Feb 14,2014 I.
Topic
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II.
Objectives:
GEOMETRIC RELATION
At the end of the learning period, the student should be able to; A. Determine relationships that exist between the angles and sides of triangles . B. Determine the triangle inequality theorem C. Solve Triangle- Angle Sum Theorem and Exterior Angles Theorem with
fluency Subject Matter A. Properties of Triangle (Triangle Inequality Theorem, Triangle- Angle Sum Theorem and Exterior Angles Theorem) B. Reference : www.mathwarehouse.com C. Materials : toothpicks, cartolina, marker, chalk, board Procedure: A. Classroom Management
III.
IV.
B. Lesson Recall ( Recap on one of the properties of Triangle) identifying which
Theorems B Theorem 5.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
Theorem 5.11 If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
5
3 A
D
C
60º
40º
E
F C. Motivation Done in Drill Students will use toothpicks to explore the properties of triangles. They will organize their discoveries into a chart. Their explorations should lead them to conclude that the sum of two sides of a triangle must be equal to or greater than the measure of the third side. The exercise should help students define and recognize isosceles, equilateral, scalene, obtuse, right, and equilateral triangles.
A. Group the class into 4 and give them about a dozen toothpicks and have them discard any that are not the same size as the others.
B. Have them take out a piece of paper, pencil, and ruler to set up a chart like the chart below. C. As a whole class, instruct students to pull three toothpicks from the pile, repeat the process with four toothpicks. Then five toothpicks, six toothpicks, and so on until time is up. Remind students that they can only count triangles in which all toothpicks are end to end.. D. Give students about five minutes to work Guiding Questions Can you form a triangle using all three ( four toothpicks,then five toothpicks, six toothpicks, and so on toothpicks) placed end to end in the same plane? Can a different triangle be formed? What kinds of triangles are possible? What is your findings on each sides of triangles formed? How many toothpicks consumed on both side of triangle formed? What is your finding if you compare it to the base of triangle formed?
Lesson Proper Properties of Triangles A. The sum of all the angles of a triangle will always be 180 degrees. B. The sum of any two sides of a triangle will always be greater than the third side. C. Any external angle of a triangle will be equal to the sum of the two internal angles, which are not adjacent to it.
What you did is one of the properties of triangle proving that the sum of any 2 sides of a triangle must be greater than the third side and we call it as a;
TRIANGLE INEQUALITY THEOREM This theorem states that the sum of the lengths of any 2 sides of a triangle must be greater than the third side When sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle! The Formula
Note: This rule must be satisfied for all 3 conditions of the sides. In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle .
Look at the example below, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)
Example 2:
Could a triangle have side lengths of Side 1: 4 Side 2: 8 Side 3: 2
Answer is No ! Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.
Example 3:
Could a triangle have side lengths of Side 1: 5 Side 2: 6 Side 3: 7
Yes Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .
small + small > large
example 4:
because 5 + 6 > 7
Could a triangle have side lengths of Side 1: 6 Side 2: 8 Side 3: 15
No Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .
small + small > large
because 6 + 8 ≯ 16
Triangle-Angle-Sum Theorem
This may be one the most well known mathematical rule--The sum of all 3 interior angles in a triangle is 180°. If you add up all of the angles in a triangle the sum must equal 180°..
Let's do some examples involving the Triangle Sum Theorem to help us see its utility. Examples (1) Find the measure of angle C.
Solution:
As with all problems, we must first use the facts that are given to us. Using the diagram, we are given that
Since our goal is to find the measure of ?C, we can use the Triangle Angle Sum Theorem to solve for the missing angle. So we have Using the angle measures we were given, we can substitute those values into our equation to get.
Exterior Angle Theorem Any exterior angle of a triangle will be equal to the sum of the two interior angles, which are not adjacent to it. The Exterior, Interior and Remote Interior Angles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure An exterior angle of a triangle is formed by extending one of the sides of the triangle. In a triangle, each exterior angle has two remote interior angles (see picture below). The remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle.
Formula for Remote Interior Angles and Exterior Angles As the picture below shows, an exterior angle (A) equals the sum of the remote interior angles. To rephrase it, the angle 'outside the triangle' (exterior angle A) equals D + C (the sum of the remote interior angles).
However, the exterior angle is supplemental to the adjoining interior angle. Example 1:
If the exterior angle, 1+
2= 180
1 = 110°what is m
2?
2 =180-110 = 70
Example 2: Use the formula to calculate the values of x (a remote interior angle) and of Y.
Since X and J are remote interior angles in relation to the 120° angle, you can use the formula. 120 = 45 + x 120- 45 = x 75° = x. Now, since the sum of all interior angles of a triangle is 180°. You can solve for Y 75 + 45 + y = 180 120 + Y = 180 Y = 60
V.
Generalization
Let the student generalized the theorems on properties of triangles
VI.
APPLICATION:
2. Solve for x
3. Do the side lengths below satisfy the triangle inequality theorem Side 1: 5 Side 2: 5 Side 3: 10 Answer :
No
Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .
small + small > large
because 5 + 5 ≯ 10
4. Do the side lengths below satisfy the triangle inequality theorem Side 1: 7 Side 2: 9 Side 3: 15 Answer:
Yes
Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .
small + small > large
because 7 + 9 >15
5.
VII. Evaluation (1) Find the measures of angle1 and angle2 in the figure .
(2) Find measurement angle B.
Solution:
First, we can solve for m?1 since we are given the measure of two angles of that triangle. This part of the problem is similar to the examples we have already done above. Let's begin with the statements of what we are given, which are:
Now, we can solve for m?1 by using the Triangle Angle Sum Theorem. So we have
In order to solve for the measure of ?2, we will need to apply the Exterior Angle Theorem. We know that the two remote interior angles in the figure are ?S and ?A. Thus, by the Exterior Angle Theorem, the sum of those angles is equal to the measure of the exterior angle. We have
While not always necessary, we can check our solution using our previous knowledge of lines. We see that ?1 and ?2 make up ray AK. And since straight lines have 180° measures, we know that the sum of ?1 and ?2 must be 180. Let's check to make sure:
So, we know we have worked this problem out correctly.
(2) Find measurement angle B.
Solution: Let's take a look at the information we have been given first. We know that
Right off the bat, we can apply the Exterior Angle Theorem to help us solve the problem. We have
This does not answer the question, however. The question asked for m angle B. The variable x alone does not tell us what the measure of the angle is. So, we must plug x = 4 into our equation for m angle B:
. Now we have found that the measure of angleB is 39°.
