ABC ~ DFE

In Fig. 4.1, if the matching ABC ~ DEF gives a congruence, we can state i that AC and DF are corresponding sides, and LBCA and LEFD are corre-

i

sponding angles. Can you find the other pairs of corresponding sides and corresponding angles?

Two scalene triangles can have only a single one-to-one correspondence which will give a congruence. Two isosceles triangles can have two one-to-one correspondences which will give congruence.

T M

R

s

K L

Fig. 4.2.

In Fig. 4.2, if RT == ST and KM == LM, the two correspondences RST ~ KLM and RST ~

LKM might give congruences. We will determine late

what additional conditions must be known before the triangles can be provec congruent to each other.i I

The order in which matching pairs of vertices are given is not importan~

in expressing a congruence and the vertex you start with is not importanti

In Fig. 4.3, we could describe the one-to-one correspondence in one line a

_'

DEFG

^~

HKjI or EFGD

^~ KjIH. There are two others. Can you fin4 them? All that matters is that corresponding points be matched.

I t should be evident that a triangle can be made to coincide with itself.

F J

~r:~

^{D E H}

Fig. 4.3.

K

-

109

one-tn-one correspondence in which every vertex is matched with itself is called the identity congruence.

Thus

T

ABC ~

ABC

s

is an identity congruence.

For the isosceles triangle RST (Fig. 4.4), where

RT

⁼⁼

ST, it can be shown that, under the one-to-one

correspondence RST ~ SRT, the figure can be made to coincide with itself.

R

Fig. 4.4.

Exercises

I. Draw a 6CH] and a 6KLM. List all the possible matchings of the second triangle with the ordered sequence CH] of the first triangle.

2. If the matching RST

^~

LMK gives a congruence between 6RST and

. 6LMK, list all the pairs of corresponding sides and corresponding angles

of the two triangles.

3. Write down the six matchings of equilateral 6ABC with itself, beginning with the identity congruence ABC ~ ABC.

4. Write down the four matchings of rectangle ABCD with itself.

5. In matching 6ABC with 6RST, AC and RT were matched as correspond- ing sides. Does it then follow that (I) LB and LS are corresponding angles? (2) BC and ST are corresponding sides?

6. Which of the following figures form matched pairs that are congruent to each other?

(a) (b) (c)

(d) (e) (l)

Prob.6.

--

110 FUNDAMENTALS OF COLLEGE GEOMETRY

7-12. In each of the following use ruler and protractor to find which triangles seem to be congruent. Then indicate the pairs of sides and angles in the triangles which seem to match in a congruence.

c

A _D B ^A

Ex. 7.

F E

A D

Ex. 9.

D

c

A B

EX.n.

D

B Ex. 8.

A

Ex.lO.

f]

A

D Ex.l2.

4.17. The triangle is a rigid figure. Much of our study of congruence of geometric figures deals with triangles. The triangle is the most widely used of all the geometric figures formed by straight lines. The triangle is rigid in structural design. If three boards are bolted together at A, B, and C, as

CONGRUENCE - CONGRUENT TRIANGLES III shown in Fig. 4.5, the shape of the triangle

is fixed. It cannot be changed without bending or breaking the pieces of wood.

However, if we bolt together four (or more) boards, forming a four-sided figure as shown in Fig. 4.6, the shape of the frame can be changed by exerting a force on one of the bolts. The measures of the angles formed by the boards can be changed in size even though the lengths of the sides of the figure remain the same. The frame of Fig. 4.6 can be made rigid by bolting a

board across D and F (or E and G), thus forming two rigid triangles.

The rigidity of triangles is illustrated in the practical applications of this property in the construction of many types of structures, such as bridges, fowers, and gates (Fig. 4.7).

4.10. Congruence of triangles. The engineer and the draftsman are con- tinually using congruence of triangles in their work. By applying their know- ledge of congruent triangles, they are able to study measures of the three sides and the three angles of a given triangle and to compute areas of triangles.

Often they apply this knowledge in constructing triangular structures which will be exact duplicates of an original structure.

Definition: If there exists some correspondence ABC ~ DEF of the ver- tices of L.ABC with those of L.DEF such that each pair of corresponding sides are congruent-amt--eaeh-pair--of-coll cspul1diltg angles al e COllgl uellt, tile correspondence ABC ~

DEF is called a congruence between the triangles.

The triangles are congruent triangles. Or we may state that L.ABC is con-

gruent to L.DEF, written L.ABC

₌₌

L.DEF.

Fig. 4.5.

Fig. 4.6.

- ---

Fig. 4.7.

