7 e’s DETAILED LESSON PLAN
I.
II.
III.
IV.
OBJECTIVES
At the end of the lesson the students must be able to:
1. illustrate theorems on triangle inequalities
2. apply theorems on triangle inequalities
3. take and pass the test with mastery level of 75%
CONTENT
A. Topic: Axiomatic System
B. Subject Integration: English
LEARNING RESOURCES
A. Materials Needed: Power point presentation, videos
B. References: Haidee P. Rosette & Michelle S. Sonza, (2015). Math World 8, C&e
Publishing, Inc. pp.211-218.
PROCEDURE
Teachers Activity
Students Activity
A. Introductory Activity
- Greetings
Good afternoon Class
How are you?
Do you feel good? Or you are now tired? Because it is almost your final examination
that will start on next week.
So, are you now ready?
Good Afternoon
Ma’am
Each students how
they feel
Okay good to hear class
Are you ready to listen for our class discussion this afternoon class?
What are you expecting on this class and about the subject to be presented?
Okay will see if your expectation will come into reality
Are you Ready?
Now let us bow down our head and put ourselves in the presence of the Lord.
(The Teacher will present a Prayer and ask a student to lead the Prayer.)
-Checking of attendance
Now for your attendance let me check if you are present or not and say present
as I call your name.
Yes Ma’am we are
ready
B. ELICIT
-Review
- Reviewing a previous lesson
- Who can recall your previous lesson?
- Thank you Very Good
We have here an example;
Consider the given figure. Which vertices should be paired to show that the
triangles are congruent?
Write the triangle congruence statement using the proper notation.
Solutions:
∆𝑃𝐴𝑆 ↔ ∆𝐵𝐿𝑀
<
C. ENGAGE
Have you ever wondered
how artists utilize triangles
in their artworks? Have you
ever asked yourself how
contractors, architects, and
engineers make use of
triangular features in their
designs? What mathematical concepts justify all the triangular intricacies of
their designs?
The concepts and skills you will learn from this lesson on the axiomatic
development of triangle inequalities will improve your attention to details,
shape your deductive thinking, hone your reasoning skills and polish your
mathematical communication. In short, this module unleashes that mind power
that you never thought you ever had before
D. EXPLORE
Activity :ARTISTICALLY YOURS
Our topic last
meeting is about
Congruent triangles
Ma’am
Answer
1. Different
triangles
2. Yes, studen
will explain
their answe
3. Triangles
have energy
and power
associated
with them a
they can
point out
direction,
depending
on where
their base is
placed. Tria
gles can giv
a feeling of
action,
tension or
even
aggression.
On the one
hand, they
can
symbolize
strength
while on th
other,
conflict.
4. The greater
angle and
greater side
theorem
states that
within a
triangle,
longer sides
lie opposite
larger angle
and the
triangle
inequality
theorem
states that
the sum of
any two
sides of a
triangle mu
be greater
than the thi
side.
E. EXPLAIN
This afternoon we were going to discuss about Inequalities triangle. Yesterday
we discuss about congruent triangles meaning they are equal in sides. But this
time we are going to illustrate theorems on triangle inequalities, and then apply
theorems on triangle inequalities, and to prove inequalities in a triangle.
In this lesson you will explore cases in which sides ore angles of triangles are
unequal in measure.
An exterior angle is an angle formed outside a polygon when a side of the
polygon is extended.
In figure 5-10, if C is between B and D, then <ACD is an exterior angle of
∆𝐴𝐵𝐶.
In other words, if one side of a triangle is extended like → in the figure, the
𝐶𝐷
angle that → makes with another side of the triangle (AC in this case) is an
𝐶𝐷
exterior angle at vertex C. Another exterior angle can be made at vertex C by
extending side AC. The angle that it makes with side BC is also an exterior
angle of ∆𝐴𝐵𝐶 at vertex C.
A remote interior angle is an interior angle of a triangle that is not adjacent to
a given exterior angle of the triangle. It is also called nonadjacent interior
angle.
Base on the figure shown,< 𝐴 𝑎𝑛𝑑 < 𝐵 are the remote interior angles of the exterior
angle< 𝐴𝐶𝐷.
Exterior Angle Theorem
The measure of an exterior angle is greater than the measure of each of its remote
interior angles.
To prove this Theorem, consider figure 5-10. The angles < 𝐴𝐶𝐷 𝑎𝑛𝑑 < 𝐴𝐶𝐵 form a
linear pair. Thus, they are supplementary. This means that
𝑚 < 𝐴𝐶𝐷 + 𝑚 < 𝐴𝐶𝐵 = 180°.
