Congruent Triangle - Unit 2 Part 2

Jenny Secor, Alyssa Noland, Sep 21, 2019

Table of Contents

  1. Theorems Involving Angles
    1. Triangle Angle Theorems
    2. Triangle Angle Theorems (V2)
    3. Triangle Angle Theorems (V3)
    4. Triangle Angle Sum Theorem
    5. Exterior Angles of a Triangle
  2. Triangle Theorems (General)
    1. Isosceles Triangle (Properties)
    2. Midsegment of a Triangle
    3. A Special Theorem: Part 1 (V1)
    4. A Special Theorem: Part 2 (V1)
    5. Another Special Theorem: Part 1 (V1)
    6. Another Special Theorem: Part 2 (V1)
    7. Similar Right Triangles (V1)
    8. Another Special Triangle and its Properties (II)
    9. Another Special Right Triangle (Guided Discovery)
    10. A Special Triangle & Its Properties (I)
    11. Special Right Triangle (I)
    12. Acute, Right, or Obtuse?
    13. Acute, Right, Obtuse (III)
    14. Napoleon Motion!
    15. Isosceles Triangle Medians
  3. Points of Concurrency
    1. Incenter Exploration (A)
    2. Incenter Exploration (B)
    3. Orthocenter Exploration
    4. Circumcenter (& Questions)
    5. Medians Centroid Theorem (Proof without Words)
    6. Euler Line (Informal Investigation)
    7. Midpoint of HYP
    8. Incenter & Incircle Action!
    9. Medians and Centroid Dance
    10. Circumcenter & Circumcircle Action!
  4. Points of Concurrency - Extension Activities
    1. 9-Point Circle (Informal Investigation)
    2. 3 Special Points!
    3. 9 Point Circle Action
    4. 9-Point Circle Action (Part 2)
    5. 9-Point Circle Action (Part 3A)
    6. 9-Point Circle Action (Part 4)
    7. 9-Point Circle Action (Part 3B)
  5. Older (Earlier) Applets
    1. Perpendicular Bisector Theorem (Part 1)
    2. Perpendicular Bisector Theorem (Part 2)
    3. Angle Bisector Theorem (Part 1)
    4. Angle Bisector Theorem (Part 2)

1.

Theorems Involving Angles

  • 1. Triangle Angle Theorems
  • 2. Triangle Angle Theorems (V2)
  • 3. Triangle Angle Theorems (V3)
  • 4. Triangle Angle Sum Theorem
  • 5. Exterior Angles of a Triangle

1.1.

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

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2.

Triangle Theorems (General)

  • 1. Isosceles Triangle (Properties)
  • 2. Midsegment of a Triangle
  • 3. A Special Theorem: Part 1 (V1)
  • 4. A Special Theorem: Part 2 (V1)
  • 5. Another Special Theorem: Part 1 (V1)
  • 6. Another Special Theorem: Part 2 (V1)
  • 7. Similar Right Triangles (V1)
  • 8. Another Special Triangle and its Properties (II)
  • 9. Another Special Right Triangle (Guided Discovery)
  • 10. A Special Triangle & Its Properties (I)
  • 11. Special Right Triangle (I)
  • 12. Acute, Right, or Obtuse?
  • 13. Acute, Right, Obtuse (III)
  • 14. Napoleon Motion!
  • 15. Isosceles Triangle Medians

2.1.

Isosceles Triangle (Properties)

[b]Definition:[/b][br][br]An [b][color=#c51414]ISOSCELES TRIANGLE[/color][/b] is a triangle that has [b][color=#c51414]AT LEAST 1 PAIR OF CONGRUENT SIDES[/color][/b]. [br][br][b]Directions:[/b][br][br]1) Click on the [color=#c51414]red checkbox[/color] to illustrate this definition. [br]2) Move any 1 (or more) of the vertices of this triangle around. Does it remain isosceles?[br]3) Click on checkbox 2. Move the vertices of the triangle around again. What do you notice? [br]4) Click on checkbox 3. This will draw the line that bisects the vertex angle of the isosceles triangle.[br]5) Click on checkbox 4. What else do you notice? [br][br]6) Please answer the questions that appear below this applet.
[b]Questions: [/b][br][br]Fill in the following blanks (based upon your observations).  The words to complete these statements can be found in the [b]word bank[/b] below.  [br][br]1) If [color=#c51414]two sides of a triangle are congruent, then the _____________________ opposite those sides are ____________________[/color]. [br][br]2) The[color=#1551b5] ___________________ of the _________________ angle[/color] of an isosceles triangle is the ________________________ ________________ of the base (3rd side).[br][br][b]Word Bank:  [br][br]bisector   angles     congruent     perpendicular    vertex    bisector [/b]
– Tim Brzezinski

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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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3.

