Grade -9 NCERT -Maths Made Easy

Maths Made Easy Suresh, Aug 19, 2019

NCERT GRARDE 9 6. LINES AND ANGLES 7. TRIANGLES 8. QUADRILATERALS 9. AREAS OF PARALLELOGRAMS AND TRIANGLES 10. CIRCLES 13. SURFACE AREAS AND VOLUMES

Table of Contents

  1. Writing Prompts: Definitions & Theorems
    1. Congruent Segments: Quick Exploration
    2. Congruent Angles: Definition
    3. Midpoint Definition
    4. Segment Bisector Definition
    5. Perpendicular Bisector Definition
    6. Perpendicular Bisector Definition
    7. Angle Bisector Definition (I)
    8. Complementary Meaning?
    9. Supplementary Angles (Quick Exploration)
    10. Triangle Altitude Illustrator and Definition Writing Prompt
    11. Triangle Median Illustrator & Definition Writing Prompt
    12. Equilateral Action!!!
    13. Equiangular Action!!!
    14. Regular Polygon Action!
  2. Parallel Lines & Related Angles
    1. Vertical Angles Theorem
    2. Exploring Corresponding Angles (V2)
    3. Alternate Interior Angles Theorem (V1)
    4. Exploring Alternate Interior Angles (V2)
    5. Alternate Interior Angles Theorem (V3)
    6. Alternate Exterior Angles Theorem (V1)
    7. Exploring Alternate Exterior Angles (V2)
    8. Alternate Exterior Angles Theorem (V3)
    9. Same Side Interior Angles Theorem
    10. Exploring Same Side Exterior Angles
    11. Solution to Parallel Lines & Angles Problem
  3. Polygons & Angles: Interior & Exterior Angles
    1. Triangle Angle Theorems
    2. Triangle Exterior Angle
    3. Angles of a Right Triangle
    4. Quadrilateral Angle Theorems
    5. Pentagon Angle Theorems
    6. Hexagon Angle Theorems
    7. Heptagon Angle Theorems
    8. Octagon Angle Theorems
  4. Quadrilaterals: Theorems & Properties
    1. Parallelogram: Theorem 1
    2. Parallelogram: Theorem 2
    3. Parallelogram: Theorem (3)
    4. Parallelograms (I)
    5. Parallelogram Diagonals
    6. Quad Midpoints Action!
    7. Rhombus Action!
    8. Rhombus Diagonals
    9. Rectangle Action!
    10. Rectangle Diagonals
    11. Square Action!
    12. Isosceles Trapezoid Action!
    13. Trapezoid Median: 2 Discoveries
    14. Isosceles Trapezoid Diagonals
    15. Isosceles Trapezoid Midpoints
    16. Kite Action!
  5. Triangles: Theorems & Properties
    1. SAS: Dynamic Proof!
    2. SSS: Dynamic Proof!
    3. SSS: Dynamically Illustrated
    4. ASA Theorem?
    5. Similar Right Triangles (V1)
    6. Similar Right Triangles (V2)
    7. AA Similarity Theorem
    8. SAS ~ Theorem
    9. SSS ~ Theorem (V1)
    10. SSS ~ Theorem (V2)
    11. Parallel Lines Proportionality Theorem
    12. Triangle-Angle Bisector Theorem
    13. A Special Triangle & Its Properties (I)
    14. Converse of IST (V1)
    15. Isosceles Triangle Medians
    16. Triangle Midsegment Action!
    17. Triangle Midsegment
    18. Another Special Triangle and its Properties (II)
    19. Converse of IST (II)
    20. Incenter & Incircle Action!
    21. Incenter + Incircle Action (V2)!
    22. Medians and Centroid Dance
    23. Circumcenter & Circumcircle Action!
    24. Midpoint of HYP
    25. 3 Special Points!
  6. Circle Theorems
    1. Congruent Circles: Definition
    2. Properties of Tangents Drawn to Circles (A)
    3. Properties of Tangents Drawn to Circles (B)
    4. Diameters & Chords I (VA)
    5. Diameters & Chords I (VB)
    6. Congruent Chords (I)
    7. Inscribed Angle Theorem (V1)
    8. Inscribed Angle Theorem Dance: Take 2!
    9. Inscribed Angle Theorem: Corollary 1
    10. Thales' Theorem (VA)
    11. Cyclic Quadrilaterals (IAT: Corollary 3)
    12. Cyclic Quadrilateral: Proof Hint
    13. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
    14. Angle Formed by Chord & Tangent (I)
    15. Crossing Chords: Proof Hint
    16. Congruent Chords Action (A)
    17. Equidistant Chord Action
    18. Butterfly Theorem Action!
  7. Geometric Constructions
    1. Isosceles Triangle (Quick Construction Technique) V1
    2. Equilateral Triangle Construction (Dynamic Illustration)
    3. Congruent Angle Construction Action!
    4. Angle Bisector Construction Action!
    5. Alternate Angle Bisector Construction
    6. Angle Trisection by Paper Folding!
  8. Transformations
    1. Dilating a Point (Intro)
    2. Dilating a Line: HSG.SRT.A.1.A
    3. Dilating a Segment: HSG.SRT.A.1.B
    4. Properties of Dilations
  9. Area, Surface Area, Volume
    1. Parallelogram: Area
    2. Triangle Area Action! (V1)
    3. Area of a Triangle (Discovery)
    4. Trapezoid: Area (I)
    5. Trapezoid: Area (2)
    6. Area of a Trapezoid (Discovery)
    7. Right Circular Cylinder!

