Chapter Geometric Relations
2
Lesson 2.1
Relations Involving Segments an Angles
I. Mathematical Concecpts and SkillsA. The absolute value of x denoted by is nonnegative, and is the opposite of x if x is negative.
B. 
There is a one-to-one correspondence between the points of a line and the set of real numbers such that the distance between any two points of the line is the aabsolute value of the difference between the corresponding numbers.
There is a one-to-one correspondence between the points of a line and the set of real numbers such that the distance between any two points of the line is the aabsolute value of the difference between the corresponding numbers.C. 
Segment is congruent to segment if and only if the length of line AB is equal to the length line CD .
Segment is congruent to segment if and only if the length of line AB is equal to the length line CD .D. M is the midpoint of line AB if and only if (a) M lies between A and B and (b) AM = MB.
E. Given two points P and S on a line, a coordinate system can be chosen in such a way that the coordinate of P is 0 and the coordinate of S is greater than ).
F. On a ray AB, there is exactly one point P that lies at a distance x from
A.
A.
G. 
Every segment has exactly one midpoint.
Every segment has exactly one midpoint.H. Ray ET is the bisector of <BEH if and only if T is in the interior of <BEH and <BET <HET .
I. Congruence for angles is reflexive, symmetric, and transitive.
I. Objectives
A. To find the coordinate of a point on a number line
B. To find the distance between two points in a number line
C. To solve equations involving absolute values
D. To find the coordinate of the midpoints of a segment
E. To plan and write two-column proofs
F. To apply the properties of equalities in proving
G. To apply the definition of angle bisector in computations
H. To prove the congruence for angles is reflexive, symmetric, and transitive
II. Values Integration
Being open and honest
III. Materials
ruler
sticks
IV. Instructional Strategy
Discussion
V. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of between and the definition of collinear points.
B. Lesson Proper
1. Draw a line on the board and ask the students to describe it. Tell the students that to each of the point on the number line there corresponds a real number and to every real number there corresponds a point on the number line.
Ask the students to give examples of real numbers between any two numbers. Tell them to plot their corresponding points on the same number line.
2. Ask the students to define absolute value. Then tell them to do the following exercises.
a. l-6l b. l-8l+l-3l
c. –l-6l+l+8l e. -4l-8-(-3)+1l
d. -6-l-10l
3. 
Ask the students to find the number of units from 5 to -2 on the number line by counting the number of units from -2 to 5.
Ask the students to find the number of units from 5 to -2 on the number line by counting the number of units from -2 to 5.4. Illustrate how to use the concept of absolute value in finding the distance between two points on a number line.
5. Illustrate the meaning of congruent segments by asking the students to measure two sticks with equal lengths. Next, draw a line similar to the one below.
6. 
Draw this number line on the board.
Draw this number line on the board.Ask the following questions.
a. Are points A, M, and B collinear? Why?
b. Which point is between A and B?
c. What is the distance from B to M? from M to A?
d. What is the midpoint of segment AB?
e. When is a point the midpoint of a segment?
7. Practice Exercises
a. Answer Mental Mathematics of Exercise 2.1 (numbers 1-10) on page 116 of the textbook.
b. Solve Written Mathematics of Exercise 2.1 (numbers 1-6) on page 116 of the textbook. Tell the students to be honest. They should refrain from looking at their seatmates’ paper.
C. Assignment
Solve Written Mathematics of Exercise 2.1(numbers 7-12) on page 117 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Review the past lesson.
B. Follow-up Lesson
1. 
Draw the same number line used in Session 1 on the board. The time replace the coordinate of the midpoint by x and ask the following questions:
Draw the same number line used in Session 1 on the board. The time replace the coordinate of the midpoint by x and ask the following questions:a. What is the distance from B to M? from M to A?
b. Are the two distances equal ? Why?
c. Write an equation and solve for x. What is the coordinate of the midpoint?
