Download Triangles, Part 4

Prove It
How do we create truth?
2:1:26: Triangles, Part 4: Congruent Triangles
TITLE OF LESSON
Geometry Unit 1 Lesson 26 – Triangles, Part 4: Congruent Triangles
Prove it! What’s on the outside? What’s on the inside? Of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding
parts of congruent triangles.
MATERIALS
Pairs of three pieces of string. See below (Activities #4) for more description. Have at least 3 pairs of three
pieces of string. Each pair should have 3 sets of congruent lengths. For example, have one pair with two pieces
measuring 4 feet, two pieces measuring 6 feet and two pieces measuring 7 feet. Have one set with all six pieces
measuring 3 feet, and so on.
LESSON OBJECTIVES
To be introduced to the following concepts:
• Corresponding angles
• Corresponding sides
• Included angle
• Included side
• Congruent Triangles
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Stamp/initial complete homework assignment. Pass back graded work and have
students place in the appropriate sections of their binders.
2) Agenda – Have students copy the agenda.
3) Homework Review – (10 minutes) Review homework from Lesson 23. Students were to make drawings to
describe the following properties of triangles. Each drawing should have been labeled completely so that an
observer could understand what is being described. When you go over this homework in class, have
someone other than the person whose homework it is interpret the drawing.
• Base
• Perpendicular
• Leg
• Altitude
• Base of an isosceles triangle
• Hypotenuses
• Legs of an isosceles triangle
• Legs of a right triangle
• Base Angle
• Median
• Vertex Angle
• Angle Bisector
Have each student show one of her drawings to the class. One of the other students will interpret the drawing and
point out what it is that this drawing is supposed to demonstrate. For instance if the drawing is an angle bisector it
should be apparent that a line segment is bisecting an angle – both which angle is being bisected and what is the
line segment used as the bisector. Then collect the assignments.
ACTIVITIES – INDIVIDUAL AND GROUP
1
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
1.
2:1:26: Triangles, Part 4: Congruent Triangles
Lecture: Congruent Triangles – (15 minutes) Introduce the concept of congruence of two triangles. We say two
triangles are congruent if they have the same size and shape. This means that the three sides and angles of one
triangle must be identical to the three sides and angles of another triangle. This does not mean the triangles have to
appear the same—they could have different bases, which could make them appear different.
These two triangles could appear to be different, but are, in fact, congruent based on the definition.
2.
Lecture Continued – When we say two triangles are congruent we use the following symbol: ≅ or we write that
∆ABC ≅ ∆DEF (using the triangles below) implying that triangle ABC is congruent to triangle DEF. It does not
matter how you draw the two triangles on paper or on the board the important thing is that for each side of the first
triangle you can associate a side of equal length on the other triangle and that each angle on the first triangle
matches an angle of equal measure on the other triangle.
In the case below, we say that ∆ABC ≅ ∆DEF implying that AB = DE, AC = DF, BC = EF and m∠ABC =
m∠DEF, m∠BAC = m∠EDF and m∠ACB = m∠DFE
A
B
D
C
E
F
Congruent Triangles
We designate the equality of AB and DE by drawing a single slash through each of the sides. We designate the
equality of BC and EF by drawing 2 slashes through each of the sides. We designate the equality of angles ABC
and DEF by drawing a single curved line in the interior of the angle. How did we show the equivalence of angles
BAC and EDF? (By drawing two curved lines in interior of the angle.) How would we designate the equivalence of
angles BCA and EFD? (3 curved lines)
However it is important to realize the drawing of each triangle does not have to be from the same perspective for
the two triangles to be congruent. It is hard to tell from the drawing because all of the sides look to be fairly similar
in size, as do all of the angles. It is possible that AB = DF, AC = DE and BC = EF and m∠ABC = mDFE, ∠mBAC
= ∠mFDE, and ∠mBCA = m∠FED. These triangles would still be congruent triangles. Ask student why. How
would we designate the equivalent triangles if this were the case? Ask students to redraw the two triangles in their
notebooks and to designate the equal sides and angles by slashes and curved lines but have the equivalence
correspond to the second set or equivalence. When they have finished, have them pair up with another student to
compare. Circulate to make sure they’ve all got it.