Thus, if !'iABC ₌₌ !'iDEF (Fig. 4.8), we know six relationships between thej

mAB

= mDE mBC = mEF mAC = mDF mLA = mLD mLB = mLE mLC = mLF

AB

₌₌

DE BC

₌₌

EF AC

==

DF LA

==

LD LB

==

LE LC

==

LF

The equations in the left column and the congruences in the right column!

mean the same thing. They can be used interchangeably.

In Section 9.2 we will introduce a third way to indicate congruency of segments."

c

~

A B D

Fig. 4.8.

~

^E

CONGRUENCE - CONGRUENT TRIANGLES

113

4.19. Basic congruence postulate. Although we defined two triangles as

congruent if three pairs of sides and three pairs of angles are congruent, triangles can be proved congruent if fewer pairs of corresponding parts are known to be congruent. We must first accept a new postulate.

postulate 17 (the S.A.S. postulate). Two triangles are congruent

if

two sides and the included angle of one are, respectively, congruent to the two sides and the included angle of the other.

This postulate states that, in Fig. 4.9, if AB

₌₌

ED, AC

₌₌

EF, and LA

==

LE, then !'iABC

==

!'iDEF.

C

A~R

F

D E

Fig. 4.9,

The student often will find that he is aided in making a quick selection of the congruent sides and congruent angles in the two triangles by designating them with similar check marks for the congruent pairs of congruent sides and congruent angles. In thIS text we WIlt trequently llse_ul1ash marks to IndI-

cate "given" congruences. Thus, in Fig. 4.10, if it is given that AC

₌₌

DE, AB

₌₌ DB, AC 1- AD and DE 1- AD, the student can readily see which are the congruent pairs.

It will also be helpful if, in proving a congruence for two triangles, the stu- dent names the triangles in such a way as to indicate the matching vertices.

For example, in Fig. 4.10, since ABC ~ DEE can be proved a congruence, it would be more explicit to refer to these

triangles as "!'iABC and !'iDBE" rather ,

than, say, "!'iABC and !'iDEB." Al- Flf{.4.1O,'

though the sentence "!'iABC

⁼⁼

!'iDEB" can be proved correct, the sentence

"LABC == !'iDBE" will prove more helpful since it aids in picking out the corresponding parts of the two figures.

.It is important that the student recognize, in using Postulate 17 to prove tnangles congruent, that the congruent angles must be between (formed by)

c

A,'l

B

E

114

FUNDAMENTALS OF COLLEGE GEOMETRY

the corresponding congruent sides. If the congruent angles are not between the two known congruent sides, it does not

necessarily follow that the correspon-

^\ dence will give a congruence. In L.RST and L.KLM (Fig. 4.11) note that, though

RS

==

KM, ST

⁼⁼

ML, and LR

⁼⁼

LK, the

triangles certainly are not congruent.

4.20. Application of Postulate 17. In Postulate 17 we have stated that two tri- angles, each made up of three sides and three angles, are congruent if only three particular parts of one triangle can be shown congruent respectively to the three corresponding parts of the second triangle. Hereafter, when we are given any two triangles in which we know, or can prove, two sides and the included angle of one triangle congruent respectively to two sides and the included angle of the other, we can quote Postulate 17 as the reason for stating that the two triangles are congruent.

It is essential that the student memorize, or can state the equivalent in his own words, the statement of Postulate 17 because he will be required fre- quently in subsequent proofs to give it as a reason for statements in these proofs. After the student has shown competence in stating the postulate, the instructor may permit him to refer briefly to it by the abbreyiation S.A.S. ,j

(side-angle-side). Thisabbteviatlc)tiWillbe usecthereafterin [his text:

^.~

Once Postulate 17 is accepted as true, it becomes possible to prove various congruence theorems for triangles. We will next consider a theorem and two other examples of how this postulate can be used in proving other con- gruences.

T

R~S

L

K

Fig. 4.11.

Theorem 4.13

4.21. If the two legs of one right triangle are congruent respectively to the two legs of another right triangle, the triangles are congruent.

Given: L.ABC and L.DEF with AC

₌₌

DF, BC

==

EF; LC and LF are right ,1;.

~;:~m;on ^MBC~

6/)m~

~

Theurem 4.13. A C D 11

--

STATEMENTS

CONGRUENCE - CONGRUENT TRIANGLES

115

REASONS

I. AC ==

DF;BC

₌₌

EF.

2. LC and LF are right ,1;.

3. LC ==

LF.

4. LABC

⁼⁼

L.DEF.

1. Given 2. Given.

3. Right angles are congruent.

4. S.A.S.

C 4.22. Illustrative Example 1: The bisector of the

vertex of an isosceles triangle divides it into two congruent triangles.

Given: Isosceles triangle ABC with AC

₌₌

BC; CD

bisects LA CB.

Conclusion: L.ADC

₌₌

L.BDC.