2. Show that Postulate 11 can be used as an alternative method in determining the distance between two points.
3. Prove Theorem 2-1 and Theorem 2-2 on paragraph form.
4. Practice Exercises
Solve Written Mathematics of Exercise 2.1 (numbers 13 and 14) on page 117 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.1 (numbers 15 and 16) on page 118 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignment.
2. Recall proving statements in paragraph form.
B. Follow-up Questions
1. Write several number sentences on the board. Ask the students to write opposite each sentence the property of equality being illustrated.
2. 
Present the following illustrations.
Present the following illustrations.3. Tell the students to observe the angles and ask the following questions.
a. What do you observe?
b. Is point T in the interior of each angle?
c. Is it correct to say that m<BET + m<HET = m<BEH?
4. Present the following illustration:
5. Ask the following questions:
a. Is the point S in the interior of <ABC?
b. Is m<ABC = m<ABS + m<CBS?
6. Tell the students that ray ET is an angles bisector whereas ray BS is not. Ask the students to define an angle bisector in their own proofs.
7. Discuss the steps in writing a two-column proof. Then prove Theorem 2-3.
8. Practice Exercise
Solve Written Mathematics of Exercise 2.1 (number 17) on page 118 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.1 (number 19) on page 118 of the textbook.
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Lesson 2.2
Adjacent and Conplementary Angles
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(Textbook pages 119-128, 2 sessions)
I. Mathematical Concepts and skills
A. Adjacent angles are two angles which have a common vertex and a common side but have no interior point in common.
B. Angles are complementary if the sum of their measures is 90. Each angles is called a complement of the other.
C. Of two angles are complementary of congruent angle, then the two angles are congruent.
D. If two angles are complementary to the same angles, then they are congruent.
II. Objectives
A. To illustrate, identify, and define adjacent and complementary angles.
B. To find the measures of the complements of given angles
C. To state, prove, and apply the following:
1. Theorem2-4.If two angles are complements of congruent angles, then the tow angles are congruent.
2. Corollary 2-4.1 If two angles are complementary to the same angle, then they are congruent.
D. To differentiate a corollary from a postulate.
Being friendly to other people
IV. Materials
manila paper
marker
V.Instuctional Strategy
Discussion
VI.Procedure
Session 1
A. Preliminary Activities
1. Check the assignments.
2. Recall the meaning of postulate and the properties of equalities.
B. Lesson Proper
1. Introduce the terms adjacent angles and complementary angles by doing the following activity.
a. Draw the following angles on the manila paper. Post the manila paper on the board.
Ask the following questions:
· What is the common vertex of <ABD and <CBD ?
· What is the common side of <ABD and <CBD ?
· Do they have common interior points?
b. Draw this figure on manila paper. Post the manila paper on the board.
Ask the following questions:
· What is the common vertex of <EFG and <EFH?
· Do <EFG and <EEEFH have a common side?
· Do they have interior points in common?
2. Ask the students to define in their own words the terms adjacent and complementary angles.
3. Explain the proof of Theorem 2-4.
4. Ask the students to complete the proof of Corollary 2-4.1 Direct the students’ attention to the proof of this corollary on page 122 of the textbook.
5. Practice Exercises
a. Answer Mental Mathematics of Exercise 2.2 (numbers 1-4) on page 124 of the textbook.
b. Solve Written Mathematics of Exercise 2.2 (numbers 1-4 of the textbook in page 125.
C. Assignment
Solve the Written Mathematics of Exercise 2.2 (numbers5-10) on page 126 of the textbook)
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Review the past lesson.
B. Follow-up Lesson
1. Illustrative Example
Explain the proof of this problem.
Given <BAD (congruent) <CAE
Prove: <BAE (congruent) <CAD
2. Practice Exercises
Solve Written Mathematics of Exercise 2.2 (numbers 11 and 12) on page 127 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 2.2 (numbers 13-16) on page 127 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 2.2 (numbers 17-20) on page 127 of the textbook.
If there is enough time, solve the challenge Problems of Exercise 2.2 on page 118 of the textbook.
