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CARAGA REGIONAL SCIENCE HIGH SCHOOL
San Juan, Surigao City
Syllabus in Math II-B
Geometry
SY 2011-2012
I – Course Description
Math II-B (Geometry) is a 1.5 – unit offered to Regional Science High Schools students in the
Philippines. It is taken in addition to Math II-A (Advanced Algebra).
The subject includes both the theoretical and applied study of measurement, angles,
triangles, lines, polygons and circles. Emphasis is given in developing proofs. The practical
application of the subject further enhances the students’ appreciation of the subject.
II – Credit Units
1.5 units
III – Time Duration
45 hours every grading period
IV – Mission/Vision
MISSION
Towards its goal, the CARAGA REGIONAL SCIENCE HIGH SCHOOL is committed to provide
quality education that is equitably accessible to the intellectually gifted and science inclined
Youth who understand and internalize the value of scientific knowledge towards the
advancement of our country.
VISION
The CARAGA REGIONAL SCIENCE HIGH SCHOOL aims to develop a core of Youth who are
scientifically inclined science oriented and competent whose scientific efforts shall lead the
country to progress and development.
V – Values Integration
The values of the students can derive this course are the following:
1. Diligence, patience, tolerance, self-reliance and self-discipline especially in doing group
activity and reports;
2. Cooperation, teamwork, unity, participation during group work activities;
3. Logical, creative and reflective thinking in the presentation of outputs; love for
mathematics and awareness on the practical value of the subject radiating the same
feeling to peer groups
4. Scientific and scholarly research output.
5. Relate the importance of solid figures in our daily life and activities and to name some
things which may represent these common solids and which are very useful to man.
VI – Course Objectives:
At the end the course, the students are expected to:
1. Apply skills of logical reasoning in dealing with different geometric facts.
2. Manifest the use of the skills developed to prove relationships among the different
geometric figures.
3. Apply the different conditions that will guarantee congruence and similarities of triangles
and quadrilaterals.
4. Demonstrate skills in solving problems involving real life situations such as area and
volumes of polygonal figures, distance and many others.
5. Reason logically and abstractly after familiarizing oneself with the underlying concepts
and useful areas of the Geometry.
VII – Course Outline
FIRST QUARTER
I – Introduction to Geometry
1. Underlined Terms
2. Deductive and Inductive Reasoning
3. Logic in Mathematics
4. Distance Between Two Points
5. Betweenness
6. Subsets of a Line
II – Lines and Planes in Space
1. Collinear and Coplanar Points
2. Convex sets and Separation
3. Angles
3.1 Parts of an Angle
3.2 Measure of an Angle
3.3 Angle Pairs
3.4 Kinds of Angles
3.5 Perpendicular Lines
3.6 Congruence of Angles
III – Lines and Planes in Space Congruent Figures
1. Congruent Triangles
2. Congruence as an Equivalence Relation
3. Medians, Angle Bisectors, Perpendicular Bisectors, Altitudes
4. Basic Congruence Postulates
5. Kinds of Triangles and Polygons
6. Isosceles Triangle and its Converse
7. The Angle Bisector Theorem
IV – Circles
1. Define a circle and its parts
1.1 radius, diameter and chord
V – Measurement
1. Parts of solid
1.1 cone, pyramid, sphere, cylinder, rectangular prism
2. Perimeter of a triangle, square and rectangle
3. Circumference of a circle
FIRST PERIODIC EXAMINATIONS
SECOND QUARTER
VI – Inequalities in Geometry
1. Exterior Angle
2. The Exterior Angle Theorem
3. SAA Theorem
4. Theorems on Congruence of Right Triangles
5. Inequalities in Two triangles
VII – Parallelism
1. Necessary Conditions for Parallelism
2. The parallel postulate
3. Sum of the Angles in a polygon
VIII – Angles and sides of triangles
1. Relationships among the sides and angles of a triangle
IX – Angles Formed by Parallel lines Cut by a Transversal
1. Relationships between pairs of angles
alternate interior angles
alternate exterior angles
corresponding angles
angles of the same sides of the transversal
X – Conditions for Oblique Triangle Congruence
SECOND PERIODIC EXAMINATIONS
THIRD QUARTER
XI – Polygonal Regions and Their Area
1. Polygonal Regions
2. Area of Triangles and Quadrilaterals
3. Special Right Triangles
XII – Conditions for right triangle congruence
1. HyL congruence
2. LL congruence
3. HyA congruence
4. LA congruence
XIII – Quadrilaterals
1. Different Types of Quadrilaterals and their properties
Inductive and deductive skills to derive its properties
diagonals bisect each other
diagonals of a rectangle
diagonals of a square
diagonals of a rhombus
1.2 Conditions that guarantee that a quadrilateral is a parallelogram
2. Special Parallelograms and Transversals to Many Lines in a Plane
THIRD PERIODIC EXAMINATIONS
FOURTH QUARTER
XIV - Similarity
1. The Idea of Similarity
2. Proportionality
3. Similar Polygons
4. Similarity Between Triangles
5. The Basic Proportionality Theorem and its converse
6. The Basic similarity theorem
7. Similarities in Right Triangles
8. Area of Similar Triangles
XV – Circles
1. Parts of a circle
2. Central Angles and Arcs of Circles
3. Inscribed Angles
4. Chords and Tangents to a circle
5. Angles Formed by Secants and Tangents
6. The Power Theorems
XVI – Area of Circles and Sectors
1. Polygons
2. Regular Polygons
3. The Circumference of a Circle
4. The Area of a Circle
5. Lengths of Arcs
6. Area of Sectors
XVII – Plane Coordinate Geometry
1. Cartesian Coordinate system
2. Slope, distance and midpoint
FOURTH PERIODIC EXAMINATIONS
VIII – Teaching Strategies
First Quarter (Lines and Planes in Space, Triangles, Circles and Measurement)
1. Multiple Intelligence
Ask the students to identify things around them which represent a point, line, plane, segment or ray.
To tap the bodily/kinesthetic intelligence of the students let the students illustrate given angles using their body parts.
To tap the visual intelligence of the students, ask them to cite things around them which illustrate polygons
In page 12, ask the students to work in pairs as they try to come up with the definition of a regular polygon
To tap the visual/spatial intelligences, encourage the students to visualize the figures in their imagination before sketching
Use a geoboard to construct a square, rectangle and triangle. Let the students visualize the solid figures using the
geoboard.
