Basic Engineering Mathematics, Fifth Edition

176 Basic Engineering Mathematics

Problem 25. In Figure 20.36, trianglePQRis

isosceles withZ, the mid-point ofPQ. Prove that

trianglesPXZandQYZare congruent and that

trianglesRXZandRYZare congruent. Determine

the values of anglesRPZandRXZ

P Q

R

(^678) Y
288 288
X
Z
Figure 20.36
Since trianglePQRis isosceles,PR=RQand thus
∠QPR=∠RQP.
∠RXZ=∠QPR+ 28 ◦and∠RYZ=∠RQP+ 28 ◦(ext-
erioranglesofatriangleequalthesumofthetwointerior
opposite angles). Hence,∠RXZ=∠RYZ.
∠PXZ= 180 ◦−∠RXZ and ∠QYZ= 180 ◦−∠RYZ.
Thus,∠PXZ=∠QYZ.
TrianglesPXZandQYZare congruent since
∠XPZ=∠YQZ,PZ=ZQand∠XZP=∠YZQ(ASA).
Hence,XZ=YZ.
TrianglesPRZandQRZare congruent since
PR=RQ, ∠RPZ=∠RQZ and PZ=ZQ (SAS).
Hence,∠RZX=∠RZY.
TrianglesRXZandRYZare congruent since
∠RXZ=∠RYZ,XZ=YZand∠RZX=∠RZY(ASA).
∠QRZ= 67 ◦and thus∠PRQ= 67 ◦+ 67 ◦= 134 ◦.
Hence,∠RPZ=∠RQZ=
180 ◦− 134 ◦
2
= 23 ◦.
∠RXZ= 23 ◦+ 28 ◦= 51 ◦ (external angle of a tri-
angle equals the sum of the two interior opposite
angles).
Now try the following Practice Exercise
PracticeExercise 79 Congruent triangles
(answers on page 349)

1. State which of the pairs of triangles in
   Figure 20.37 are congruent and name their
   sequences.

A

C

BD

M

(a) (b) (c)

(d) (e)

QSR

T

V

W

U

X

Y

Z

K

L

O

P

N

E

I

G

H

J

F

Figure 20.37

2. In a triangleABC,AB=BCandDandE
   are points onABandBC, respectively, such
   thatAD=CE. Show that trianglesAEBand
   CDBare congruent.

20.5 Similar triangles

Two triangles are said to besimilarif the angles of one

triangle are equal to the angles ofthe other triangle. With

reference to Figure 20.38, trianglesABCandPQRare

similar and the corresponding sides are in proportion to

each other,

i.e.

p

a

=

q

b

=

r

c

658 588

578

c

B a C

A

b

658588

r^578

Q p R

P

q

Figure 20.38

Problem 26. In Figure 20.39, find the length of

sidea