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Lesson 2.3
Supplementary Angles
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(Textbook pages 129-139, 2 sessions)
J. Mathematical Concepts and Skills
A. Two angles are supplementary if the sum of their measures is 180.
B. If two angle are supplements of congruent angles, then the two angles are congruent.
C. If two angles are supplementary to the same angle, then the two angles are congruent.
D. If two angles are both congruent and supplementary, then each is a right angle.
II. Objectives
A. To illustrate, identify, and define supplementary angles
B. To state, prove, and apply the following theorems:
1. Theorem 2-5.If two angles are supplement of congruent angles then the two angles are congruent.
2. Corollary2-5.1 If two angles are supplementary to the same angle, then the two angles are congruent.
3. Theorem 2-6. If two angles are both congruent and supplementary then each is a right angle.
Being helpful
IV. Materials
manila paper
marker
V.Instructional Strategies
A. Group work
B. Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignments
2. Recall the definitions. of congruent angles and right angles,
B. Lesson Proper
1. Draw the figures:
a. What is the sum of the two angles in figure 1? in figure 2?
b. Compare the two angles on Figure 1 with those on figure 2.
2. After telling the students that the two angles in each figure are supplementary angles, ask them to define supplementary angles in their own words.
3. Ask the students to compare supplementary angles with adjacent angles and with complementary angles.
4. Ask the students whether the common side of supplementary, and at the same time adjacent angles, is always in the same plane as the other two sides. Ask them to state the theorem or postulate that can be the reason for their answer.
5. Explain the proof of Theorem 2-5.
6. Divide the class into small groups. Tell the groups to complete the proof Corollary 2-5.1 and Theorem 2-6. Tell the members of each group to help each other in completing the proof.
7. Practice Exercise
Answer the Mental Mathematics of Exercise 2.3 (numbers 1-10) on pages 134-135.
C. Assignment
Solve Written Mathematics of Exercise 2.3 (numbers 1-4) of the textbook 134-135.
Session 2
A. Preliminary Activities
1. Check the assignments.
2. Review the past lesson.
B. Follow-up Lesson
1. Illustrative Example
Explain the proof of this problem.
Given: ABD is supplementary to DBE.
CBE is supplementary to DBE.
Prove: ABD is congruent to CBE
2. Practice Exercises
Solve Written Mathematics of Exercise 2.3 (numbers 5-8) on page 136 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 2.3 (numbers 9-11) on page 137 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 2.3 (numbers 12-15) on page 138 of the textbook.
If there is enough time, solve the Challenge Problems of Exercise 2.3 on page 139 of the textbook.
Lesson 2.4
Linear Pairs
(Textbook pages 140-146, 2 sessions)
I. Mathematical Concepts and Skills
A. Two angles form a linear pair if and if only if they are adjacent angles and their uncommon sides are opposite rays.
B. If two angles form a linear pair, then they are supplementary angles.
II. Objectives
A. To illustrate, identify, and define a linear pair
B. To state and apply the Linear Pair Postulate
III. Values Integration
Refine civic consciousness to the definition of adjacent angles
IV. Procedure
manila paper
markers
V. Instructional Strategies
A. Individualized Instructions
B. Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of adjacent angles and opposite rays.
B. Lesson Proper
1. Present the following pairs of angles.
2. Ask the students to compare the angles in the figure 1 with those in figure 2.
3. Tell the students that the two angles in the first figure from a linear pair.
4. Ask the students to define a linear pair in their own words.
5. Present at least three figures on the board that form a linear pair.
6. Ask the students to complete this statement, “If two angles form a linear pair, then __________________.”
7. Illustrate how to solve this problem.
If two angles are a linear pair and the measure of one is four times the measure of the other, what is the measure of each angle?
8. Practice Exercises
a. Answer Mental Mathematics of Exercises 2.4 (numbers 1-3) on page 141 of the textbook.
b. Solve Written Mathematics of Exercise 2.4 (numbers 1-4) on page 142 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.4 (numbers 5-10) on page 142 of the textbook.
