5.2.4 Congruent triangles
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144 161➤We often need to compare pairs of triangles. Two triangles are calledcongruentif they
are identical. They may notlookidentical, because they may be rotated with respect to
each other, so we need a test to determine when two triangles are identical. For this, the
corresponding angles of the two triangles must be identical, and the corresponding sides
must be of the same length. However, it is not necessary to check all these conditions
to ensure that two triangles are congruent. The following provide alternative tests for
congruent triangles:
- three sides of one triangle must be equal to three sides of the other
- two sides and the angle between them in one triangle must be equal to
two sides and the enclosed angle in the other triangle - two angles and one side in one triangle must be equal to two angles
and the corresponding side in the second triangle - for right-angled triangles, the right angle, hypotenuse and side in the
first triangle must be equal to the right angle, hypotenuse and the corre-
sponding side in the second triangle.
Solution to review question 5.1.4(i) In Figure 5.4(i)ADis equal toBC,andDCis equal toAB.ACis
a common side. SoACDandABCare congruent triangles, since all
corresponding sides are equal.
(ii) WithAC as a common side and two identical sidesAB,AD,the
trianglesABCandACDin Figure 5.4(ii) have two sides of identical
length. They also have two equal angles DCAand ACB. However,
these angles are not those between the pair of identical sides in the
two triangles (second test in the text above) so the triangles are not
necessarily congruent.5.2.5 Similar triangles
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145 162➤If one triangle is simply an enlargement or rotation of another triangle, then we say the
two triangles aresimilar. So the three angles of one triangle are equal to the three angles
of the other, similar triangle, and the corresponding sides of the two triangles are in the
same ratio. However, we don’t need to test both of these conditions, since it can be shown
that they are equivalent, i.e. thatif two triangles contain the same angles then their
corresponding sides are in the same ratio.
Solution to review question 5.1.5
SinceABCandDEFin Figure 5.5 are similar, then the ratios of corre-
sponding sides are equal.(i) In this case the ratio is equal toBC/FE= 5 /3.
(ii) Thus,AB/DE= 4 /DE= 5 /3 whenceDE= 12 /5 cm. Similarly(!)
AC/DF= 3 /DF= 5 /3, soDF= 9 /5cm.