Use cut out figures to present the lesson to help students visualize the solid figures
Use paper folding activities
2. Cooperative work activities
Do the activity for them to develop the definition of median, Distribute cut out triangles to the students and give them the
following instructions:
1. choose one side of a triangle, then bisect it by folding it upon itself so that the ends coincide
2. Unfold the triangle to locate the midpoint on the side.
3. Fold the triangle through the midpoint and the opposite vertex
Ask the students the following questions:
1. Why do we have to let the endpoints of the side of the triangle coincide?
2. How does the midpoint relate to the measure of the chosen side?
3. Based on the activity, how will you define a median?
4. How will you differentiate the angle bisector from a median?
5.
Prepare cut out triangles which students can classify as well as discuss in terms of classification. This can be done in
groups.
Divide the class into groups and ask each group to work on page 23, number 25-34 and the present their groupwork on
Manila paper, to be posted on the board.
Prepare a group puzzle on finding the areas of a square, triangle, parallelogram and trapezoid
3. High Order Thinking Skills
Ask the students to do the “Think of This” problem on page 23 and page 41.
In introducing the lesson, use a picture which suggest the different solids (e.g. a mountain which suggests a triangle, the
earth and the moon which suggest a sphere, a building which suggests a prism. The given examples are the models of
geometric forms we see around us.
Define volume in your words. What are some similarities and differences between volume and surface area? In your
answer, be sure to explain the appropriate measuring unit for volume and surface area.
Make a table of the formulas for the volume of a prism, pyramid, cylinder, cone and sphere as well as the formulas for the
surface area of these.
4. Values Education
Relate the importance of solid figures in our daily life and activities. Ask them to name some things which
may represent these common solids and which are very helpful to man.
Ask the students to identify occupations where the people deal much with the surface area of solid figures.
Making connections – The volume of a three-dimensional figure measures its capacity. When you’re deciding which
product is the best buy in a grocery store or which cooler to take on a picnic, you’re using the idea of volume. Think of other
uses of one’s knowledge of the formula for the volume of solids.
Second Quarter (Inequalities in Geometry, Parallelism, Angles and sides of triangles, Angles formed
by Parallel lines Cut by a Transversal and Conditions for Oblique Triangle Congruence)
1. Multiple Intelligence
To tap the interpersonal intelligence of the students, ask them to explain at the end of the lesson, to a partner the triangle
inequality. The two students who are partners must take turn
To develop visual/spatial and naturalist intelligence, students may be asked to go outside the room and identity things
that represent transversals, such as structure, stems, spider webs, vines, etc.
To enhance the Bodily Kinesthetic and Musical/Rhythmic Intelligence of the students, use the following activity; call on
eight students. Then draw the figure below on the floor in the center of the classroom
Instruct the students to occupy a corner on angle each. Then dance music will be played. When the
teacher says “all angles”, then all 8 students will be dancing. When the teacher says “corresponding
angles”, only students standing in the corresponding angles will dance. The teacher may also call alternate
interior angles, alternate exterior angles, interior angles on the same side of the transversal, linear pairs
and vertical angles.
Instead of eight students, the teacher may involve all the students and divide them into eight to occupy the
eight angles.
2. Cooperative Work Activities
Divide the class into groups of 4’s and let them discuss the following:
Congruence of segments and congruence of angles have the reflexive, symmetric and transitive properties.
Do non-congruence (either > or <) of segments and angles have these 3 properties also? Justify your
answers.
Let the students work on the proof of the Pythagorean theorem in groups.
Ask the students to prepare a Chinese checkboard or star or to cut artworks from newspapers and magazines, by groups.
Let them prepare a report stating the corresponding parts of the triangles that are congruent.
Let the perform the following activity in pairs or triads.
Directions:
1. Draw triangle ABC on a piece of paper.
2. On another sheet of paper preferably of different color, draw triangle DEF such that side DE is
congruent to aide AB, angle E is congruent to angle B and side EF is congruent BC
3. Measure angle A, angle C and side AC.
4.
Measure angle D, angle F and side DF
5. Cut the triangular regions and compare them by placing triangle DEF upon triangle ABC.
Lead the students to see that the two triangles are congruent based on the condition that the two sides and
the included angle are congruent based on the condition that the two sides and the included angle are
congruent, respectively, to the two sides and the included angle of the other triangle. The following
illustration will be helpful.
B
E
A
D
C
F
ABC is congruent to DEF by SAS Congruence Postulate
State the SAS Congruence Postulate: Given a correspondence between two triangles. If two sides and
the included angle of one triangle are congruent to their corresponding parts in the second triangle, then
the triangles are congruent.
3. Higher Order Thinking Skills
Prove angle DEF > angle CAB if side BE and side CF bisect each other at D and B is between A and D.
Ask the students to discuss the following question and to explain their arguments: Are all converse true?
If not, give an example of a converse which is not true?
In proving corresponding parts of congruent triangles congruent, choose a problem which requires analysis
like the one given below:
In the figure below, if side DC is congruent to side DE, angle CBD is congruent to angle EDF and Angle
BDA is congruent to angle FDA, prove that side AE is congruent to side AC
C
B
D
A
F
E
4. Values Education
Try to bring in the following thought: A leader may act like a transversal. He should be able to meet (intersect) at certain
point with his subordinates.
Try to instill in the minds of the students that through patience and perseverance, any problem, no matter how difficult it
may seem, can be solved.
Third Quarter (Polygonal Regions and Their Area, XII – Conditions for right triangle congruence
and Quadrilaterals)
1. Multiple Intelligence
Ask the students to perform the following activity:
1. Draw a trapezoid on a tracing paper. Pinch to locate the midpoints of the legs. Draw the median.
2. Place another tracing paper over the first and make a copy of the trapezoid and the median.
3. On the original trapezoid, extend the larger base to the right by at least the length of the shorter leg.
4. Slide the 2nd tracing paper under the 1st. Show the sum of the lengths of the 2 bases by marking a
point on the extension of the larger base.
5. Check how many times the median fits on the segment representing the sum of the lengths of the 2
bases.
What do you notice about the lengths of the median and the sum of the lengths of the 2 bases? Compare
your results with the results of your seatmate.
1.2 To show that each diagonal of a parallelogram separates it into two congruent triangles and that the
opposite sides are congruent, you may use any of the following activities:
1. Paper-cutting – Let the students draw a parallelogram and cut it along its diagonal
2. Translation – Ask the students to draw a parallelogram on a tracing paper. Copy this on another
tracing paper. Place this copy over the original and slide the paper sideways or upwards/downwards
until the opposite sides overlap.
Using a graphing paper, ask the students to draw different parallelograms and their diagonals. For each
figure, they will measure each diagonal and its portions. They must show that the diagonals bisect each
other.
To tap the interpersonal intelligence of the students, ask them to identify a special quadrilateral and talk
about its properties to a partner. The 2 students who are partners must take turns.
Ask the students to draw a Venn diagram which shows how quadrilaterals are related to one another.
