ ASA ® AAS © SAS ® SSS

3. In the figure below, /:,.ABE:= !iDCE.

Ai"":"

B

oIL -c


/:"CDB:= /:"BAC?

@ ASA ® SSS © SAS ® AAS

4. In the figure below, PQ II SR.

I • AQ

sv I I

Which additional information would be enough

to prove /:"PQS:= /:,.RSQ?

@ PQ:=PS

© PQ:=SR

® SR:=QR

® PS:=QR

California Standards Review and Practice18 Geometry Standards

5. In the figure below, HI bisects LKHI and LKII.

H

K

J


/:"HKI:= /:"HIJ?

@ ASA

© SAS

® AAS

® SSS

6. In the figure below, L P := LX .

X

P

R~

Z

Q y

Which of the following would be sufficient to

prove the triangles are similar?

®

®

RP

ZX

RP

ZX

PQ

XY

RQ

ZY

7. In the, figure below, ED ..1DF, HG..l GF, F is

the midpoint of DG.

E

H


/:"DEF:= /:,.GHF?

@ ASA ® SSS © SAS ® HL

>-cco0..

Eou

c

€

~c

.8..r::Ol

::>o::c

Cic

o'0;

'>'6

co

Triangle Congruence and Similarity - Mr. Montrella · Name _ Date _ Example 2 Example 3 Determine...

Documents

Transcript of Triangle Congruence and Similarity - Mr. Montrella · Name _ Date _ Example 2 Example 3 Determine...