c. Construct two nonoverlapping triangles with different bases between the two parallel lines.
d. Represent the altitude of each triangle by h and their respective bases by b1 and b2.
e. Find the ratio of the areas of the two triangles.
f. What can you conclude?
Tell the students that in doing the given activities, they should rely on their own abilitiesand judgment.
2. Guide the students in proving Theorem 5-1 and 5-2 and in completing the proof of Basic Proportionality Theorem.
3. Illustrative Examples
Use the examples on page 366 of the textbook.
4. Practtice Exercises
a. Answer Mental Mathematics of Exercise 5.2 (numbers 1-10) on page 368 of the textbook.
b. Solve Written Mathematics of Exercise 5.2 (numbers 1-7) on page 368 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 5.2 (numbers 8-10) on page 369 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the Law of Proportion
B. Follow-up Lesson
1. Guide the students in proving the Converse of the Basic Proportionality Theorem.
2. Practice Exercises
Solve Written Mathematics of Exercise 5.2 (numbers 11 and 12) on page 369 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 5.2 (numbers 13-16) on page 369 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 5.2 (numbers 17-20) on page 370 of the textbook.
Lesson 5.3
Similarity Between Triangles
I. Mathematical Concepts and Skills
A. Similar polygons are polygons in which the corresponding angles are congruent and the ratios of the lenghts of the corresponding sides are equal.
B. Two triangles are similar, if and only if the corresponding angles are congruent and the lenghts of the corresponding sides are proportional.
C. If there exists a correspondence between the vertices of two triangles such that three angles of one triangle are congruent to the corresponding angles of the second triangle, respectively, then the two triangles are similar.
D. If two angles of a triangle are congruent to two angles of the second triangle, respectively, then the two triangles are similar.
E. Similarity between triangles is an equivalence relation.
F. If a triangle is similar to a second triangle and a second triangle is congruent to a third triangle, then the first triangle is similar to the third triangle.
G. If two pairs of corresponding sides of two triangles are propotional and the included angles are congruent , then the two triangles are similar.
H. If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.
II. Objectives
A. Illustrate and define similar polygons and similar triangles
B. Prove and use the AAA similarity and AA Similarity Theorems to draw conclusions about triangles
C. Show that similarity between triangles is an equivalence relation
D. Prove that if a triangle is similar to to a second triangle and a second triangle is congruent to a third triangle, then the first triangle is similar to the third triangle
E. Prove and use the SAS Similarity and SSS Similarity Theorems to draw conclusions about triangles
III. Values Integration
Self-confidence is necessary to accomplish desired result
IV. Materials
Picture of similar figures
Manila paper
Marker
V. Instructional Strategies
A. Whole Class Discussion
B. Practice
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the Assignment.
2. Recall the congruence Postulate for Trianlges
B. Lesson Proper
1. Show the class pictures having the same shape but differnt sizes, Ask the students afterward to describe the pictures.
2. Then present the following geometric figures:
a. C.
 |
b. 
D.
D. 
| |
Ask the students to describe the geometric figures.
3. Tell the students that the last pairs of geometric figures are examples of similar polygons. Ask them to define similar polygons.
4. Consider the two triangles below .
Ask the students to describe the triangles.
5. Ask the students to complete the proofs of AAA Similarity and AA Similarity Theorems. Tell the students that they should have self-confidence in accomplishing the task given to them.
6. Illustrative Examples
Use the examples on page 372 of the textbook.
C. Practice Exercises
a. Answer Mental Mathematics of Exercises 5.3 (numbers 1-10) on pages 380-382 of the textbook.
b. Solve Written Mathematics of Exercises 5.3 (numbers 1-6) on pages 382 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 5.3 (numbers 7 and 8) on pages 383 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the Poin Plotting Theorem and the Line Postulate .
B. Follow-up Questions
1. Ask the students to complete the proof of the SAS Similarity Theorem.
2. Guide the students in proving the SSS Similarity Theorem.
3. Practice Exercises
Solve Written Mathematics of Exercise 5.3 (number 11) on page 383 of the textbook.
C. Assignment
Solve the Written Mathematics of Exercise 5.3 (number 11) on page 383 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignments
2. Recall the following:
a. AA Similarity
b. SAS Similarity Theorem
c. SSS Similarity Theorem
B. Follow-up Questions
1. Illustrative Example
Show how to prove this problem.
Given : AB is perpendicular to BD
EC is perpendicular to CD
Prove : triaangle ABD is congruent to triangle ECD
2. Practice Exercises
Solve written mathematics of exercise 5.3 (numbers 12 and 13) on page 384 of the text book.
C. Checking for Understanding
Solve written mathematics of exercise 5.3 (numbers 14-20) on page 384 of the text book.
D. Assignments
Study the Proof of the right Triangle Similarity Theorem.
Lesson 5.4
Similarities in Right Triangle
I. Mathematical Concepts and Skills
A. In any right triangle, the altitude to the hypotenuse divides the triangle into two right triangles which are similar to each other and to the given right triangle.
B. In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse and each of the leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
C. In an isoceles right triangle, the lenght of the hypotenuse is equal to the lenght of the leg times square root of 2.
D. In a 30-60-90 triangle, the length of the hypotenuse is twice the lenght of the shorter leg and the lenght of the longer leg is square root of 3 times the length of the shorter leg.
II. Objectives
A. Find the lengths of sides and altitudes of rifgt triangles
B. Apply relationships in Isoceles Right Triangle Theorem and right 30-60-90 degrees triangles.
III. Values Integration
Impotance of cooperation
IV. Materials
marker
Manila paper
V. Instructional Strategies
A. Whole Class Discussion
B. Small Essay Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check assignments.
2. Recall the Pythagorean Theorem
B. Lesson Proper
1. Ask the students to solve the following problems.
If in triangle ABC, angle ACB is a right angle , line CD is the hypotenuse to line AB, A is a 60 degrees angle and angle ACD is 30 degrees, find angle DCB and angle B.