The outer circle represents all quadrilaterals. Circle A represents trapezoids, which are quadrilaterals with
exactly two parallel sides. Give the circle or set of the region that represents each type of quadrilateral:
square, parallelogram, rectangle and rhombus.
A
B
C
D
2. Cooperative Work Activities
2.1 Instruct the students to work with a partner and do the following activity:

Draw an isosceles trapezoid wxyz.

Using the protractor, measure angle z and angle y. Record the measurements.

Measure angle w and angle x. Record the measurements.

Make a conclusion
Perform activity 1, 2 and 3. Then ask the students to work in groups of 4 or 5.
1.
2.
3.
4.
Activity 1
Get a rectangular piece of paper
Fold along its diagonals
Measure the diagonals. What conclusions can you make?
Measure the angles at the intersection of the diagonals. Do the measurements suggest that he
diagonals are perpendicular?
1.
2.
3.
4.
5.
6.
Activity 2
Cut out a rhombus and label it RHOM.
Fold on diagonal HM.
Measure angle RHM and angle MHO. Compare the two angles. Do the same for angle RMH and
angle OMH. What can be concluded about the angles formed when the diagonals are drawn?
Fold on diagonal RO and do the same discovery.
Label the intersection of the two diagonals as B.
Measure angle RBH. What can be concluded about the diagonals of a rhombus?
Activity 3
Cut out a square
Fold along its diagonals
Measure the diagonals. Are they congruent?
Does the diagonal of the square bisect its angles? If so, what is the measure of the angle formed by
two intersecting diagonals?
5. Are the diagonals perpendicular? Why
Divide the class into groups. Task each group to state the geometric principles applied in each of the
following situations:
a. A girl trying to locate the center of a rectangular bed sheet, folds the sheet “on the bias” twice. The
creases on the cloth trace two diagonal lines whose intersection is the center.
b. A boy wants to find the center of a wall in his room. Can you help him find the center of the wall?
1.
2.
3.
4.
3. Higher Order Thinking Skills
3.1 Let the students do the following problem: Diamond in the Round
In the diagram below, HCEG is a rhombus formed by connecting the midpoints of a rectangle ABDF. Find
the length of a side of the rhombus if OG = 10 and CK = 8. Assume that O is the center of the circle
circumscribing ABDF.
K
B
C
D
.O
E
H
G
A
F
Let the students do the following problem: Eddie needs to replace a panel on her garage door. He found a
rectangular sheet of wood of the right size. He made a design by locating the midpoint of each side of the sheet
and connecting the points. Explain why XYZW is parallelogram.
Y
X
Z
W
3.2 How would you rearrange the cross AB and CD of this kite in order to redesign it in the shape of a
parallelogram?
A
D
C
B
4. Values Education
4.1 Link characteristics of special quadrilaterals to characteristics of special people.
* what are the characteristics of special people like students who are being admired in school.
4.2 In discussing the different quadrilaterals and their respective properties, find a way to bring individual
differences and uniqueness of individuals.
Fourth Quarter (Similarity, circles, Area of circles and sectors and Plane coordinate
geometry)
1. Multiple Intelligence
To tap the visual/spatial intelligence of the students show the following figure and let them color the parts that show
proportional segments.
1.2 To develop the verbal/linguistic intelligence, the students may be assigned to make a research on
eclipses and explain how the concept on tangent lines maybe applied to them.
1.3 To sharpen the kinesthetic intelligence of students, have them create dances (native or modern) where
tangent circles and tangent lines are visibly exhibited. This may serve as their project and culminating
activity on circles.
Try to visualize an object and then draw this on the Cartesian plane. This activity can enhance the students’ visual or
spatial intelligence. Then name the coordinates of the vertices.
Another activity can consist of plotting pre-identified points and then asking the students to connect the
points in alphabetical order and identify the figure.
1.
2.
3.
4.
5.
A(0,0), B(9,0), C(5,0)
D(2,2), E(2,-4), F(-7,-4), G(-7,2)
L(-6,1), M(-2,-4), N(2,-9)
H(7,-4), I(5,-3), J(8,4), K(10,3)
O(-10,13), P(-4,13), Q(-2,11), R(-3,8), S(-7,9)
To develop intrapersonal intelligence of the students, let them reflect and solve the following puzzle.
“Loop the Word”
QEVI TI CA LAT L X A
UUERTSORG LI N E B
ADASLOP ED IVM AC
AUADRANT ANEOS I
NI G I SET RACSED S
Were you able to spot the following terms above?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
line – the graph of a linear equation
slope – is the ratio of the difference between two ordinates and difference between two abscissas
vertical – line parallel t the x-axis and has no slope
negative – sign of the slope if the direction of inclination of the line graph is to the left
positive – sign of the slope if the direction of inclination of the line graph is the right
Descartes – a French mathematician who developed analytical geometry.
origin – has the coordinate (0,0)
abscissa – another name for the x-coordinate
ordinate – another name for the y-coordinate
quadrant – region formed when the x-axis and y-axis divide the plane
To tap the logico-mathematical intelligence of the students, ask them to solve this puzzle
NUMBER SPELLING
In the following activity you are supposed to form a word or group of words for ach number. Each
Word or group of words is mathematical in nature.
Use the code below to identify the letter corresponding to the number below the blanks where
numbers are written. Since two letters correspond to a number, choose the letter carefully. For
the other blanks place a letter and its corresponding value below the blank. As a hint, the
numbers below the blank must total to the indicated sum.
A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13
Z Y X W V U T X R Q P O N
1.
_
8
_
8
_ _
3 1
_ _
9
_ _
_ _ _ _ _ _
8
13 8 = 51
_ _ _ _ _ _ _
12 12
12 13 = 79
_ _ _ _ _ _ _ _ _ _ _ _
1 13 11 12
5 = 98
_ _ _ _ _ _ _ _ _ _
5
5
12 13
_ _ _ _ _ _ _ _ _ _ _
5
5
12
8 = 82
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
12
9 5 10
13 = 120
2.
3.
4.
5.
6.
2. Cooperative Works Activities
2.1 Group the students then perform the activity for discovering the Pythagorean relation
Square B
b
a
Square
A
c
Square
C
1.
2.
3.
4.
Enlarge the figure. The triangle in the middle is a right triangle with legs of lengths a and b.
Squares are formed from the sides of the triangle. What is the area of each square
Cut out the 3 squares from each side of the triangle.
Students are to form a square congruent to the largest square C from the other two
squares. Tell the students to cut out square B to form the 4 triangles.
With the 4 triangles and square A, form a figure congruent to square C.
