Your SlideShare is downloading. ×
0
GRADE 9 MATHEMATICS
QUARTER 3
Module 5
May 2014
FIGURE, PICTURE,
QUOTATION β€˜n ONE
Instruction:
Arrange the jigsaw puzzle to form a picture/figure and
paste it in the mani...
Competencies
1. Identifies quadrilaterals that are parallelograms.
2. Determines the conditions that guarantee a
quadrilat...
Parallelograms
Definition:
A quadrilateral whose opposite sides are parallel.
(Garces, et. Al., 2007)
Properties of Parallelograms
1. The diagonal divides it into two congruent triangles.
2. Opposite angles are congruent.
3....
Parallelograms
The diagonal divides it into
two congruent triangles.
Proof:
Given that β–‘ABCD is a parallelogram. We have t...
Parallelograms
Opposite angles are congruent.
Proof:
Given that β–‘ABCD is a parallelogram. We have to prove
that ∠𝐡𝐴𝐷 β‰… ∠𝐷𝐢...
Parallelograms
Consecutive angles are supplementary.
Proof:
Given that β–‘ABCD is a parallelogram. We have to prove
that ∠𝐴𝐡...
Parallelograms
Diagonals bisect each other.
B
A D
C
Proof:
Given that β–‘ABCD such that 𝐴𝑃 β‰… 𝐢𝑃 and 𝐡𝑃 β‰… 𝑃𝐷. We
to prove tha...
Parallelograms
Opposite sides are congruent.
Proof:
Given that β–‘ABCD such that 𝐴𝐡 β‰… 𝐢𝐷 and 𝐡𝐢 β‰… 𝐴𝐷. We
to prove that β–‘ABCD...
Parallelograms
The opposite sides are parallel
and congruent.
Proof:
Given that β–‘ABCD such that 𝐴𝐷 βˆ₯ 𝐡𝐢 and 𝐴𝐷 = 𝐡𝐢. We
to...
Parallelogram
Find the values of x and y in the parallelogram ABCD.
B
A
x0
C
y0
400
600
Answer: y = 1200 and x = 200
Parallelogram
In the figure below, β–‘AECG is a parallelogram. If 𝐡𝐸 β‰…
𝐺𝐷, prove that β–‘ABCD is also a parallelogram.
A
B C
D...
Statement Reason
1. Draw 𝐴𝐢 such that 𝐴𝐢 intersect 𝐸𝐺 at X 1. Construction
2. 𝐴𝐢 is diagonal to β–‘AECG and β–‘ABCD 2. Def. of...
Kinds of Parallelogram
Rectangle – All angles are congruent
Rhombus – All sides are congruent
Square – All angles and side...
Theorems on Rectangle
1. In a rectangle, the two diagonals are congruent.
Statements Reasons
1. β–‘ABCD is rectangle 1. Give...
Theorems on Rectangle
2. If one angle of a parallelogram is right, then it is a
rectangle.
Proof:
Given that ∠𝐴 of paralle...
Theorems on Rhombus
1. The diagonals of a rhombus are perpendicular and
bisect each other.
Given: 𝐴𝐢 and 𝐡𝐷 are diagonals ...
Midline Theorem
The segment that joins the midpoints of two sides of a
triangle is parallel to the third side and half as ...
Midline Theorem
The segment that joins the midpoints of two sides of a
triangle is parallel to the third side and half as ...
Theorems on Trapezoids
1. The median of a trapezoid is half of the sum of its
bases.
2. The base angles of an isosceles tr...
Theorems on kite
1. The diagonals are perpendicular to each other.
2. The area is half of the product of its diagonals.
Upcoming SlideShare
Loading in...5
×

Module5 dodong2

309

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
309
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
14
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Transcript of "Module5 dodong2"