1.

Writing Prompts: Definitions & Theorems

  • 1. Congruent Segments: Quick Exploration
  • 2. Congruent Angles: Definition
  • 3. Midpoint Definition
  • 4. Segment Bisector Definition
  • 5. Perpendicular Bisector Definition
  • 6. Perpendicular Bisector Definition
  • 7. Angle Bisector Definition (I)
  • 8. Complementary Meaning?
  • 9. Supplementary Angles (Quick Exploration)
  • 10. Triangle Altitude Illustrator and Definition Writing Prompt
  • 11. Triangle Median Illustrator & Definition Writing Prompt
  • 12. Equilateral Action!!!
  • 13. Equiangular Action!!!
  • 14. Regular Polygon Action!

1.1.

Congruent Segments: Quick Exploration

[color=#000000]The following app illustrates what it means for segments to be classified as [b]congruent segments.  [br][/b][/color]
Interact with the applet below for a few minutes. Be sure to move the LARGE POINTS around each time before you drag the slider! Then answer the questions that follow.
What transformations have we learned about so far? List them.
What transformations did you observe here while interacting with this app?
What does it mean for segments to be classified as [b]congruent segments[/b]? Explain.
– Tim Brzezinski

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

1.11.

–

1.12.

–

1.13.

–

1.14.

–

2.

Parallel Lines & Related Angles

  • 1. Vertical Angles Theorem
  • 2. Exploring Corresponding Angles (V2)
  • 3. Alternate Interior Angles Theorem (V1)
  • 4. Exploring Alternate Interior Angles (V2)
  • 5. Alternate Interior Angles Theorem (V3)
  • 6. Alternate Exterior Angles Theorem (V1)
  • 7. Exploring Alternate Exterior Angles (V2)
  • 8. Alternate Exterior Angles Theorem (V3)
  • 9. Same Side Interior Angles Theorem
  • 10. Exploring Same Side Exterior Angles
  • 11. Solution to Parallel Lines & Angles Problem

2.1.