Session 2
A. Preliminary Activities
1. Illustrative Example
2. Review the past lesson.
B. Follow-up Lesson
1. Illustrative Example
Explain the proof of this problem.
Given: <1 and <2 form a linear pair.
<4 and <3 form a linear pair.
Prove: <1 is congruent to <4
2. Practice Exercises
Solve Written Mathematics of Exercise 2.4 (numbers 11-13) on page 143 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 2.4 (numbers 14 and 15) of the textbook on pages 144-145.
D. Assignment
Solve Written Mathematics of Exercise 2.4 (number 16 and 17) on page 145 of the textbook.
If there is enough time, solve the Challenge Problems of Exercise 2.4 on page 146 of the textbook.
Lesson 2.5
Vertical Angles
(Textbook pages 147-151, 2 sessions)
I. Mathematical Concepts
A. Two angles are vertical angles if and if only if they are nonadjacent angles formed by two intersecting lines.
B. Vertical angles are congruent.
II. Objectives
A. To illustrate, identify, and define vertical angles
B. To state, prove, and apply the Vertical Angle Theorem
III. Values Integration
Being open and honest
IV. Materials
manila paper
marker
protractor
V. Instructional Strategy
Discussion
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of adjacent angles.
B. Lesson Proper
1. Introduce the concept of vertical angles by drawing two intersecting lines on manila paper. Post the manila paper on the board.
2. Ask one student to measure the angles using a protractor.
3. Ask the students to make a general statement.
4. Guide the students in proving the Vertical Angle Theorem.
5. Practice Exercises.
Solve Written Mathematics of Exercise 2.5 (numbers 1-6) on page 149 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.5 (number 7-10) on page 150 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of perpendicular bisector of a segment.
B. Follow-up Lesson
1. Illustrative Example
Explain the proof of this problem.
Given: m BET =60
M HET =30
MA MR
Prove: BEH is congruent to AMR
2. Practice Exercises
a. Answer Mental Mathematics of Exercise 2.6 (numbers 1-4) on page 156 of the textbook.
b. Solve Written Mathematics of Exercise 2.6 (numbers 8-10) on page 158 of the textbook.
C. Checking for Understanding
Solve Written Mathematics of Exercise 2.6 (number 11) of the textbook on page 158 of the textbook.
D. Assignment
Solve Written Mathematics of Exercise 2.6 (number 12-20) on page 158- 160 of the textbook.
If there is enough time, solve the Challenge Problems on page 160 of the textbook.
Lesson 2.7
Angles and Sides of a Triangle
(Textbook pages 161-174, 5 sessions)
I. Mathematical Concepts and Skills
A. An exterior angle is an angle l, which is adjacent and supplementary to one of the angles of a triangle.
B. Remote interior angles are which are not adjacent to the given exterior angle of the triangle.
C. An adjacent interior angle is an interior angle, which forms a linear pair with the given exterior angle of a triangle.
D. The measure of an exterior angle is equal to the sum of its remote interior angles.
E. AB > CD if and if only if AB > CD.
F. AB < CD if and if only if AB < CD.
G. A > B if and if only if m A > m B.
H. A < B if and if only if m A < m B.
I. The whole is greater than any of its parts.
J. The measure of an exterior of a triangle is greater than the measure of either of the two remote interior angles.
K. In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
L. In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.
II. Objectives
A. To illustrate, identify, and define exterior angles, remote interior angles, adjacent interior angles, greater than segments and angles, and less than for segments and angles.
B. To state and apply the properties of inequalities.
C. To prove and apply the theorems about exterior angles of a triangle.
D. To prove and apply the theorem, which says that, "The whole is greater than any of its parts"
E. To show that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side
F. To apply the Triangle Inequality Theorem in computations
G. To prove and apply the Pythagorean Theorem
III. Values Integration
Concern for others
IV. Materials
manila paper
marker
cutouts
V. Instructional Strategies
A. Discussion
B. Group Work
VI. Procedure
Session 1
A. Preliminary Activities
1. Check the assignment.
2. Recall the definition of a triangle.
B. Lesson Proper
1. Draw a triangle like the one below:
2. Use the figure, guide the students in identifying and defining exterior angles and remote interior angles.
3. Guide the students in defining less than for segments, greater than for segments, less than for angles and greater than for angles.
