GRADE 9 MATHEMATICSQUARTER 3Module 5

May 2014

FIGURE, PICTURE, QUOTATION ‘n ONE

Instruction:Arrange the jigsaw puzzle to form a picture/figure and paste it in the manila paper. Shout your yell after you made this activity. Present the puzzle, describe the figure and explain briefly the Mathematics quotation found therein.

Competencies1. Identifies quadrilaterals that are

parallelograms.2. Determines the conditions that guarantee a

quadrilateral a parallelogram.3. Uses properties to find measures of angles,

sides and other quantities involving parallelograms.

4. Proves theorems on the different kinds of parallelograms (rectangle, rhombus, square)

5. Proves the Midline Theorem.6. Proves theorems on trapezoids and kites.7. Solves problems involving parallelograms,

trapezoids and kites.

ParallelogramsDefinition:A quadrilateral whose opposite sides are parallel. (Garces, et. Al., 2007)

Properties of Parallelograms1. The diagonal divides it into two congruent

triangles.2. Opposite angles are congruent.3. Two consecutive angles are supplementary.4. The diagonals bisect each other.5. Opposite sides are congruent.6. The opposites sides are parallel and

congruent.

ParallelogramsThe diagonal divides it into two congruent triangles.

Proof:Given that □ABCD is a parallelogram. We have to prove that ∆ABD ∆CDB.By definition , and , then we can say that and by Alternate Interior Angles theorem. This implies that ∆ABD ∆CDB by ASA congruence theorem.

B

A D

C

ParallelogramsOpposite angles are congruent.

Proof:Given that □ABCD is a parallelogram. We have to prove that .Because ∆ABD ∆CDB by ASA congruence theorem, then by CPCTC.

B

A D

C

ParallelogramsConsecutive angles are supplementary.

Proof:Given that □ABCD is a parallelogram. We have to prove that .Since the sum of interior angles of quadrilaterals is 3600 and the opposite angles of parallelogram are congruent, then we can say that . thus the last equation is

ParallelogramsDiagonals bisect each other.

B

A D

C

Proof:Given that □ABCD such that and . We to prove that □ABCD is a parallelogram.By Vertical Angle Theorem , . This implies that ∆APD∆CPB by SAS theo., and by CPCTC. This means that . Similarly, . Hence □ABCD is a parallelogram. QED

P

ParallelogramsOpposite sides are congruent.

Proof:Given that □ABCD such that and . We to prove that □ABCD is a parallelogram.Since and , then we can say that ∆ABD∆CDB by SSS theo. is transversal line and , and (CPCTC) then by the Converse of Alternate Interior Angles. Hence □ABCD is a parallelogram.QED

B

A D

C

ParallelogramsThe opposite sides are parallel and congruent.

Proof:Given that □ABCD such that and We to prove that □ABCD is a parallelogram.The diagonal BD is a transversal of and And by Alternate Interior angle theo., . is a common side between ∆ADB and ∆CBD which are congruent by ASA. By CPCTC, AB=CD. Therefore, □ABCD is a parallelogram.

B

A D

C

ParallelogramFind the values of x and y in the parallelogram

ABCD.

B

Ax0

C

y0

400 60

0

Answer: y = 1200 and x = 200

ParallelogramIn the figure below, □AECG is a parallelogram. If , prove that □ABCD is also a parallelogram.

A

B C

D

E

G

Statement Reason

1. Draw such that intersect at X 1. Construction

2. is diagonal to □AECG and □ABCD 2. Def. of diagonal line

3. 3. Prop. of parallelogram

4. 4. Given

5. ; 5. Segment Addition Postulate

6. ; 6. Substitution

7. 7. Transitive PE

8. 8. Prop of Parallelogram (□AECG)

9. is a parallelogram 9. Prop of Parallelogram (7 and 8)

x

Kinds of ParallelogramRectangle – All angles are congruentRhombus – All sides are congruentSquare – All angles and sides are congruent

Theorems on Rectangle1. In a rectangle, the two diagonals are

congruent.

Statements Reasons

1. □ABCD is rectangle 1. Given

2. 2. Def. of rectangle

3. 3. Prop. Of rectangle

4. ∆ABC∆DCB 4. SAS Theorem

5. 5. CPCTC

A D

CB

Given: □ABCD is a rectangleProve:

Theorems on Rectangle2. If one angle of a parallelogram is right, then it is a rectangle.

Proof:Given that of parallelogram ABCD is right, we need to prove that the parallelogram is rectangle.

Since the consecutive angles of a parallelogram are supplementary, we can say and 900 since they are supplementary. From the properties of parallelogram that the opposite angles are congruent, then and are congruent. And since is opposite to . Therefore parallelogram ABCD is a rectangle.

Theorems on Rhombus1. The diagonals of a rhombus are perpendicular and bisect each other.

Given: and are diagonals intersect at P. , .Prove: □ABCD is a rhombus.

A

B C

D

P

Statements Reasons

1. 1. Given

2. 2. Def. of

3. , 3. Given

4. ∆A∆APD 4. SAS Theorem

5. 5. CPCTC

6. □ABCD is a rhombus 6. Prop of rhombus

Midline TheoremThe segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long.

Given: In ∆ABC, P and Q are midpoints of and respectively.Prove: and PQ=.

A

P

B

Q

C

R

Proof: Let R be a point in such that . Since Q is midpoint . And by Vertical AngleTheom., so ∆AQP∆CQR by SAS cong. Theo.which implies that by CPCTC. This means that □PBCR is parallelogram because (converse of AIA) and by transitivity PE. Therefore .

Midline TheoremThe segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long.

Given: In ∆ABC, P and Q are midpoints of and respectively.Prove: and PQ=.

A

P

B

Q

C

R

Since □PBCR is parallelogram, . We can say thatPR=PQ+QR by Segment Addition postulate. Since PQ=QR and PR=BC, then BC=2PQ. Simplify, the results will be PQ=

Theorems on Trapezoids1. The median of a trapezoid is half of the sum

of its bases.2. The base angles of an isosceles trapezoid

are congruent.3. Opposite angles of an isosceles trapezoid

are supplementary.4. The diagonals of an isosceles trapezoid are

congruent.

Theorems on kite1. The diagonals are perpendicular to each

other.2. The area is half of the product of its

diagonals.