5. Relate the areas of square A and B to that of square C.
2.2 Conduct an outdoor group activity. Let each group locate an object (such as a tree, a flagpole, building,
tower, creek, etc.), which they would like to measure indirectly. This can be done outside the classroom
but within the school campus or in any nearby place such as town plaza, playground, etc. Before this
lesson, teacher should conduct an ocular inspection to set up the conditions for the activity. For example
placing a ladder against a wall, etc. This activity may consume the whole period.
Let group leaders discuss and illustrate the methods that they used in measuring the heights/distance of
Objects.
You may wish to divide the tasks among the members, which include taking measurements, recording the
Measurements, performing the calculations, and being responsible for the equipment. Materials include:
meter stick or measuring tape, notebook for recording the measurements, mirror, straightedge, grease
pencil to make 2 lines in the middle of your mirror.
Before the groups begin, let them make tables like the one below:
Table for Measurement by the Shadow Method
Object
Height
of person
Shadow length
of person
Shadow length
of object
Calculated
height of object
Table for Measurement by Mirror Method
Object
Height
of observer’s eye
Distance from observer
to mirror
Distance from
object to mirror
Calculated
height of object
2.3 “Guess my Rule” contest
Number of players: two teams with 5 members each
To play the game:
The object of the game is to be able to be the first to give the rule that works for all pairs of numbers in a
given set as indicated on a table.
The object of the game is to be able to be the first to give the rule that works for all pairs of numbers in a
given set as indicated on a table.
The contest master (teacher) starts the game by giving a number. He writes the number on the board and
writes another number beside it. After a while the board may have something like this.
1st number
2nd number
5
0
3
7
2
-3
0
4
(5,2)
(0,-3)
(3,0)
(7,4)
The contest master then asks the team to figure out the rule that works for all the pairs of numbers
(In the given example the rule is, “first number less second number is three”). The contest master
continues to add entries on the table until a team gets the correct answer.
3. Higher Order Thinking Skills
3.1 In introducing proportional segments ask the students to do the following activity.
Materials: lined paper, patty paper (or onion skin)
A sheet of lined paper can be used to divide a segment into equal parts. Draw a segment on a piece
of patty paper and divide the segment into five equal parts positioning it onto lined paper or graph paper.
What conjecture explains why this works?
3.2 In introducing the theorem, let the students do the following investigations:
An altitude has been constructed to the hypotenuse of each right triangle below. This construction creates
two more right triangles, each within the original right triangle. For the cases below, calculate the measure
of the measure of the acute angles in each right triangle.
1. a = ?
b=?
c=?
2. a =?
b=?
c =?
3. a =?
b =?
c =?
b
b
a
a
a
C
b
c
c
Do you notice anything special about the two small right triangles in each large right triangle? How do the
two small right triangles compare with each other? How do they compare with the original right triangle? If
you have trouble visualizing how the three triangles in each case relate to another, study the figures below.
The two smaller right triangles have been separated from the original right triangle.
B
D
D
A
A
C
Let the students discuss their observation with groupmates.
B
B
C
3.3 Before introducing the Pythagorean Theorem, let the students investigate the floor tiles show I the figure:
i.
ii.
what geometric figures do they see?
Find any relationship among the Geometric figures
3.4 Ask the students to choose one challenging problem from a Geometry book, and solve it. Ask them to
Document it by writing about why they choose it, what made it challenging, what strategies they used to
solve it and what they learned from it.
Example: In the figure, LAND and MARK are similar rhombi. Find area of LAND
area of MARK
A
20
N
A
16
L
R
20
25
D
M
K
3.5 The students are confused with a slope of 0 for horizontal lines with no slope for vertical lines. Let them
verify this by drawing a vertical line parallel to the y-axis where x2 –x1 = 0 and x1 = xx. Let them discover
the reason why vertical lines do not have slope, because division by 0 is undefined and the formula
y2 – y1
does not apply.
X2 – x1
3.6 Why is it true if the three lengths 3,4 and 5 were put end to end they will form a right triangle, while other
Sets of three lengths like 7,8,9 or 3,5,7 will not produce the same result?
4. Values Education
4.1 Pose this problem: If you own an equilateral lot, how will you divide this into 4 equal parts using the basic
Proportionality theorem? Illustrate and explain why. If you are to donate this lot, to whom will give it and
why?
4.2 Let the students realize that similarity is important in solving sophisticated problems like the problems in
surveying and engineering. Similarity is the concept behind scale drawing, which make possible the
Construction of buildings and sky scrapers.
4.3 Ask the student to think you, “My science teacher says that we get a total eclipse of the sun because the
ratio of the moon’s diameter to its distance from the earth is about the same as the ratio of the sun’s
diameter to its distance to the earth. But I don’t understand how this works. How can you help your
friend? (Draw a diagram and use similar triangles, to explain how this works).
“PROBLEMS ARE MADE TO BE SOLVED, SO EVERY PROBLEM HAS A SOLUTION”
Ask the students to share with a partner whether this statement is true or not true for him and to explain
Why?
PHYTAGORAS (ca.540 B.C.)
Very little is known about the man Phytagoras. He was a leader of a secret society or school founded in
the 6th century B.C. This group, known as the Pythagoreans, considered it impious for a member to claim
any discovery for himself. Instead, each new idea was attributed to their founder, Pythagoras. This was
more than a school; it was a philosophy and a way of life. Every evening, each member of the
Pythagorean society had to reflect on three questions:
1. What good have I done?
2. Where have I failed?
3. What have I not done that I should have done?
Ask the students to reflect on the same questions. Every student can share his answers to a partner.
4.6 Ask the students to share with a classmate how the circle can be a symbol for community building
IX – Course Requirements
1. Class attendance and participation
2. Quarterly exams
3. Assignments, projects, unit tests and quizzes
X – Analytical Scoring Rubric on Making a Project
A) Strategic Intervention Materials
Criteria/Level
Time Element
10 pts.
Functionality/Workability
25 pts.
Usability
(Depends on type of
objective/project)
15 pts.
Resourcefullness/
Creativity
25 pts.
Application of Math
Concept/Principles
25 pts.
Total = 100
Exemplary
100%
Submits project
before the
deadline
Completely
working (all parts
are properly
fixed)
Lasting for
instructional
materials
Completely made
of locallyavailable/recycle
d materials
Utilized at least 3
(or more related
math concepts)
Standard
75%
Submits
project on the
deadline
Working
(not all parts
are properly
fixed)
Submits project
after deadline
Novice
25%
Submit projects
after several
reminders
Working for 2 or
more trials
There is an
attempt to work
no project
Short span of
time
Can be used
within the rating
period
Made of below
50%
locally/recycled
materials
Can not be used
as instructional
materials
Made of below
25%
locally/recycled
materials
no project
Utilized only 1
math concept or
principle
Not utilized the
math concept or
principle
Made of at
least 75% of
locally/recycle
d available
materials
Utilized only 2
math concepts
or principles
Apprentice
50%
No
Achievement
no project
no project
no project
Total
XI – Evaluation
15% Periodic Exams; 15% Unit Test; 15% Quizzes; 30% Class Participation; and 25%
Projects
XII – Performance Contract
The students are expected to fulfill all the requirements of the course including projects and
assignments. They are expected to give good analysis, interpreting, describing and proving of the
different geometric figures and relations of the classification of polygons considering the conditions
given that postulate and theorems.