  1. 1. GRADE 9 MATHEMATICS QUARTER 3 Module 5 May 2014
  2. 2. FIGURE, PICTURE, QUOTATION β€˜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.
  3. 3. Competencies 1. 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.
  4. 4. Parallelograms Definition: A quadrilateral whose opposite sides are parallel. (Garces, et. Al., 2007)
  5. 5. Properties of Parallelograms 1. 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.
  6. 6. Parallelograms The diagonal divides it into two congruent triangles. Proof: Given that β–‘ABCD is a parallelogram. We have to prove that βˆ†ABD β‰… βˆ†CDB. By definition , 𝐴𝐡 βˆ₯ 𝐢𝐷 and 𝐡𝐢 βˆ₯ 𝐴𝐷, then we can say that ∠𝐷𝐡𝐢 β‰… ∠𝐡𝐷𝐴 and ∠𝐢𝐷𝐡 β‰… ∠𝐴𝐡𝐷 by Alternate Interior Angles theorem. This implies that βˆ†ABD β‰… βˆ†CDB by ASA congruence theorem. B A D C
  7. 7. Parallelograms Opposite angles are congruent. Proof: Given that β–‘ABCD is a parallelogram. We have to prove that ∠𝐡𝐴𝐷 β‰… ∠𝐷𝐢𝐡. Because βˆ†ABD β‰… βˆ†CDB by ASA congruence theorem, then ∠𝐡𝐴𝐷 β‰… ∠𝐷𝐢𝐡 by CPCTC. B A D C
  8. 8. Parallelograms Consecutive angles are supplementary. Proof: Given that β–‘ABCD is a parallelogram. We have to prove that ∠𝐴𝐡𝐢 + ∠𝐡𝐢𝐷 = 180. Since the sum of interior angles of quadrilaterals is 3600 and the opposite angles of parallelogram are congruent, then we can say that 2∠𝐴𝐡𝐢 + 2∠𝐡𝐢𝐷 = 360. thus the last equation is ∠𝐴𝐡𝐢 + ∠𝐡𝐢𝐷 = 180
  9. 9. Parallelograms Diagonals bisect each other. B A D C Proof: Given that β–‘ABCD such that 𝐴𝑃 β‰… 𝐢𝑃 and 𝐡𝑃 β‰… 𝑃𝐷. We to prove that β–‘ABCD is a parallelogram. By Vertical Angle Theorem , βˆ π΄π‘ƒπ· β‰… βˆ π΅π‘ƒπΆ. This implies that βˆ†APDβ‰…βˆ†CPB by SASβ‰… theo., and βˆ π·π΄π‘ƒ β‰… βˆ π΅πΆπ‘ƒ by CPCTC. This means that 𝐴𝐷 βˆ₯ 𝐡𝐢. Similarly, 𝐴𝐡 βˆ₯ 𝐷𝐢. Hence β–‘ABCD is a parallelogram. QED P
  10. 10. Parallelograms Opposite sides are congruent. Proof: Given that β–‘ABCD such that 𝐴𝐡 β‰… 𝐢𝐷 and 𝐡𝐢 β‰… 𝐴𝐷. We to prove that β–‘ABCD is a parallelogram. Since 𝐴𝐡 β‰… 𝐢𝐷 and 𝐡𝐢 β‰… 𝐴𝐷, then we can say that βˆ†ABDβ‰…βˆ†CDB by SSSβ‰… theo. 𝐡𝐷 is transversal line 𝐴𝐷 and 𝐡𝐢, and ∠𝐷𝐡𝐢 β‰… ∠𝐴𝐷𝐡 (CPCTC) then 𝐴𝐡 βˆ₯ 𝐡𝐢 by the Converse of Alternate Interior Angles. Hence β–‘ABCD is a parallelogram. QED B A D C
  11. 11. Parallelograms The opposite sides are parallel and congruent. Proof: Given that β–‘ABCD such that 𝐴𝐷 βˆ₯ 𝐡𝐢 and 𝐴𝐷 = 𝐡𝐢. We to prove that β–‘ABCD is a parallelogram. The diagonal BD is a transversal of 𝐴𝐷 and 𝐡𝐢. And by Alternate Interior angle theo., ∠𝐴𝐷𝐡 β‰… ∠𝐢𝐡𝐷. 𝐡𝐷 is a common side between βˆ†ADB and βˆ†CBD which are congruent by ASA. By CPCTC, AB=CD. Therefore, β–‘ABCD is a parallelogram. B A D C
  12. 12. Parallelogram Find the values of x and y in the parallelogram ABCD. B A x0 C y0 400 600 Answer: y = 1200 and x = 200
  13. 13. Parallelogram In the figure below, β–‘AECG is a parallelogram. If 𝐡𝐸 β‰… 𝐺𝐷, prove that β–‘ABCD is also a parallelogram. A B C D E G