Vertical Angles Theorem

[b]Definition:[/b] [b][color=#b20ea8]Vertical Angles[/color][/b] are angles whose sides form 2 pairs of opposite rays. [br][br]When 2 lines intersect, 2 pairs of vertical angles are formed. [color=#b20ea8]One pair of vertical angles is shown below. [/color] [br][color=#888](Click the other checkbox on the right to display the other pair of vertical angles.) [/color][br][br]Interact with the following applet for a few minutes, then answer the questions that follow.
Directions & Questions: [br][br]1) Complete the following statement (based upon your observations). [br] [br] [color=#b20ea8][b]Vertical angles are always __________________________.[/b] [/color] [br][br]2) Suppose the pink angle above measures 140 degrees. What would the measure of its vertical angle? What would be the measure of the other 2 (gray) angles?
– Tim Brzezinski

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

2.8.

–

2.9.

–

2.10.

–

2.11.

–

3.

Polygons & Angles: Interior & Exterior Angles

  • 1. Triangle Angle Theorems
  • 2. Triangle Exterior Angle
  • 3. Angles of a Right Triangle
  • 4. Quadrilateral Angle Theorems
  • 5. Pentagon Angle Theorems
  • 6. Hexagon Angle Theorems
  • 7. Heptagon Angle Theorems
  • 8. Octagon Angle Theorems

3.1.

Triangle Angle Theorems

Interact with the app below for a few minutes.  [br]Then, answer the questions that follow.  [br][br]Be sure to change the locations of this triangle's vertices each time [i]before[/i] you drag the slider!
What is the [b]sum of the measures of the interior angles of this triangle? [/b]
What is the [b]sum of the measures of the exterior angles [/b]of this triangle?
– Tim Brzezinski

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

4.

Quadrilaterals: Theorems & Properties

  • 1. Parallelogram: Theorem 1
  • 2. Parallelogram: Theorem 2
  • 3. Parallelogram: Theorem (3)
  • 4. Parallelograms (I)
  • 5. Parallelogram Diagonals
  • 6. Quad Midpoints Action!
  • 7. Rhombus Action!
  • 8. Rhombus Diagonals
  • 9. Rectangle Action!
  • 10. Rectangle Diagonals
  • 11. Square Action!
  • 12. Isosceles Trapezoid Action!
  • 13. Trapezoid Median: 2 Discoveries
  • 14. Isosceles Trapezoid Diagonals
  • 15. Isosceles Trapezoid Midpoints
  • 16. Kite Action!

4.1.

Parallelogram: Theorem 1

Interact with the applet below for a few minutes. [br]Then, answer the questions that follow. [br][br]Feel free to move the BIG WHITE POINTS anywhere you'd like! [br][color=#ff00ff]You can also adjust the size of the pink angle by using the slider. [/color]
1.
What special type of quadrilateral was formed in the first half of your sliding-the-slider? How do you know this?
2.
What else can you conclude about this special type of quadrilateral? Be specific!
3.
Write a 2-column, paragraph, or coordinate geometry proof of what you've informally observed here. (Hint: If you choose a 2-column or paragraph proof, this proof will involve a pair of congruent triangles!)
Quick Demo: 0:00 sec to 0:33 sec (BGM: Andy Hunter)
– Tim Brzezinski

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

4.10.

–

4.11.

–

4.12.

–

4.13.

–

4.14.

–

4.15.

–

4.16.

–

5.

Triangles: Theorems & Properties

  • 1. SAS: Dynamic Proof!
  • 2. SSS: Dynamic Proof!
  • 3. SSS: Dynamically Illustrated
  • 4. ASA Theorem?
  • 5. Similar Right Triangles (V1)
  • 6. Similar Right Triangles (V2)
  • 7. AA Similarity Theorem
  • 8. SAS ~ Theorem
  • 9. SSS ~ Theorem (V1)
  • 10. SSS ~ Theorem (V2)
  • 11. Parallel Lines Proportionality Theorem
  • 12. Triangle-Angle Bisector Theorem
  • 13. A Special Triangle & Its Properties (I)
  • 14. Converse of IST (V1)
  • 15. Isosceles Triangle Medians
  • 16. Triangle Midsegment Action!
  • 17. Triangle Midsegment
  • 18. Another Special Triangle and its Properties (II)
  • 19. Converse of IST (II)
  • 20. Incenter & Incircle Action!
  • 21. Incenter + Incircle Action (V2)!
  • 22. Medians and Centroid Dance
  • 23. Circumcenter & Circumcircle Action!
  • 24. Midpoint of HYP
  • 25. 3 Special Points!