2. Ask the following questions:
a. Into how many triangles is the given triangle divided by its hypotenuse?
b. Are the two triangles congruent? Similar? Why?
c. Are the two triangles similar to the original triangle? Why?
3. Divide the class into small groups.Remind them about the importance of cooperation in doing a task. Then tell them to prove the Right Triangle Similarity theorem and the Geometric Mean Theorem.
4. Illustrative Examples.
Use the examples on page 388 of the book
5. Practice Exercises
a. Answer Mental Mathematics of Exersice 5.4 (numbers 1-10) on page 390 of the textbook.
b. Solve Written Mathematics of Exercise 5.4 ( numbers 1-9) on page 390 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 5.4 (numbers 10-19) on page 391 0f the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the Pythagorean Theorem.
B. Follow-up Lesson
1. Prove the Isoceles Rigth Triangle Theorem and the 30 dergee, -60 degree, -90 degree Triangle Theorem.
2. Illustrative Example
Find the perimeter of triangle ABC.
3. Practice Exercises
Solve Written Mathematics of Exercise 5.4 (number 20) on page 319 of the textbook.
C. Checking Understanding
Solve Written Mathematics of Exercise 5.4 (number 21) on page 391 of the textbook.
D. Assignment
Solve Written mathematics o f Exercises 5.4 (number 22) on page 391 of the textbook.
Lesson 5.5
Consequences of the Basic Proportionality Theorems; Areas and Perimeters of Similar Figures
I. Mathematical Concepts and Skills
A. If each of the three or more coplanar parallel lines are each cut by two transversals, the intercepted segments on the two trnsversals are proportional.
B. The bisector of an angleof a triangle separates the opposite sides into segments whose lengths are proportional to the lengths of the other two sides.
C. Two corresponding altitudes of similar triangles are proportionalto the corresponding sides.
D. Any two corresponding angle bisectors of similar triangles are proportional to the corresponding sides.
E. Any twocorresponding medians of similar triangles are proportional to the corresponding sides.
F. The ratio of the perimeters of two similar triangles is equal to the ratio of any pair of corresponding sides.
G. If two triangles are similar, then the ratio of their areas equals the square of the ratio of the lrngths of any corresponding sides.
H. The ratio of the areas of two similar triangles is equal to the square of the ratio of the two corresponding perimeters.
II. Objectives
A. To prove and apply the theorems about three or more parallel lines cut by a transversal
B. To prove and apply the Theorem about the bisector of an angle of a triangle
C. To prove and apply the theaorems about proportional segments in triangles
D. To prove and apply the theorems about perimeters and areas of similar triangles
III. Values Integration
The need for self-confidence in performing a task
IV. Materials
Manila paper
Marker
V. Instructional Strategies
A. Whole class discussion
B. Practice
VI. Procedure
Session 1
A. Preliminary Activities
1. Check assignments
2. Recall the basic Proportionality Theorem
B. Lesson Proper
1. Ask the students to solve the following problems
2. Suppose in the above figur, the transversal t1 is moved the right, will the value of QRr change as well as a result of the movement?
3. Guide the students in proving that if each three or more coplanar parallel lines are cut by two transversals, the intercepted segments of the two transversals are proportional.
4. Illustrative Example
5. Practice Exercises
a. Answer Written Mathematics of Exercises 5.5 (numbers 1-10) on page 406 of the textbook.
b. Answer Written Mathematics of Exercises 5.5 (numbers 1-2) on page 406 of the textbook.
C. Assignment
Solve written Mathematics of Exercise 5.5 (number 3) on page 406 of the textbook.
Session 2
A. Preliminary activities
1. Check the assignment.
2. Recall the definition of an angle bisector.
B. Follow-up Questions
1. Guide the students in proving the following theorem:
The bisector of an angle of a triangle separates the opposite sides into segments whose lengths are proportional to the lengths of the two other sides.
2. Illustrative Example
In triangle ABC, ray AD bisects angle BAC.
If AB is 8,AC is 10, and BC is 6 find BD and DC
3. Practice Exercises
Solve written mathematics of exercises 5.5 (number 4)on page 406 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 5.5 (number 5) on page 406 of the textbook.
Session 3
A. Preliminary Activities
1. Check assignments
2. Recall the AA Similarity Theorem
B. Follow-up Questions
1. Guide the students in proving the following theorems:
a. Any two corresponding altitudes of similar triangles are proportional to the corresponding sides.
b. Any two corresponding angle bisectors of similar triangles are proportional to the corresponding sides.
2. Tell the students that they should have sel-confidence whenever they are asked to prove any geometric statement
Prove the following:
a. Any two corresponding medians of similar triangles are proportional to the corresponding sides.
b. The ratio of the perimeters of two similar triangles is equal to the ratio of any pair of corresponding sides.
C. Complete the proof of Theorem 5-20.
Session 4
A. Preliminary Activities
1. Check assignment.
2. Recall the Geometric mean Theorem.
B. Follow-up Lesson
1. Prove th pythagorean Theorem using the geometric Mean theorem.
2. Introduce the Pythagorean Triples.
3. Illustrative Example
Find the perimeter and area of triangle ABC.
4. Practice exercise
Solve Written Mathematics of Exercise 5.5 (number 7) on page 407 of the text book.
C. Checking for Understanding
Solve Written Mathematics of Exercise 5.5 (number 8) on page 407 of the textbook.
D. Assignment
Solve Written Mathematics Exercise 5.5 (numbers 9 and 10) on page 407 of the textbook.
No comments:
Post a Comment