Lesson 5: Triangle Congruence

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Lesson 5 Triangle Congruence
Congruency of geometrical objects The
mathematical term to describe two identical
objects is called congruency. Two geometrical
objects (segments, angles, etc.) are congruent to
each other if they have the same shape and same
size.
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Congruence
Discussion When are two segment
congruent? When are two angles congruent?
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Triangle Congruence
Definition ?ABC is congruent to ?DEF, denoted
by ?ABC ?DEF, if (1) m?A m?D, m ?B
m?E, m?C m?F, and (2) ABDE, ACDF,
BCEF.
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Triangle Congruence
Note When we write ?ABC ?DEF, A corresponds
to D B corresponds to E C corresponds to
F Therefore ?ABC ?DEF and ?ABC ?DFE
are not equivalent In general.
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Congruency Tests Note that if two triangles are
congruent, we have six pair of equal quantities.
Do we need to know all six equalities before we
can conclude that two triangles are congruent?
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Activity I Constructing triangles Apparatus
Compass, ruler and pencil 1. Using only compass,
ruler and pencil, construct the following
triangles on a piece of paper. a. A right-angled
triangle. b. A triangle with an acute angle of 45
?. c. A triangle with an acute angle of 22.5
?. d. An equilateral triangle. 2. Construct
triangle ?ABC with the given condition. a. AB
15 cm, BC 10 cm, and m?B 30?. b. AB 15
cm, BC 10 cm, and m?A 30?. Compare the
triangles with your friends, do you always get
the same triangles?

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Activity I Constructing triangles Apparatus
Compass, ruler and pencil 1. Using only compass,
ruler and pencil, construct the following
triangles on a piece of paper. a. A right-angled
triangle. b. A triangle with an acute angle of
45o. c. A triangle with an acute angle of
22.5o. d. An equilateral triangle. 2. Construct
triangle ?ABC with the given condition. a. AB
15 cm, BC 10 cm, and m?B 30?. b. AB 15
cm, BC 10 cm, and m?A 30?. Compare the
triangles with your friends, do you always get
the same triangles?

Answers 2 a. Always get the same triangle. b.
There are two possible answers. What can you
deduce from this activity?
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Congruency Tests From Activity 2 (a), we see
that we need only three pieces of information to
determine congruence. Activity 2 (b) tells us
that not any three values determine
congruency. - 3 equalities is sufficient to
conclude that two triangles are congruent,
provided they are the correct ones.
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Congruency Tests Side-Angle-Side (SAS) Test
Axiom If two sides and the included angle of a
triangle are congruent to, respectively, two
sides and the included angle of another triangle,
then the two triangles are congruent.
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Congruency Tests Side-Side-Side (SSS) Test If
three sides of a triangle are congruent to,
respectively, three sides of another triangle,
then the two triangles are congruent.
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Congruency Tests Angle-Angle-Side (AAS) Test If
two angles and the corresponding side of a
triangle are congruent to, respectively, two
angles and the corresponding side of another
triangle, then the two triangles are congruent.
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Group Discussion Is AAA a congruency condition?
Why?
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Example 1 Show that in an isosceles triangle, the
two base angles are equal. Or ?ABC is an
isosceles triangle, with AB AC, use a
congruency test to show that m?B m ?C.
Proof


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Group Discussion ?ABC is an isosceles triangle,
with AB AC, (use SAS test for congruency
to) show that m?B m ?C.
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Example Show that the opposite sides of a
parallelogram are congruent.
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• Group Discussion
• In ?ABC, given that m? BAC m? BCA, show that
  ?ABC is an isosceles triangle.
• 2. If a quadrilateral has congruent opposite
  sides, show that it is a parallelogram.