Points of Concurrency

  • 1. Incenter Exploration (A)
  • 2. Incenter Exploration (B)
  • 3. Orthocenter Exploration
  • 4. Circumcenter (& Questions)
  • 5. Medians Centroid Theorem (Proof without Words)
  • 6. Euler Line (Informal Investigation)
  • 7. Midpoint of HYP
  • 8. Incenter & Incircle Action!
  • 9. Medians and Centroid Dance
  • 10. Circumcenter & Circumcircle Action!

3.1.

Incenter Exploration (A)

[color=#000000]Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly one point. [br][br]The angle bisectors of a triangle's 3 interior angles are all concurrent. [br]Their point of concurrency is called the I[b]NCENTER[/b] of the triangle. [br][br]In the applet below, [b]point I [/b]is the triangle's [b]INCENTER[/b]. [br]Use the tools of GeoGebra in the applet below to complete the activity below the applet. [br][i]Be sure to answer each question fully as you proceed. [/i] [/color]
[color=#000000][b]Directions: [/b][br][br][/color][left][color=#000000]1) In the applet above [/color][color=#38761d]construct a line passing through I and is perpendicular to [i]AB[/i][/color][color=#000000]. [br]2) Use the [/color][b][color=#000000]Intersect[/color][/b][color=#000000] tool to plot and label a point [/color][i]G[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (1)[/color][color=#000000] intersects [/color][i][/i][color=#000000][i]AB[/i].[/color][color=#000000][br]3) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]BC[/i].[/color][color=#000000] [br][/color][color=#000000]4) Plot and label a point [i]H[/i] where [/color][color=#38761d]the line you constructed in (3)[/color][color=#000000] intersects [i]BC[/i].[/color][br][color=#000000]5) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]A[/i][i]C[/i]. [/color][color=#000000][br][/color][color=#000000]6) Plot and label a point [/color][i]J[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (5)[/color][color=#000000] intersects [/color][i][color=#000000]AC[/color][/i][color=#000000]. [br]7) Now, use the [b]Distance[/b] tool to measure and display the lengths [i]IG[/i], [i]IH[/i], and [i]IJ[/i]. What do you notice?[br][br][br]8) Experiment a bit by moving any one (or more) of the triangle's vertices around[br] Does your initial observation in (7) still hold true? [br] Why is this? (If you need a hint, refer back to the worksheet found [url=https://tube.geogebra.org/m/tU3ZqhjN]here[/url]. [/color][/left][color=#000000][br]9) Construct a circle centered at I that passes through [i]G[/i]. What else do you notice? [br] Experiment by moving any one (or more) of the triangle's vertices around. [br] This circle is said to be the triangle's [i]incircle[/i], or [i]inscribed circle[/i]. [br] It is the largest possible circle one can draw [i]inside[/i] this triangle. [br] Why, according to your results from (7) is this possible? [br][br]10) Do the angle bisectors of a triangle's interior angles also bisect the sides opposite theses angles? [br] Use the [b]Distance[/b] tool to help you answer this question. [/color][br][br][color=#000000]11) Is it ever possible for a triangle's [b]INCENTER[/b] to lie OUTSIDE the triangle?[br] If so, under what condition(s) will this occur? [br][br]12) Is it ever possible for a triangle's [b]INCENTER[/b] to lie ON the triangle itself?[br] If so, under what condition(s) will this occur? [/color]
– Tim Brzezinski

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4.

Points of Concurrency - Extension Activities

  • 1. 9-Point Circle (Informal Investigation)
  • 2. 3 Special Points!
  • 3. 9 Point Circle Action
  • 4. 9-Point Circle Action (Part 2)
  • 5. 9-Point Circle Action (Part 3A)
  • 6. 9-Point Circle Action (Part 4)
  • 7. 9-Point Circle Action (Part 3B)

4.1.