XIII – Table of Specifications (TOS) 1st to 4th quarter
FIRST QUARTER
Learning Competencies
1. Undefined Terms
1.1 Describe the ideas of point line and plane
1.2 Describe, identify and name the subsets of a
line
2. Angles
2.1 Illustrate, name, identify and define an angle
2.2 Name and identify the parts of an angle
2.3 Read or determine the measure of an angle
using a protractor
2.4 Illustrate, name, identify and define different
kinds of angles
* acute, right and obtuse
3. Polygons
3.1 Illustrate, identify, and define different kinds
of polygons according to the number of sides
* Illustrate and identify convex and nonconvex polygons
* Identify the parts of a regular polygon
(vertex angle, central angle, exterior angle)
3.2 Illustrate name and identify a triangle and its
basic and secondary parts (e.g., vertices, sides,
angles, median, angle bisector, altitude)
3.3 Illustrate, name and identify different kinds of
triangles and their parts (e.g. legs, base,
hypotenuse)
3.4 Illustrate, name and define a quadrilateral
and its parts
3.5 Illustrate, name and identify the different
kinds of quadrilaterals
3.6 Determine the sum of the measures of the
interior and exterior angles of polygon
* sum of the measures of the angles of a
triangle is 180
* sum of the measures of the exterior angles
of a polygon is 360
* sum of the measures of the interior angles
of an n-sided convex polygon is (n-2) 180
4. Circle
4.1 Define a circle
4.2 Illustrate, name, identify, and define the
terms related to the circle (radius, diameter and
chord)
5. Measurement
5.1 Identify the following common solids and
their parts; cone, pyramid, sphere, cylinder,
rectangular prism
5.2 State and apply the formulas for the
measurement of plane and solid figures
* perimeter of a triangle, square and
rectangle
* circumference of a circle
5.3 State and apply the formulas for the
measurement of plane and solid figures
* area of a triangle, square, parallelogram,
trapezoid, and circle
* surface area of a cube, rectangular prism,
square pyramid, cylinder, cone and a
sphere
* volume of a rectangular prism, triangular
prism, pyramid, cylinder, cone and a sphere
6.6 Illustrate and identify the perpendicular bisector
of a segment
total
# of
Days
%
Wt
Re
me
mbe
ring
30%
Under
standi
ng
30%
Appl
ying
Analy
Zing
Crea
Ting
Evalua
ting
Total
20%
10%
5%
5%
100%
4
8
1
1
1
3
5
12
1
1
1
1
14
31
4
4
2
1
4
8
1
1
1
13
29
4
4
2
1
1
1
13
5
45
12
100
1
12
1
12
1
8
1
4
2
2
4
40
4
1
1
13
3
SECOND QUARTER
Learning Competencies
# of
Days
%
Wt
Re
me
mbe
ring
30%
Under
standi
ng
30%
Appl
ying
Analy
Zing
Crea
Ting
Evalua
ting
Total
20%
10%
5%
5%
100%
1. Angles and Sides of a triangle
1.1 Derive or apply the relationships among the
sides and angles of a triangle
* exterior and corresponding remote interior
angles of a triangle
* Pythagorean Theorem for Right Triangles
18
40
5
5
3
2
1
1
17
2. Angles Formed by Parallel Lines Cut by a
Transversal
2.1 Illustrate and define a transversal
2.2 Identify the angles formed by parallel lines
cut by a transversal
2.3 Determine the relationship between pairs of
angles formed by parallel lines cut by a transversal
* alternate interior angles
* alternate exterior angles
* corresponding angles
* angles on the same sides of the transversal
16
36
4
4
3
1
1
1
14
3. Conditions for Triangle Congruence
3.1 Define and illustrate congruent triangle
3.2 Apply deductive skills to show congruence
between triangle
* SSS Congruence * SAS congruence
* ASA congruence * SAA congruence
6
13
2
2
1
1
4. Applying the conditions for triangle congruence
4.1 Prove congruence and inequality properties
in an isosceles triangle using the congruence
conditions
* congruent sides in a triangle imply that the
angles opposite them are congruent
* congruent angles in a triangle imply that the
sides opposite them are congruent
* non-congruent angles in a triangle imply
that the sides opposite them are not
congruent
4.2 Use the definition of congruent triangles and
the conditions for triangle congruence to prove
segments and congruent angles between two
triangles
Total
5
11
1
1
1
45
100
12
12
8
4
2
2
40
# of
Days
%
Wt
Analy
Zing
Crea
Ting
Evalua
ting
Total
20%
1
10%
1
5%
5%
13
Under
standi
ng
30%
2
Appl
ying
6
Re
me
mbe
ring
30%
2
100%
6
5
11
1
1
1
10
22
3
3
2
1
12
27
3
3
2
1
6
3
THIRD QUARTER
Learning Competencies
1. Congruence of Right triangle
2. Different types of quadrilaterals and their
properties
2.1 Recall previous knowledge on the different
kinds of quadrilaterals and their properties (square,
rectangle, rhombus, trapezoid)
2.2 Apply inductive and deductive skills to derive
certain properties of the trapezoid
* median of a trapezoid
* base, angles and diagonals of an isosceles
trapezoid
3. Define similar triangles
3.1 each diagonals divides a parallelogram into
two congruent triangles
3.2 opposite angles are congruent
3.3 non-opposite angles are supplementary
3.4 opposite sides are congruent
4. Different types of quadrilaterals and their
properties
4.1 Apply inductive and deductive skills to derive
the properties of a parallelogram
* diagonals bisect each other
4.2 Apply inductive and deductive skills to derive
the properties of the diagonals of special
quadrilaterals
* diagonals of a rectangle
* diagonals of a square
* diagonals of a rhombus
3
9
1
1
11
5. Conditions that guarantee that a quadrilateral is
a parallelogram
5.1 verify sets of sufficient conditions which
guarantee that a quadrilateral is a parallelogram
5.2 Apply the conditions to prove that a
quadrilateral is a parallelogram
Total
12
27
3
3
2
1
1
1
11
45
100
12
12
8
4
2
2
40
# of
Days
%
Wt
Re
me
mbe
ring
30%
Under
standi
ng
30%
Appl
ying
Analy
Zing
Crea
Ting
Evalua
ting
Total
20%
10%
5%
5%
100%
FOURTH QUARTER
Learning Competencies
1. Similarity
1.1 State and apply the fundamental law of
proportion
* product of the means is equal to the
product of the extremes
1.2 Apply the definition of proportional segments
to find unknown lengths
1.3 State and verify the Basic Proportionality
Theorem and its Converse
1.4 Apply the definition of similar triangle
* determining if two triangles are similar
* finding the length of a side or measure of
an angle of a triangle
1.5 Apply the properties of similar triangles and
the proportionality theorems to calculate lengths of
certain line segments, and to arrive at other
properties