Proo!'

STATEMENTS

B

A D

Illustrative Example 1.

REASONS 1. AC == BC.

2. CD == CD.

3. CD bisects LACB.

4. La == LfJ- 5. LA DC == L.BDC.

I. Given.

2. Reflexive theorem of segments.

3. Given.

. 4~A~tor-divides -au an.gkinto.

two congruent angles.

5. S.A.S.

C

B

E 4.23. Illustrative Example 2:

Given: The adjacent figures with AD and CE bisecting each other atB.

Conclusion: L.ABC== L.DBE.

Proo!,

ST ATEMENTS

I. AD and CE bisect each other at B.

2.

BA

_==BD.

3. BC

₌₌

BE.

4. LABC

== LDBE.

5. 6ABC

₌₌

L.DBE.

--.

Illustrative Example 2.

REASONS

1. Given.

2. Definition of bisector.

3. Reason 2.

4. Vertical angles are congruent.

5. S.A.S.

D 4.24. Use of figures in geometric proofs. Every valid geometric proof should be independent of the figure used to illustrate the problem. Figures are used merely as a matter of convenience. Strictly speaking, before Example 2 could be proved, it should be stated that: (1) A, B, C, D, and E are' five points lying in the same plane; (2) B is between A and D; and (3) B is between C and E.

To include such information, which can be inferred from the figure, would make the proof tedious and repetitious. In this text it will be permis- sible to use the figure to infer (without stating it) such things as betweenness, collinearity of points, the location of a point in the interior or the exterior of an angle or in a certain half-plane, and the general relative position of points, lines, and planes.

The student should be careful not to infer congruence of segments and angles, bisectors of segments and angles, perpendicular and parallel lines.

just because "they appear that way" in the figure. Such things must be included in the hypotheses or in the developed proofs. It would not, for example, be correct to assume LA and LD are right angles in the second example because they might look like it.

Exercises (A)

The triangles of each of the twelve following problems are marked to show congruent sides and angles. Indicate the pairs of triangles which can be proved congruent by Postulate 17 or Theorem 4.13.

c F

₀

M~N

A ^E R~r

B D

s

EX.i. Ex. 2.

c

F E

A~/~l A _B

Ex. 4.

Ex. 3.

---

A 1~/C B

Ex.5.

,,07

Ex. 7.

Axn

B

Ex. 9.

s

IV

R

Ex. 11.

Exercises (B)

Prove the following exercises:

13. Given: AC .1 AB; DE .1 BD;

AC ₌₌

DE; B bisects AD.

Conclusion: 6.ABC

₌₌

6.DBE.

T

~s p~

Ex.6.

MlZlN

Ex.8.

Z

D E

Ex. 10.

D

A

c

EX.i2.

C E

A~~n _B

Ex.13.

- -- ---

118 FUNDAMENTALS OF COLLEGE GEOMETRY

14. Given: AD and BE intersecting at C;

CE ₌₌

CB;AC

==

DC.

Conclusion: ,6.ABC

==

,6.DEe.

^A

X ^"

_EX.14.

^c

^B

ⁿ

15. Given: QS

..1 RT; S bisects RT.

Conclusion: ,6.RSQ

₌₌

,6.TSQ.

Q

R~T _S

Ex.15.

D 16. Given: LDAB == LCBA; EA == BF.

Conclusion: ,6.ABE ₌₌

,6.BAF.

^E

A

EX.16.

s

17. Given: RS == QT,PS == PT;

LRTP == LQSP.

Conclusion: ,6.RTP

₌₌

,6.QSP.

T

Q Ex. 17.

18. Given: AC == AD; BC == BD;

La == L(}; L{3 == L y.

Conclusion: ,6.ABC ₌₌

,6.ABD.

c

$

⁽³

A B

IJ 'Y D Ex.18.

c

F

19. Given: Isosceles ,6.ABC with

AC

==

BC; D the mid- point of AC; E the midpoint of Be.

Conclusion: ,6.ACE

₌₌

,6.BCD.

CONGRUENCE - CONGRUENT TRIANGLES

g

C

A B

Ex.19.

119

s 20. Given: ,6.QRS with LSQR

==

LSRQ;

T the midpoint of QS;

W the midpoint of RS;

QS

⁼⁼

RS.

Conclusion: ,6.TQR

₌₌

,6.WRQ.

Q R

Ex. 20.

Theorem 4.14

4.25. If two triangles have two angles and the included side of one con- gruent to the corresponding two angles and the included side of the other, the triangles are congruent.

A~B ^D _E

Theorem 4.14.

Given: ,6.ABCand,6.DEFwithLA

== LD,LB ==

LE,AB

⁼⁼

DE.