4. Illustrate the properties of inequalities. Give at least two examples of each property.
5. Explain the proof pf Theorem 2-9.
6. Illustrative Example
Use the example on page 162 of the textbook.
7. Practice Exercises
Solve Written Mathematics of Exercise 2.7 (numbers 1-3) on page 172 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.7 (numbers 4 and 5) on page 172 of the textbook.
Session 2
A. Preliminary Activities
1. Check the assignment.
2. Recall the properties of inequalities.
B. Follow-up Lesson
1. Explain the proof of Theorem 2-10.
2. Illustrative Examples
Use the examples on page 165 of the textbook.
3. Practice Exercises
Solve Written Mathematics of Exercise 2.7 (numbers 6 and 7) on page 172 of the textbook.
B. Assignment
Solve Written Mathematics of Exercise 2.7 (numbers 8 and 9) on page 172 of the textbook.
Session 3
A. Preliminary Activities
1. Check the assignment.
2. Recall the Exterior Angle Equality Theorem.
B. Follow-up Lesson
1. Prove Theorem 2-11.
2. Illustrative Examples
3. Practices Exercises
Solve Written Mathematics of Exercise 2.7 (numbers 11 and 12) on page 172 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.7 (number 13) on page 173 of the textbook.
Session 4
A. Preliminary Activities
1. Check the assignment.
2. Recall Theorems 2-11 and 2-12.
B. Follow-up Lesson
1. Ask the students to work on Explore on page 167 of the textbook.
2. Illustrative Example
Use the examples on page 168 of the textbook.
3. Practice Exercises
a. Answer Mental Mathematics of Exercise 2.7 (number 1-10) on page 171 of the textbook.
b. Solve Written Mathematics of Exercise 2.7 (number 14) on page 173 of the textbook.
C. Assignment
Solve Written Mathematics of Exercise 2.7 (numbers 15 and 16) on pages 173-174 of the textbook.
Session 5
A. Preliminary Activities
1. Check the assignment.
2. Recall the Area Addition Postulate.
B. Follow-up Lesson
1. Tell the students something about Pythagoras.
2. Tell the students that one of the proofs of the Pythagorean Theorem was attributed to Bhaskasa, a Hindu mathematician of the twelfth century.
3. Draw a figure similar to figure 1 on a cardboard. Cut out the four congruent triangles and the square. Make several sets of these cut-outs. Distribute one set to each group.
4. Divide the class into small groups. Ask each group to form a bigger (figure 1) using all the cutouts. Figure 1 should not be shown to the students.
5. Ask the groups to label the sides of their triangles as in figure 2.
6. Ask the following questions:
a. In terms of a and b, what is the length of the side of the inner square?
b. In terms of a and b, what is the area of one of the triangles?
c. What is the area of the inner square?
d. What is the area of the bigger square in terms of c?
e. If you add all the areas of the four congruent triangles and the area of the inner square, will the sum be equal to the area of the bigger square? Why?
7. Ask the students to form an equation.
8. Ask the students to simplify the left side of the equation to arrive at the Pythagorean Theorem.
9. Ask the students to remove the inner square.
10. Ask the students to write an equation indicating the removal of the inner square. The students should be able to write this equation:
11. Tell the students to simplify the equation to arrive at the Pythagorean Theorem.
C. Checking for Understanding
1. Find x.
2. Jay-jay left his house and walked 6 km due east and then 8km due month. How far is he from the starting point?
3. Find the range of the possible lengths of LM.
D. Assignment
Solve Written Mathematics of Exercise 2.7 (number 17-19) on page of the textbook.
If there is enough time, solve the Challenge Problems of Exercise 2.7 on page 174 of the textbook.
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