1.6 Apply the AA similarity theorem to determine
similarities in a right triangle
* In a right triangle the altitude to the
hypotenuse divides it into two right triangle
which are similar to each other and to the
given right triangle
1.7 Derive the relationships between the sides of
an isosceles triangle and between the sides of a
30-60-90 triangle using the pythagorean theorem
1.8 Apply knowledge and skills related to similar
Triangles to word problems
2. Circles
2.1 Define and identify a minor and major arc of
a circle
2.2 Determine the degree measure of an arc of a
circle
2.3 Determine the measure of an inscribed angle
2.4 State and apply the properties of a line
tangent to a circle
* if a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency
* If two segments from the same exterior
point are tangent to a circle, the two
segments are congruent
2.5 Illustrate and identify externally and internally
tangent circle illustrate and identify a common
internal tangent or a common external tangent
3. Plane Coordinate Geometry
3.1 Review of the Cartesian coordinate system,
linear equations and systems of linear equations in
2 variables
3.2 Name of parts of a Cartesian plane
3.3 Represent ordered pairs on the Cartesian
plane and denote points on the Cartesian plane
3.4 Define the slope of a line and compute for
the slope given the graph of a line
3.5 Apply the distance and midpoint formulas to
find or verify the lengths of segments and find
unknown vertices or points
3.6 Determine the equation of a circle given its
center and radius
Total
18
40
5
5
3
2
1
1
17
16
36
4
4
3
1
1
1
14
11
24
3
3
2
1
45
100
12
12
8
4
9
2
2
40
XIIV – PERIODIC EXAMS (1st to 4th quarter)
FIRST QUARTER
I – Matching Type: Match Column B with Column A. Write the letter of the correct answer on the space provided
before the number.
COLUMN A
COLUMN B
_____1. Has an infinite length and width but no thickness; is a flat surface
a) Angle bisector
_____ 2. A figure formed by two rays with a common endpoint and which
b) Sphere
are not on the same line
c) Median
_____ 3. The sum of the degree measures of the angles of a triangle
d) Geometry
_____4. Is a segment that divides any angle of a triangle Into two angles
e) Plane
of equal measures
f) The pi theorem
_____5. Is a segment drawn from any vertex of a triangle to the midpoint
g) trapezoid
of the opposite side
h) pyramid
_____6. Is a quadrilateral with exactly a pair of parallel sides
i) angle
_____7. Is an angle formed by two radii of a circle with its vertex in the
j) cube
center of the circle
k) central angle
_____8. A solid where very point of its square is equally distant from its center l) angle sum theorem
_____9. The ratio of the circumference to the measure of the diameter is the
m) linear pair
same for all circles
____ 10. A solid shape which has one base. It has 5 vertices, 5 faces and 8 edges.
____ 11. A solid shape having 6 square faces. It has 8 vertices and 12 edges.
_____12. Two angles which are adjacent and whose non-common sides are opposite rays. .
II – Identification: Identify each of the following then write the correct answer on the space provided before the
number.
_____13. Has no length, width or thickness; occupies no space
_____14. Is a closed figure made up of three or more line segments joined at their endpoints.
_____15. A quadrilateral with exactly one pair sides parallel.
_____16. a quadrilateral with two pairs of sides parallel and congruent.
_____17. A line segment which divides an angle into two angles of equal measures.
_____18. A line segment drawn from any vertex perpendicular to the opposite side of triangle.
_____19. Is a set of points on a given plane, which is equidistant from a fixed point called the center.
_____20. Defined as the sum of the areas of the outer surfaces of a solid
_____21. The amount surface in a region or plane.
_____22. Is used to measure space.
_____23. Three-dimensional figure.
_____24. A point on the segment that divides the segment into two congruent parts.
III – Practical Applications
25. Draw and label a picture of two lines l and m intersecting at point o in a plane.
26. Use the figure given below to measure the angle AOC using protractor.
27. A square garden is to be fenced. One side is 5 1/3 meters. How long is the fence needed to surround it on all sides.
28. A triangle has an area of 45 square centimeters and a base of 5 cm. What height corresponds to this base?
29. Draw and label, then give the measure of the angle determined by the hands of a clock at four o’clock.
30. Eddie is using a bed which is 2.0 meters long and 1.3 meters side. What is the size of the bed in square centimeters?
31. A cylindrical water tank is 2.2 meters high. If the radius of its base is 0.8 meter, what is the volume of the tank?
32. The measure of angle A is 9 more than twice the measure of angle B. If angle A and Angle B are supplementary angles,
what is the measure of angle A?
IV – Think of this. . . . then investigate
33. How many nonzero angles are formed in a fan of 30 rays
34. Figure given below. Find the shape of a rectangle and parallelogram then investigate why rectangle is not a square and
parallelogram is not a rectangle.
35. The two rhombuses at the right have the same perimeter. Knowing that diagonals of a rhombus are perpendicular and
bisect each other. Investigate the area of each rhombuses.
36. Draw and illustrate the measure of angles if two angles with the same measure are complementary.
V – Give what is asked:
A. Determine whether the statement is true or false. Then justify your answer (Refer to questions 37 and 38).
37. Three sides of a triangle measure 8 cm., 9 cm., and 10 cm., respectively. The triangle is an isosceles triangle. (Justify your
answer).
38. A circle whose diameter is 5 cm. has the same area of a circle whose radius is 2 ½ cm. (justify your answer).
B. Construct then prove the parallelograms are rectangles
39. Ronald is building a barn for his horse. Eddie measures the diagonals of the door opening to make sure they bisect each
other and they are congruent. How does Ronald know that the measure of each corner is 90?
C. Formulate and develop
40. Formulate the concept on how the volume of solids developed?
SECOND QUARTER
I – Matching Type: Match Column B with Column A. Write the letter of the correct answer on the space provided
before the number.