Conclusion: ,6.ABC

₌₌

,6.DEF.

Proof

ST ATEMENTS REASONS

1. AB

== DE, LA ==

LD.

2. On

l5F

there is a point H such

that mDH = mAC.

3. Draw HE.

1. Given.

2. Point plotting postulate.

3. Two points determine a line.

- - - -- - ---

--n__-

--- - - - ---

6. LB:::= LE.

7. LDEH:::= LE.

8.

Eli

^and

EF

are the same ray.

9. H=F.

4. ,6.ABC:::=,6.DEH. 4. S.A.S. i

5. LDEH:::= LB. 5. Corresponding ,§ of congruen~;

&. are :::=to each other. I

6. Given.

7. Congruence of ,§ is transitive.

8. Angle construction postulate.

9. Two lines intersect in at mos"

one point.

10. Replacing H of Statement 4 by (from Statement 9).

10. ,6.ABC:::= ,6.DEF.

It will be noted that in drawing the figure for the proof of Theorem 4.14J the point H is shown between D and F. The point could just as well drawn with F between Hand D. This would not alter the validity of th proof. The abbreviation for the statement of this theorem is A.S.A.

Theorem 4.15

4.26. If a leg and the adjacent acute angle of one right triangle are con, gruent respectively to a leg and the adjacent acute angle of another, the rig triangles are congruent.

H

~~

A ^, C

E

D~~F

Theorem 4.15.

Given: Right &. ABC and DEF with LA :::=

LD, leg AC

:::=

leg DF, LC and L are right angles.

Conclusion: ,6.ABC

:::=

,6.DEF.

Proof

ST ATEMENTS REASONS

1. LA :::=

LD.

2. LC and LF are right 1:;.

3. LC:::= LF.

4. AC :::=DF.

5. :. ,6.ABG :::=,6.DEF.

1. Given.

2. Given.

3. Right angles are congruent.

4. Given.

5. A.S.A.

---

4.27. Illustrative Example 1:

Giz'en: CD bisects LACB; CD ..L AB.

Conclusion: ,6.ADC

:::=

,6.BDG.

Proal'

STATEMENTS

1. CD bisects LA GB.

2. LA CD

:::=

LBCD.

3: CD ..LAB.

4. LADC and LBDC are right angles.

5. LADG :::=LBDG.

6. CD :::=

CD.

7. ...6ADG

:::=

,6.BDC.

- 121

c

Illwtrative Example 1. A. [) R

REASONS

1. Given.

2. A bisector divides an angle into two congruent angles.

3. Given.

4. Two perpendicular lines form right angles.

5. Right angles are congruent.

6. Congruence of segments is reflexive.

7. A.S.A.

The student will note how the method of modus ponens has been applied in the <lbo\e proof. The logic used could be written:

(a) I. A bisector divides an angle into two congruent angles.

2.

CD

bisects LACB.

3. LACD :::=

LBCD.

(b) l. Two perpendicular lines form right angles.

2. CD

is perpendicular to

AB.

3. LADC and LBDG are right angles.

(c) I. All right angles are congruent.

2. LADC and LBDC are rig-ht ang-Ies.

3. LA DC :::=

LBDC.

(eI) 1. If two triangles have two angles and the included side of one congruent to the corresponding two angles and the included side of the other, the triangles are congruent.

2. LACD

:::=

LBCD;

CD :::= CD; LADC :::=

LBDC.

3. ,6.ADC :::= ,6.BDC.

122

FUNDAMENTALS OF COLLEGE GEOMETRY

4.28. Illustrative Example 2:

Given: AB J.. BC, DC J.. Be, LABD ₌₌ LDCA.

Prove: ,6,ABC ₌₌

,6,DCB.

Proof

A D

Illustrative Example 2.

ST ATEMENTS REASONS

1. AB J.. BC; DC J.. Be.

2. LABC is a right angle;

LDCB is a right angle.

3. LABC ₌₌

LDCB.

4. LABD

₌₌

LDCA.

5. LDBC

₌₌

LACB.

6. LACB

==

LDBC.

7. BC

==

BC.

1. Given.

2. Perpendicular lines meet to form right angles.

3. Right angles are congruent.

4. Given.

5. Subtraction of angles theorem.

6. Symmetric theorem of

₌₌

angles.

7. Reflexive theorem of congruent segments.

8. A.S.A.

8. :. ,6,ABC

₌₌

,6,DCB.

Exercises (A)

The triangles of each of the following ten problems are marked to show congruent sides and angles. Indicate the pairs of triangles that can be proved congruent by Theorem 4.14 or Theorem 4.15. (See figures for.

Exercises 1 through] 0.) ~

A

A~

D

Ex.I. B

_{Ex. 2.}

N C