Feb 17, 2014
I.
Topic
:
GEOMETRIC RELATION
II.
Objectives: At the end of the learning period, the student should be able to; A. Identify interior, exterior and remote interior angles of triangles B. Find the missing measurement by Triangle- Angle Sum theorem and Exterior Angle theorem using the formulas.
III.
Subject Matter A. Properties of Triangle (Triangle –Angle Sum and Exterior Angle Sum Theorem) B. Reference : C. Materials
IV.
:
www.mathwarehouse.com cartolina, manila paper, marker, chalk, board
PROCEDURE: A. Preliminaries 1. Classroom Management 2. Assignment Checking 3. Lesson Recall ( from assignment checking just a random, recall the concept of Triangle inequality identifying and proving the relationship re lationship between angles and sides) B. Motivation Each student will get a triangle cut c ut out of paper. I will ask students to tear all thre e corners of their tr iangle any size they want. We will then line up the angles. I will ask students what they notice when the angles are put together. They form a line which which is 180 degrees, so the sum sum of all the angles will be 180 degrees. This leads to our first theorem Triangle Sum Theorem – The The sum measures of the interior angles of a triangle is 180 degrees I will then get students to get their triangle back out. I will get students to put back one angle in the corner that it came from. Students will then line up the other two missing angles. Students will be able to see that the other two torn angles make up the exterior angle. This leads to the next theorem. Exterior Angle Theorem – the the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (nonadjacent) interior angles
C.
Lesson Proper Triangle-Angle-Sum Triangle-Angle-Sum Theorem
This may be one the most well-known mathematical rule--The rule--The sum of all 3 interior angles in a triangle is 180°. If 180°. If you add up all of the angles in a triangle tri angle the sum must equal 180°...
Let's do some examples involving the Triangle Sum Theorem to he lp 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 A ngles As the picture below shows, an exte rior angle (A) equals the sum of t he 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+
1 = 110°what is m 2= 180
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 Have the students generate concept in Triangle Angle Theorem and what makes Exterior Angle Theorem. Angle Sum Theorem: The sum of the measures of the interior angles is Exterior Angle Theorem: An exterior angle is formed by one side of a _________________ and the extension of another __________. Remote interior angles are the angles of a triangle that are not ________________ to a given __________________ angle. The measure of an exterior angle of a triangle is ____________ to the sum of the measures of the two ________________ interior angles
VI.
APPLICATION
2. Solve for x
VII.
EVALUATION 1. Find the missing angle measure
2.
VIII.
Find the measures of angle1 and angle2 in the figure. ASSIGNMENT
Find the measure of each of the following angles. 1)
2)
B
D
20
C
55 56 A
C
A D
27 B
Solution for #2:
First, we can solve form? 1 since we are given the measure of two angles of that triangle. This part of the problem is similar to the ex amples we have already done above. Let's begin with the statements of what we are given, which are:
Now, we can solve form? 1 by using the Triangle Angle Sum Theorem. S o 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 chec k 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.