COLUMN A
COLUMN B
_____1. In a right triangle, the square of the length of the hypotenuse is
a) angle sum theorem
equal to the sum of the squares of the legs
b) parallel postulate
_____2. The sum of the measures of two sides of a triangle is greater than
c) congruent
the measure of the third side
d) exterior angle theorem
_____3. The sum of the measure of the angles in a triangle is 180 degrees
e) SAS congruence postulate
_____4. For any triangle, the measure of an exterior angle is equal to the sum
f) Converse of the isosceles triangle theorem
of the measures of its two remote interior angles.
g) Pythagorean theorem
_____5. Consider a line l and a point p not on the line. If PO is perpendicular
h) supplementary
to L at O and Q in any other point on L, then PO < PQ
i) transversal
_____6. A line that intersect two or more lines at different points.
j) triangle inequality
_____7. Given a line and a point not on the line, there is exactly one line
k) SAA congruence pos
through the point parallel to the given line.
l) Corollary
_____8. If two lines are cut by a transversal, then the exterior angles on the
m) isosceles triangle theorem.
same side of transversal are____
_____9. If two lines are cut by a transversal, then the alternate exterior angles are_____
____10. If two sides and the included angle of one triangle are congruent to two angles and a non-included side or another
triangle, then the two triangles are congruent.
____11. If two angles and a non-included side of one triangle are congruent to two angles and non-included side of another
triangle, then the two triangles are congruent.
____12. If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
II – Identification: Identify each of the following, then write the correct answer on the space provided before the
number.
_____13. Conditions of two lines that are parallel and do not intersect to each other.
_____14. A three-sided polygon
_____15. The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
_____16. A Greek mathematician and philosopher that introduced and developed the mathematical principle of Pythagorean
Theorem.
_____17. The Ration of the measure of an exterior angle and the measure of its two remote interior angle.
_____18. A mathematician that introduced elements and postulate of parallel and transversal lines.
_____19. A line that intersects two or more parallel lines.
_____20. If two parallel lines are cut by a transversal, then give the sum of the measures of the interior angles on the same
side of the transversal line.
_____21. If the parallel lines are cut by a transversal, then give the sum of the measures of the exterior angles on the same
side of a transversal line.
_____22. If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another
triangle, then the triangle are congruent
_____23. If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of
another triangle, then the triangle are congruent.
_____24. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
III – Practical Applications:
25) Consider the triangle. Draw and label a line through B parallel to AC
26) Show the relation between the sum of the measures of the two sides and the measure of the third side. Given the triangle
XYZ
27) If the measure of the two sides of a triangle are 6 and 7, respectively, what may be the measure of the third side?
28) Draw and label the two parallel lines, AB and CD with the transversal line t cutting AB and CD
29) In PQ is a transversal of parallel lines MN and RS with the alternate-interior angle of O and T. Name all the angles that are
equal to / PON.
30) Two sides of a triangle measure 10 and 12. Between what numbers must the measure of the third side?
31) Give the measure of each angle of equiangular triangle.
32) Give the measure of the base angles of an isosceles triangle whose vertex angle has a measure of 50 degrees?
IV – Think of this. . . then investigate
33) Two intersecting lines divide a plane into four regions. Determine the greatest number of regions into which a plane can be
divided by 3 lines, 4 lines, 10 lines, n lines.
34) Find the length of a diagonal, when the length and width of the rectangle are given 3 / 2 and 2 / 3.
35) In accompanying figure, m//n and s is a transversal. What is the measure of each numbered angle if m /1 = 120 degrees?
Give the reason for your answer.
36) For the overlapping congruent triangles given at the right, list the corresponding congruent parts.
V – Give what is asked:
A. Determine whether the statement is true or false. Then justify your answer (Refer to questions 37 and 38)
37) Is 3,6,10 can be the measures of the sides of a triangle? Justify your answer.
38) In a line l intersect line t forming perpendicular, then a line t represents transversal? Justify your answer.
B. Construct then prove
39) Given e//p and a is a transversal. What is the measure of each numbered angle if m /1 = 50degrees? Give the reason for
your answer.
C. Formulate and develop
40) Formulate the concept on how the Pythagorean theorem a2+b2=c2? Derived?
THIRD QUARTER
I – Matching Type: Match Column B with Column A. Write the letter of the correct answer on the space provided
before the number.
COLUMN A
COLUMN B
_____1. If a leg and an acute angle of one right triangle are congruent to a leg and
a) Quadrilateral
an acute angle of another right triangle then the triangles are congruent
b) Rectangle
_____2. If an acute angle and the hypotenuse of one right triangle are congruent.
c) CPCTC
to an acute angle and the hypotenuse of the median of another
d) HyA congruence theorem
right triangle, then the triangles are congruent
e) LA congruence theorem
_____3. Is a closed plane figure consisting of four line segments or sides.
f) Definition of a trapezoid
_____4. A quadrilateral with two pairs of opposite sides parallel
g) square
_____5. A parallelogram with all four sides congruent
h) Definition of vertical angles
_____6. A rectangle with all four sides congruent
i) Definition of Isosceles trapezoid
_____7. A quadrilateral with exactly one pair of opposite sides parallel
j) Rhombus
_____8. A parallelogram with four right angles
k) ASA congruence postulate
_____9. Corresponding parts of congruent triangles are congruent
l) Trapezoid
____10. The median of a trapezoid is parallel to the bases, and its length is equal to half
m) Definition of Alternate Interior
the sum of the lengths of the bases.
angles
____11. Opposite angles are congruent.
n) parallelogram
____12. If two lines one cut by a transversal so that alternate interior angles are congruent, then the lines are parallel
II – Identification: Identify each of the following, then write the correct answer on the space provided before the number.
________________13. If a leg and the hypotenuse of one right triangle are congruent to a corresponding leg and the
hypotenuse of another right triangle, then the triangles are congruent.
________________14. If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are
congruent.
________________15. A quadrilateral with two distinct pairs of adjacent, congruent sides.
________________16. Quadrilateral when two pairs of opposite sides are both parallel and congruent.
________________17. A quadrilateral when all sides are congruent.
________________18. A quadrilateral when two sides are congruent
________________19. A quadrilateral when two sides are parallel
________________20. The formula of the area of a kite.
________________21. The sum of the measures of the angles of a quadrilateral.
________________22. A quadrilateral when congruent diagonals each of which divides the figure into two congruent isosceles
right triangle.
________________23. In parallelogram ABCD, if m/ A = 120, what is the measure
of / B?
________________24. If m / T = m / S and PT = RS, then what kind of quadrilateral PRST?
III - PRACTICAL APPLICATIONS
25) Consider two right triangles ABC and MON with right angle B and O, respectively. Draw and mark the congruence triangle
of LL theorem.
26) Illustrate and label the isosceles trapezoid ABCD, where BA = CD and AC and BD are diagonals.
27) The area of a kite is 180 sq.cm. and the length of the diagonal is 36 cm. How long is the other diagonal?
28) Draw and illustrate a nonconvex kite XWYZ.