5.1.

SAS: Dynamic Proof!

[color=#000000]The [/color][b][u][color=#0000ff]SAS Triangle Congruence Theorem[/color][/u][/b][color=#000000] states that [/color][b][color=#000000]if 2 sides [/color][color=#000000]and their [/color][color=#ff00ff]included angle [/color][color=#000000]of one triangle are congruent to 2 sides and their [/color][color=#ff00ff]included angle [/color][color=#000000]of another triangle, then those triangles are congruent. [/color][/b][color=#000000]The applet below uses transformational geometry to dynamically prove this very theorem. [br][br][/color][color=#000000]Interact with this applet below for a few minutes, then answer the questions that follow. [br][/color][color=#000000]As you do, feel free to move the [b]BIG WHITE POINTS[/b] anywhere you'd like on the screen! [/color]
Q1:
What geometry transformations did you observe in the applet above? List them.
Q2:
What common trait do all these transformations (you listed in your response to (1)) have?
Q3:
Go to [url=https://www.geogebra.org/m/d9HrmyAp#chapter/74321]this link[/url] and complete the first 5 exercises in this GeoGebra Book chapter.
Quick (Silent) Demo
– Tim Brzezinski

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

5.11.

–

5.12.

–

5.13.

–

5.14.

–

5.15.

–

5.16.

–

5.17.

–

5.18.

–

5.19.

–

5.20.

–

5.21.

–

5.22.

–

5.23.

–

5.24.

–

5.25.

–

6.

Circle Theorems

  • 1. Congruent Circles: Definition
  • 2. Properties of Tangents Drawn to Circles (A)
  • 3. Properties of Tangents Drawn to Circles (B)
  • 4. Diameters & Chords I (VA)
  • 5. Diameters & Chords I (VB)
  • 6. Congruent Chords (I)
  • 7. Inscribed Angle Theorem (V1)
  • 8. Inscribed Angle Theorem Dance: Take 2!
  • 9. Inscribed Angle Theorem: Corollary 1
  • 10. Thales' Theorem (VA)
  • 11. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 12. Cyclic Quadrilateral: Proof Hint
  • 13. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
  • 14. Angle Formed by Chord & Tangent (I)
  • 15. Crossing Chords: Proof Hint
  • 16. Congruent Chords Action (A)
  • 17. Equidistant Chord Action
  • 18. Butterfly Theorem Action!

6.1.

Congruent Circles: Definition

[color=#000000]The applet below demonstrates what it means for 2 circles to be [b]congruent circles. [/b] [br]Interact with this applet for a minute or two, then answer the writing prompt that follows. [br][i]Be sure to change the locations of the points around each time before re-sliding the slider.[/i][/color]
Complete the following sentence definition: [br][br]Two circles are said to be congruent circles [color=#1e84cc][b]if and only if...[/b][/color]
– Tim Brzezinski

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

6.13.

–

6.14.

–

6.15.

–

6.16.

–

6.17.

–

6.18.

–

7.

Geometric Constructions

  • 1. Isosceles Triangle (Quick Construction Technique) V1
  • 2. Equilateral Triangle Construction (Dynamic Illustration)
  • 3. Congruent Angle Construction Action!
  • 4. Angle Bisector Construction Action!
  • 5. Alternate Angle Bisector Construction
  • 6. Angle Trisection by Paper Folding!

7.1.

Isosceles Triangle (Quick Construction Technique) V1

The applet below illustrates a quick (and easy) way to construct an [b][color=#38761d]isosceles triangle[/color][/b] using a [b]compass and straightedge.[/b] [br][br][br][b][color=#0000ff]Can you explain why is this method of construction valid? [/color][/b]
– Tim Brzezinski

7.2.