  14. 14. Statement Reason 1. Draw 𝐴𝐢 such that 𝐴𝐢 intersect 𝐸𝐺 at X 1. Construction 2. 𝐴𝐢 is diagonal to β–‘AECG and β–‘ABCD 2. Def. of diagonal line 3. 𝐸𝑋 β‰… 𝑋𝐺 3. Prop. of parallelogram 4. 𝐡𝐸 β‰… 𝐺𝐷 4. Given 5. 𝐡𝑋 β‰… 𝐡𝐸 + 𝐸𝑋; 𝐷𝑋 β‰… 𝑋𝐺 + 𝐺𝐷 5. Segment Addition Postulate 6. 𝐡𝑋 β‰… 𝐡𝐸 + 𝐸𝑋; 𝐷𝑋 β‰… 𝐸𝑋 + 𝐡𝐸 6. Substitution 7. 𝐡𝑋 β‰… 𝐷𝑋 7. Transitive PE 8. 𝐴𝑋 β‰… 𝑋𝐢 8. Prop of Parallelogram (β–‘AECG) 9. ∴ β–‘ABCD is a parallelogram 9. Prop of Parallelogram (7 and 8) x
  15. 15. Kinds of Parallelogram Rectangle – All angles are congruent Rhombus – All sides are congruent Square – All angles and sides are congruent
  16. 16. Theorems on Rectangle 1. In a rectangle, the two diagonals are congruent. Statements Reasons 1. β–‘ABCD is rectangle 1. Given 2. ∠𝐴𝐡𝐢 β‰… ∠𝐢𝐷𝐴 β‰… ∠𝐡𝐴𝐷 β‰… ∠𝐡𝐢𝐷 2. Def. of rectangle 3. 𝐴𝐡 β‰… 𝐷𝐢, 𝐡𝐢 β‰… 𝐴𝐷 3. Prop. Of rectangle 4. βˆ†ABCβ‰… βˆ†πΆπ·π΄ β‰… βˆ†π΄π΅π· β‰…βˆ†DCB 4. SAS β‰… Theorem 5. 𝐡𝐷 β‰… 𝐴𝐢 5. CPCTC A D C B Given: β–‘ABCD is a rectangle Prove: 𝐡𝐷 β‰… 𝐴𝐢
  17. 17. Theorems on Rectangle 2. If one angle of a parallelogram is right, then it is a rectangle. Proof: Given that ∠𝐴 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 ∠𝐴 and ∠𝐡 900 since they are supplementary. From the properties of parallelogram that the opposite angles are congruent, then ∠𝐴 and ∠𝐢 are congruent. And since ∠𝐡 is opposite to ∠𝐷. Therefore parallelogram ABCD is a rectangle.
  18. 18. Theorems on Rhombus 1. The diagonals of a rhombus are perpendicular and bisect each other. Given: 𝐴𝐢 and 𝐡𝐷 are diagonals intersect at P. 𝐴𝐢 βŠ₯ 𝐡𝐷, 𝐴𝑃 β‰… 𝑃𝐢, 𝐡𝑃 β‰… 𝑃𝐷. Prove: β–‘ABCD is a rhombus. A B C D P Statements Reasons 1. 𝐴𝐢 βŠ₯ 𝐡𝐷 1. Given 2. βˆ π΄π‘ƒπ΅ β‰… βˆ πΆπ‘ƒπ· β‰… βˆ πΆπ‘ƒπ΅ β‰… βˆ π΄π‘ƒπ· 2. Def. of βŠ₯ 3. 𝐴𝑃 β‰… 𝑃𝐢, 𝐡𝑃 β‰… 𝑃𝐷 3. Given 4. βˆ†APB β‰… βˆ†πΆπ‘ƒπ΅ β‰… βˆ†πΆπ‘ƒπ· β‰…βˆ†APD 4. SAS β‰… Theorem 5. 𝐴𝐡 β‰… 𝐡𝐢 β‰… 𝐢𝐷 β‰… 𝐷𝐴 5. CPCTC 6. ∴ β–‘ABCD is a rhombus 6. Prop of rhombus
  19. 19. Midline Theorem The segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long. Given: In βˆ†ABC, P and Q are midpoints of 𝐴𝐡 and 𝐴𝐢, respectively. Prove: 𝑃𝑄 βˆ₯ 𝐡𝐢 and PQ= 1 2 𝐡𝐢. A P B Q C R Proof: Let R be a point in 𝑃𝑄 such that 𝑃𝑄 β‰… 𝑄𝑅. Since Q is midpoint 𝐴𝐢, 𝐴𝑃 β‰… 𝑃𝐡. And by Vertical Angle Theom., βˆ π΄π‘„π‘ƒ β‰… βˆ π‘…π‘„πΆ, so βˆ†AQPβ‰…βˆ†CQR by SAS cong. Theo. which implies that βˆ π‘ƒπ΄π‘„ β‰… βˆ π‘„πΆπ‘… and 𝐴𝑃 β‰… 𝑅𝐢 by CPCTC. This means that β–‘PBCR is parallelogram because 𝐴𝐡 βˆ₯ 𝑅𝐢 (converse of AIA) and 𝑃𝐡 β‰… 𝑅𝐢 by transitivity PE. Therefore 𝑃𝑄 βˆ₯ 𝐡𝐢.
  20. 20. Midline Theorem The segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long. Given: In βˆ†ABC, P and Q are midpoints of 𝐴𝐡 and 𝐴𝐢, respectively. Prove: 𝑃𝑄 βˆ₯ 𝐡𝐢 and PQ= 1 2 𝐡𝐢. A P B Q C R Since β–‘PBCR is parallelogram, 𝑃𝑅 β‰… 𝐡𝐢. We can say that PR=PQ+QR by Segment Addition postulate. Since PQ=QR and PR=BC, then BC=2PQ. Simplify, the results will be PQ= 1 2 𝐡𝐢
  21. 21. Theorems on Trapezoids 1. 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.
  22. 22. Theorems on kite 1. The diagonals are perpendicular to each other. 2. The area is half of the product of its diagonals.
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

Γ—