3.
Definitions – (10 minutes) Write the following on the board: Have the students write each definition in their binders
under Terms and Definitions.
The parts of two triangles that are equal in measurement are said to be corresponding sides or angles.
Corresponding angles are two angels, each in a different triangle, that have the same measurement. In the first
case presented we say that ∠ABC corresponds to ∠DEF since m∠ABC = m∠DEF, ∠BAC corresponds to
∠EDF since m∠BAC = m∠EDF and ∠ACB corresponds to ∠DFE since m∠ACB = m∠DFE.
Corresponding sides are two sides, each in a different triangle, with the same measurement. AB is said to
correspond to DE since AB = DE, AC is said to correspond to DF since AC = DF and BC is said to correspond to
EF since BC = EF.
An included side is the side between two angles. For instance the angles ABC and BCA have BC as the included
side. Ask the following: What are the other two included sides in this triangle? (AC is the included side for
2
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:26: Triangles, Part 4: Congruent Triangles
angles BAC and BCA. AB is the included side for angles BAC and ABC.) What are the included sides for the
triangle DEF?
An included angle is an angle between two sides. For instance angle ABC is the included angle between AB and
BC.
Ask the following: What are the two other included angles? (Angle BCA between sides BC and CA, and angle
CAB between sides CA and AB) What are the included angles for triangle DEF? Notice that the included sides are
always included in the names of the angles that surround them and the included angles are always included in the
names of the sides that surround them! E.G., ∠ABC and ∠BCA have BC as the included side, and ∠ABC is the
included angle between AB and BC.
4.
Group Work – (15 minutes) Choose two groups of three students to form groups. Give each group three pieces of
string. The students are going to form triangles with the strings by having each student hold the end of the two of
the three pieces of string in their hands (Each student will be a vertex). Each group will have the same size strings.
The three strings do not have to be the same length but it is important that the set of three strings be the same for
each group. Make sure that the strings are pulled tight and that they hold them by the end rather than in the middle.
Ask the class if the two triangles are congruent. (If you’ve cut the string correctly, they should be!) It is now up to
the rest of the class to decide which of the sides are the corresponding sides and which of the angles are the
corresponding angles. Have one student draw the two triangles on the board and follow class directions for labeling
the corresponding sides and angles. The student at the board should label the vertices with the students’ names and
should draw the triangles as closely as possible to how she sees them. This means that if they are aligned like this:
The student at the board should draw them like that. This way they’ll get practice with correct notation.
Repeat the process 3 or 4 times with different groups forming the triangles. Vary the sizes of the strings in each
experiment and challenge the students acting as vertices to make the triangles look as different as possible. Again,
they’ll get practice with correct notation and you’ll reinforce the idea that the orientation on the page or in the room
does not affect the concept of the two triangles being congruent.
4.
Homework Review – Explain the homework assignments. Answer questions.
HOMEWORK
1) Draw two triangles that are congruent. Label the equal angles and equal sides with curved lines and slashes so
that congruence is apparent. Repeat this process for two different congruent triangles.
2) Draw two sides and an included angle.
3) Draw another set of two sides and included angle in which the corresponding sides are equal and the included
angle is equal. What can we conclude?
4) Draw a side between two angles. Do not complete the triangle. It should look something like this:
5) Draw another side of the same size between two angles of same measurement as the previous drawing. What
can we conclude?
GROUP ROLES
None
DOCUMENTATION FOR PORTFOLIO
3
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:26: Triangles, Part 4: Congruent Triangles
None
4
© 2001 ESubjects Inc. All rights reserved.