9-Point Circle (Informal Investigation)

In geometry, the nine-point circle is a circle that can be constructed for any given triangle. [br]This special circle passes through the following points: [br][br]The midpoint of each side of the triangle (D, E, F in applet below)[br]The points at which the lines containing the triangle's altitudes intersect the lines containing the triangle's sides (G, H, I)[br]The midpoint of each segment connecting the triangle's vertex to its orthocenter (J, K, L). [br][br][b]There are many cool features about a triangle's 9-point circle. As you complete the investigation questions below this applet, be sure to continually MOVE VERTICES A, B, & C AROUND to informally validate that any conjecture you make STILL HOLDS TRUE (for many different possible triangles.) [/b][br][br]And let the fun begin! (See below).
9-Point Circle (Informal Investigation)
Activity Questions: [br][br]For these questions, we'll denote the circumcenter as "C", the orthocenter as "O", the centroid as "R", and the incenter as "S". Let's denote "M" as the center of the 9-point circle. [br][br]Use the tools of GeoGebra to do the following. [b] As you do, be sure to MOVE VERTICES A, B, & C AROUND to informally validate that any conjecture you make STILL HOLDS TRUE (for many different possible triangles)! [/b][br][br]1) Construct the triangle's circumcircle and measure its radius. Measure the radius of the 9-pont circle. What is the ratio of the larger radius to the smaller radius? [br][br]2) Construct a segment that connects R to O. Prove that R, M, and O are collinear. [br][br]3) For (2) above, how does RM compare to MO? [br][br]4) Construct any point "W" that lies on the circumcircle you've just constructed in (1) above. Construct a segment connecting O to W. Plot and label a point Y at which this segment intersects the 9-point circle. What seems to be true about OY and YW (regardless of where point W lies?--Try moving it around!) [br][br]5) Construct the triangle's Euler Line (line that passes through C, R, and O). Show that M is also collinear with these 3 points. In addition, find & simplify the ratio CM : MO. Also, find and simplify the ratio CM: CO. [br][br]6) Construct segments to form the quadrilateral FEJL. What special type of quadrilateral does this polygon look like? Use the tools of GeoGebra to informally prove your conjecture. [br][br]7) Repeat step (6), but this time form quadrilateral LKED. [br][br]8) Repeat step (6) again, but this time form quadrilateral FKJD. [br][br]9) Construct the triangle's incircle (inscribed circle). At how many points do the triangle's incircle and 9-point circle intersect? Where is this point of intersection?
– Tim Brzezinski

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5.

Older (Earlier) Applets

  • 1. Perpendicular Bisector Theorem (Part 1)
  • 2. Perpendicular Bisector Theorem (Part 2)
  • 3. Angle Bisector Theorem (Part 1)
  • 4. Angle Bisector Theorem (Part 2)

5.1.

Perpendicular Bisector Theorem (Part 1)

The applet below contains a segment (with endpoints A & B) and a point P that is equidistant from this segment's endpoints.
Instructions:[br][br]1) Notice how point C (in green) is the same distance (equidistant) from points A and B.[br] In fact,Point C has been programmed to always be EQUIDISTANT from the black segment's endpoints.[br][br] Use your mouse to drag point C around show show the locus (set of points) on your computer screen [br] that are equidistant from A and B. [br][br]2) What does this locus (set of points) look like? Explain.[br][br]3) Now, move point A and/or point B to change the position and length of the segment. [br] Then, click on the house button in the toolbar shown on the upper right hand side of the applet.[br] (This house button says "Back to Default View" and will erase all traces of green dots you've already made.)[br][br]4) Repeat steps (1) and (2) again, this time for this "newer" segment with endpoints A and B.[br][br]4) Click on checkbox (A) to show the locus (set of points) on your computer screen that are [br] EQUIDISTANT from A and B. [br][br]5) Use your observations to fill in each ( ) with the correct word to make a true statement: [br][br] This set of points (shown in step 4 above) looks like it is the ( ) ( ) of segment AB.[br][br]6) Click checkboxes (B) and (C) in the applet above to either check your answer to step (5) or help you fill in the [br] ( )'s for step (5).[br][br]7) Use your observations above to fill in each blank below to make a true statement: [br][br] [b]“If a point is ( ) from the ( ) of a segment, then that point[br] must lie on the ( ) ( ) of that segment.”[/b]
– Tim Brzezinski

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