Secondary 2 (Grade 8) Geometry
Guiding Question
How to copy a triangle in more than one way?
Topic
Congruency of Triangles
Software Required
Cabri-Jr. Application for the TI-83 Plus Graphics Calculator
Level
Sec 2 (Grade 8)
Duration
Five 40-minute periods
Pre-requisite Knowledge
Students have been introduced to 5 axioms of Geometry and the properties of 2D
and 3D Geometrical figures in Secondary 1.
Lesson 1:
Select 7 students to role-play the scenario. (Refer to the worksheet 1
given)
Lesson 2&3: Students to be divided into 6 groups and discuss on the 6 different
scenarios. Students can investigate the different cases using
CabriJunior on TI 83. (Refer to the worksheet 1 given)
Lesson 4:
Students to present their findings using drawings, PowerPoint, Ti83,
etc (Refer to the worksheet 2, Summary for TESTS OF
CONGRUENCY)
Lesson 5:
Teachers to bring in the concepts on congruency of triangles.
Students to complete Assignment 1 as homework.
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
Mathematics: Congruency and Triangles
Worksheet 1
Objective: To make conjecture on one congruency test of triangles
Group’s mission:
To help Alex decide if his worker is right and to make a conjecture on a short
cut to congruency test of triangles if possible.
Alex, a building contractor has just assembled two triangular trusses that will support
the school hall. Before the crane hoists them into place, Alex was told to ensure that
the two triangular trusses must be identical otherwise he has to dismantle and
rebuild!
Alex instructed one of his workers, Charlie, to measure all three sides and three
angles of the two triangular trusses to check for congruency.
Scenario 1: But Charlie told him all he need to do is to measure all three sides of the
two triangles (i.e. SSS). Charlie claimed that if the measured quantities are
congruent, the triangles must be congruent.
Is Charlie right? Should Alex agree with Charlie?
Scenario 2: But Charlie told him all he need to do is to measure all three angles
(AAA). Charlie claimed that if all three angles are congruent, the triangles must be
congruent.
Is Charlie right? Should Alex agree with Charlie?
Scenario 3: But Charlie told him all he need to do is to measure two corresponding
sides and one corresponding angle in that order (i.e. SSA). Charlie claimed that if the
two corresponding sides and one corresponding angle are congruent, the triangles
must be congruent.
Is Charlie right? Should Alex agree with Charlie?
Scenario 4: But Charlie told him all he need to do is to measure one corresponding
side, one corresponding angle and another corresponding side in that order (i.e.
SAS). Charlie claimed that if the measured quantities are congruent, the triangles
must be congruent.
Is Charlie right? Should Alex agree with Charlie?
Scenario 5: But Charlie told him all he need to do is to measure one corresponding
angle, one corresponding side and another corresponding angle in that order (i.e.
ASA). Charlie claimed that if the measured quantities are congruent, the triangles
must be congruent.
Is Charlie right? Should Alex agree with Charlie?
Scenario 6: But Charlie told him all he need to do is to measure one corresponding
side, two corresponding angles in that order (i.e. SAA). Charlie claimed that if the
measured quantities are congruent, the triangles must be congruent.
Is Charlie right? Should Alex agree with Charlie?
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
1.
In the Graphic Organizer provided,
i.
Write down the conditions stated by Charlie for two triangles to be
congruent. in the conjecture box (first box).
ii.
In order to create the sample for you to decide if Charlie is correct you
need to construct triangles. Describe how you could construct triangles
meeting the conditions stated in (i) and check on their congruency.
Record this description in the second box.
iii.
Describe /draw the samples used in the third box.
iv.
Write down results shown by sample.
1.
Check your sample used by using the guides (as stated in the fifth box) on the
Graphic Organizer.
2.
Hence, discuss if Charlie is correct, record in the last box on the Graphic
Organizer?
3.
Discuss how you are going to present to the class in the next lesson on your
group’s finding.
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
Mathematics: Congruency and Triangles
Summary
Scenario 1: But Charlie told him all he need to do is to measure all three sides of the
two triangles (i.e. SSS). Charlie claimed that if the measured quantities are
congruent, the triangles must be congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
Scenario 2 : Charlie’s Claim
Charlie told him all he need to do is to measure all three angles (AAA). Charlie
claimed that if all three angles are congruent, the triangles must be congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
Scenario 3 : Charlie’s Claim
Charlie told him all he need to do is to measure two corresponding sides and one
corresponding angle in that order (i.e. SSA). Charlie claimed that if the two
corresponding sides and one corresponding angle are congruent, the triangles must
be congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
Scenario 4 : Charlie’s Claim
Charlie told him all he need to do is to measure one corresponding side, one
corresponding angle and another corresponding side in that order (i.e. SAS). Charlie
claimed that if the measured quantities are congruent, the triangles must be
congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
Scenario 5 : Charlie’s Claim
Charlie told him all he need to do is to measure one corresponding angle, one
corresponding side and another corresponding angle in that order (i.e. ASA). Charlie
claimed that if the measured quantities are congruent, the triangles must be
congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
Scenario 6 : Charlie’s Claim
Charlie told him all he need to do is to measure one corresponding side, two
corresponding angles in that order (i.e. SAA). Charlie claimed that if the measured
quantities are congruent, the triangles must be congruent.
Is Charlie right? Should Alex agree with Charlie?
___________________________________________________________________
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
GRAPHIC ORGANIZER (Skillful Generalization)
CONJECTURE (GENERALISATION)
Two triangles are congruent if
DESCRIBE A SAMPLE THAT WOULD SUPPORT THIS GENERALISATION
DESCRIBE THE SAMPLE USED
RESULTS SHOWN BY THE SAMPLE
HAS THE SAMPLE CONSIDERED ALL POSSIBILITIES?
YES / NO
IS THE SAMPLE REPRESENTATIVE?
YES / NO
IS THIS CONJECTURE SUPPORTED?
YES / NO
IS THIS CONJECTURE UNCERTAIN?
YES / NO
WRITE DOWN YOUR GROUP’S CONCLUSION.
The conjecture made by Charlie is
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
Worksheet 2:
Summary for TESTS OF CONGRUENCY
Two triangles are congruent if :
Property
(a) ______________________________________________________
______________________________________________________
______________________________________________________
(b) ______________________________________________________
______________________________________________________
______________________________________________________
(c) ______________________________________________________
______________________________________________________
______________________________________________________
(d) ______________________________________________________
______________________________________________________
______________________________________________________
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
Name: _____________________________ (
) Class: _______
Date:______
Mathematics: Congruency and Triangles
Assignment 1
Answer all the questions in the spaces provided on this worksheet.
1. Name the triangles that are congruent. Prove that the two triangles are indeed
congruent.
a.
P
A
R
C
B
b.
Q
Given that E is the mid-point of line segments AB and CD.
B
C
E
D
A
c.
Given that AC = BD and AD = BC.
B
A
C
D
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
2.
Something for you to ponder on…….
Tom is going to build a bridge over a river. However he doesn’t know the
width of the river. Tom comes to you to seek help.
You and your friends have just learnt congruency and your sister Jess told
you that the width of the river can be easily measured by means of
congruency of triangles. However, your sister stopped short of telling you
how.
With the hint given by her, explain how congruency can be used to measure
the width of the river while remaining on the same side of the riverbank.
You may want to make use of the following sketch of the section of the river
where Tom is going to build the bridge.
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.
NIE/2004/Inquiry Learning/TCHS/Sec 2(Grade 8)/LamPL/Wonghs, Materials adapted from Jasmine
Tey.