29) If the number of degrees in one angle of an isosceles trapezoid is x and the angle opposite it contains x+20, how many
degrees are there in each angle?
GIVEN AC and BD are diagonals of parallelogram ABCD. (Refer to questions 30 to 32)
30) Given a parallelogram ABCD above, if / 1 = 45 and / 2 = 30, what is the measure of / 3?
31) Parallelogram ABCD given above, what is the measure of / ABC if / 1 = 45, / 2 = 30 and / 4 = 60?
32) After solving in questions # 30, describe the angle of supplement / 3?
IV – THINK OF THIS . . . . then investigate
33) You are to find the distance from R to the inaccessible point I. How would you lay out RST so that the length of one of its
sides would give you the distance from R to I?
34) Consider the square ABCD given below, then describe the property of a square with respect to one of its diagonals.
35) A squared rectangle is a rectangle whose interior can be divided into 4 or more squares. The given number represents the
length of a side of that square. Determine the dimensions of the unlabelled squares.
36) Describe the property of a kite with respect to the lines formed by intersecting diagonals, given kite ABCD at the right
V – GIVE WHAT IS ASKED
A.
Determine whether the statement is true or false, then justify your answer (Refer to questions 37 and 38).
37) A trapezoid has three congruent sides, then justify your answer.
38) Consider the parallelogram BEST
describe the angle indicated in / S and / T.
B.
39)
Construct then prove
In a rhombus CALM with diagonal CL. Prove that each diagonal bisects opposite angles.
a)
b)
c)
illustration
Given: Rhombus CALM with diagonal CL.
Prove: / 1 = / 4, / 3 = / 2
Proof:
Statements
C.
40)
Reasons
Formulate and develop
Formulate on how the concept of properties of the quadrilaterals derived?
FOURTH GRADING
I – Matching Type: Match Column B with Column A. Write the letter of the correct answer on the space provided
before the number.
COLUMN A
______1) The equality of two ratios
______2) Are triangles whose corresponding angles are congruent and
whose corresponding sides are proportional
______3) A line which intersects the circle of two distinct points
______4) A segment in the plane of the circle which Intersects the circle at
exactly one point
______5) An angle whose vertex is on the circle and whose sides are chords of
of the circle
______6) Is the set of all points that are at a fixed distance from a fixed
point in the plane
______7) Is a segment whose endpoints are on the circle
______8) Is a chord containing the center of the circle
______9) In 30o-60o-90o triangle, the hypotenuse is twice as long as the shorter leg,
and the longer leg is / 3 times as long as the shorter leg.
_____10) The slope-intercept form of the equation of a line.
_____11) The point-slope form of the equation of a line
_____12) The two-point form of the equation of the line
COLUMN B
a) Tangent
b) Chord
c) y = mx + b
d) Secant
e) 30o-60o-90o theorem
f) ( x1 + x2 , y1 + y2 )
2
2
g) Similar triangles
h) y2 – y1 = m( x2 – x1)
i) diameter
j) Inscribed angle
k) proportion
l) circle
m) y – y1 = y2 – y1 (x – x1)
x2 – x1
II – IDENTIFICATION: Identify each of the following, then write the correct answer on the space provided before the number.
_______________13) Two triangles are similar if and only if their corresponding angles
are congruent.
_______________14) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are
similar.
_______________15) If the square of the length of the longest side of a triangle is
Equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
_______________16) A Greek geometrician who derived a formula for the area of a triangle in terms of the length of its sides.
_______________17) A line which intersects the circle to two distinct points.
_______________18) An angle whose vertex is the center of the circle
_______________19) If AB and BC are adjacent arcs, then mAB + mBC = mABC
_______________20) The measure of an inscribed angle is half the measure of its intercepted arc.
_______________21) The father of Modern Mathematics, who bridged the gap between Algebra and Geometry.
_______________22) The formula of the slope (m) of a non-vertical line containing two points with coordinates (x1, y1) and (x2,
y2).
_______________23) The slope-intercept form of the equation of a line where m is the slope of the line and b is the y-intercept.
_______________24) The distance formula between the points A1 (x1, y1) and A2 (x2, y2)
III – PRACTICAL APPLICATIONS
25) Consider the two triangles ABC and DEF. Draw and mark the similarity triangle of AAA Similarity Postulate.
26) Assume that triangle XYZ is similar to triangle RPN with X↔ R and P↔ Y. State three proportions that are true.
27) Sketch and label if the foot of a 10-meter ladder is placed 6 meters away from a building, how high up the ladder will the
ladder reach?
28) Draw, Illustrate and label in circle O, OP, OR and OT are radii, PS is a chord, PR is a diameter, PT is a secant, QR is a
tangent line, point N is an interior point in diameter PR, and point Q is an exterior point in tangent QR.
29) Let / ABC be inscribed in circle O. Show that in / ABC = ½ mAC
30) In the accompanying figure, mAB = 112, mBC = 54, and mCD = 88,
Find m/ 1, m/ 2, and m / 3.
31) Substitute the given points in the two-point form of the equation of a line. Given that (x1, y1) = (3,5) and (x2 ,y2 ) = ( -2, -4).
We have y – y1 = y2 – y1 (x – x1)
32) Sketch the graph and find the distance between the points with coordinates (-3,4) and (5, -2).
IV – THINK OF THIS . . . . . . . . then investigate
33) The areas of two similar polygons are 120 cm 2 and 30 cm2. If the smaller polygon are 120cm2 and 30 cm2. If a side of the
smaller polygon has a length of 3 cm., find the length of the corresponding side of the larger polygon.
34) Segments AC and PQ are divided proportionally by points B and R, respectively. If AB = 5, BC = 10, and RQ = 8, find PR.
35) In the accompanying figure, m UV = 80, m VT = 130, and m TS = 20. Find the measure of each angle.
36) Consider the equation x + 2y = 6. Make a table of values of x and y, then plot the ordered pairs. Connect them with a line,
then describe the graph.
V – GIVE WHAT IS ASKED
A.
Determine whether the statement is true or false, then justify your answer (Refer to questions 77 and 38)
37) All isosceles right triangles are similar, then justify your answer.
38) There are infinitely many lines tangent to a circle at a given point on the circle, then justify your answer.
B.
Construction then prove.
39) Prove that the two triangles EPA and BSV are similar given the three conditions that makes two triangles similar..
C.
Formulate and Develop
40) Formulate on how the Cartesian Coordinates system develop?
XV– References
Geometry Based on the 2002 BEC by Antonio C. Coronel
Geometry by Edwin E. Moise and Floyd L. Downs, Jr. (Metric Edition)
Prepared by:
EDDIE P. ANAJAO
SST-1
Approved:
MATILDE J. MANLIGUIS
Principal-IV