–

7.3.

–

7.4.

–

7.5.

–

7.6.

–

8.

Transformations

  • 1. Dilating a Point (Intro)
  • 2. Dilating a Line: HSG.SRT.A.1.A
  • 3. Dilating a Segment: HSG.SRT.A.1.B
  • 4. Properties of Dilations

8.1.

Dilating a Point (Intro)

[color=#000000]The following applet illustrates what it means to dilate a point about another point.  [br][br]You can move [b]point [i]O [/i](the center of dilation)[/b] and [/color][color=#ff0000][b]point [i]A[/i][/b][/color][color=#000000] anywhere in the plane.  [br]You can also change the [b]scale factor ([i]k[/i])[/b] of this dilation by either moving the slider or[br]by typing it in the white box at the top of the applet.[br][/color][b][color=#980000][i]A' = [/i]the image of point [i]A[/i] under dilation about point [i]O[/i] with scale factor [i]k[/i].  [br][/color][/b][br][color=#000000][b]Interact with the applet below for a few minutes [i]BEFORE[/i] clicking the "Check This Out!" checkbox in the lower right corner. [/b] [i]After interacting with this applet for a bit, please answer the questions that follow the applet.  (You'll be prompted to click the "Check This Out!" checkbox in the directions below.)  [/i][/color]
[color=#000000][b]Questions:  (Please don't click the "Check This Out!" box yet!)  [/b][/color][br][br][color=#000000]1) What vocab term would you use to describe the locations of point [i]A[/i] and [i]A'[/i] with [br]    respect to [i]O[/i]?  In essence, fill in the blank:  "The [/color][color=#980000][b]image[/b][/color][color=#000000] of a [/color][color=#ff0000]point ([i]A[/i])[/color][color=#000000] under a dilation [/color][br][color=#000000]    about another point ([/color][i]O[/i][color=#000000]) is a [/color][color=#980000][b]point ([i]A'[/i])[/b][/color][color=#000000] that is _____________________ with [/color][i]O[/i][color=#000000] and [/color][i]A[/i][color=#000000].  [/color][br][br][color=#000000]2) Click the "Check This Out!" box now.  Move point(s) [/color][i]O[/i][color=#000000] and [/color][i]A[/i][color=#000000] around.  Be sure to [br]    adjust the scale factor ([/color][i]k[/i][color=#000000]) of this dilation as well.  Describe what you observe.[br][br]3) Answer the additional questions on the sheet provided to you in class.  [br][br]  [/color]
– Tim Brzezinski

8.2.

–

8.3.

–

8.4.

–

9.

Area, Surface Area, Volume

  • 1. Parallelogram: Area
  • 2. Triangle Area Action! (V1)
  • 3. Area of a Triangle (Discovery)
  • 4. Trapezoid: Area (I)
  • 5. Trapezoid: Area (2)
  • 6. Area of a Trapezoid (Discovery)
  • 7. Right Circular Cylinder!

9.1.

Parallelogram: Area

Directions:
In the app below, use the [b]filling[/b] slider to make the parallelogram light enough so that you can see the white gridlines through it. [b]But don't touch anything else yet! [/b][br][br]After you set the [b]filling[/b] slider, try to count the number of squares inside this parallelogram. [br]Be sure to include partial squares! Provide a good estimate in the question box below.
Try to count the number of squares inside this parallelogram. How many squares (i.e. square units) do you estimate to be inside this parallelogram?
Slide the "[b]Slide Me" [/b]slider now. Carefully observe what happens. What shape do you see now?
How does the area of this new shape compare with the area of the original parallelogram? How do you know this?
How many squares do you now count in the newer shape that was formed? How many squares were in the original parallelogram?
Without looking it up on another tab in your browser, describe how we can find the area of ANY PARALLELOGRAM.
– Tim Brzezinski

9.2.

–

9.3.

–

9.4.

–

9.5.

–

9.6.

–

9.7.

–

Information