From this equation,
𝑚 < 𝐴𝐶𝐷 = 180° − 𝑚 < 𝐴𝐶𝐵.(equation 1)
The three interior angles of a triangle add up to 180° . therefore,
𝑚 < 𝐵+𝑚 < 𝐴 + 𝑚 < 𝐴𝐶𝐵 = 180°
Subtracting 𝑚 < 𝐴𝐶𝐵 from both sides of the equation you get
𝑚 < 𝐵 + 𝑚 < 𝐴 = 180° − 𝑚 < 𝐴𝐶𝐵.(equation 2)
From equation 1 and equation 2, you have
𝑚 < 𝐵 + 𝑚 < 𝐴 = 𝑚 < 𝐴𝐶𝐷 (Transitive Property)
Since 𝑚 < 𝐴 and 𝑚 < 𝐵 are both positive real numbers, their sum (𝑚 < 𝐴𝐶𝐷)
is always greater than 𝑚 < 𝐴 𝑜𝑟 𝑚 < 𝐵, thus,
𝑚 < 𝐴𝐶𝐷 > 𝑚 < 𝐵 And 𝑚 < 𝐴𝐶𝐷 > 𝑚 < 𝐴.
Therefore, the measure of the exterior angle < 𝐴𝐶𝐷
Is greater than the measure of each of its remote interior angles,< 𝐴 𝑎𝑛𝑑 < 𝐵.
Inequalities in One Triangle
In the measure of two sides of a triangle are unequal, then the measures of the angles
opposite them are unequal. The larger angle is opposite the longer side.
-The greater angle and greater side theorem states that within a triangle, longer sides lie
opposite larger angles, and the triangle inequality theorem states that the sum of any
two sides of a triangle must be greater than the third side. These theorems and
inequalities come in very handy when dealing with triangles.
A
C
B
Given: ∆𝐴𝐵𝐶 𝑤𝑖𝑡ℎ 𝐵𝐴 > 𝐵𝐶. The angle opposite
BA is < 𝐴𝐶𝐵. the angle opposite BC is< 𝐵𝐴𝐶.
Prove: 𝑚 < 𝐴𝐶𝐵 > 𝑚 < 𝐵𝐴𝐶.
If the measure of two angles of a triangle are unequal, then the measures of the sides
opposite. They are unequal. The longer side is opposite the larger angle.
Example
a. List the angles in ascending order
C
5
3
A
B
7
b.
A
80°
C
40°
B
Solutions:
a. < 𝐴, < 𝐵, < 𝐶
b. AB, AC, BC
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the of
the third side.
Given ∆𝐴𝐵𝐶,the following statement are true;
𝐴𝐵 + 𝐵𝐶 > 𝐴𝐶
𝐴𝐶 + 𝐵𝐶 > 𝐴𝐵
𝐴𝐵 + 𝐴𝐶 > 𝐵𝐶
Any three segments whose measures satisfy the three inequalities above are possible
sides of a triangle.
The next corollary can help remember theorem 5-5 easily.
Corollary
The measure of the longest side of a triangle is less than the sum of the measures of the
two remaining sides of the triangles.
With this corollary, you only need to pick the longer side and check one inequality.
Example 2
Determine whether or not each set of measures shows possible side and check one
inequality.
a. {1, 1, 1}
b. {4, 3, 5}
Solution
a. Since the given number are equal, you can treat any one of them as the
“longest” and apply corollary 5-1.
The statement 1<1+1 is true. Hence, this set of numbers could be length of
the sides of a triangle. In this case, you will have an equilateral triangle.
b. The longest side measures 5 units. The statement 5<3+4 is true. Hence, this
set of numbers could be lengths if the sides of a triangle.
Inequalities Involving More than One Triangle
Hinge Theorem
Given that two sides of a triangle are congruent to two sides of another triangle. If the
included angle of the first triangle is larger than the included angle of the first triangle is
longer than the third side of the second triangle
Given: ∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆𝐷𝐸𝐹, with 𝐴𝐵 = 𝐷𝐸 𝑎𝑛𝑑 𝐴𝐶.if 𝑚 < 𝐴 > 𝑚 < 𝐷, then 𝐵𝐶 > 𝐸𝐹.
Note that the conditions should be met first before making any conclusions about the
triangles. It means that the congruence between two pairs of corresponding sides of the
triangles should be established first; otherwise, you cannot use the conclusion of the
theorem.
B
A
C
F. ELABORATE
G. EXTEND