gU5L2C: ∆ Cong. Criteria Part 3

[size=200][b][color=#ff0000][u]AAA (Angle-Angle Angle) Criteria[br][/u][/color][/b][/size][b][color=#ff0000][u][size=200][/size][br][/u][/color][/b]

There are two triangles in this applet. The triangles have the same angle measures. Click on the bottom right blue dot to resize one of the triangles.

[b][color=#0000ff]1) Is it possible to two different triangles that have the same angles but different sides?[/color][/b]

[b]No, it is not possible.[/b] If both triangles have the same angles, then they are exact copies of each other (congruent). [b]Yes, it is possible.[/b] Dilating a triangle can result in triangles with the same angles but different sides.

[b][color=#0000ff]2) Can the AAA Criteria be used to prove congruent triangles? [/color][/b]

[b]No; congruence is not guaranteed.[/b] [b]Yes; triangles with same AAA will always be congruent.[/b]

[size=200][b][color=#ff0000][u]SSS (Side-Side-Side) Criteria[br][/u][/color][/b][/size][b][color=#ff0000][u][br][/u][/color][/b]

Click Proof (or SLIDE) for the animation

[b][color=#0000ff]3) What transformations are used to map one triangle to another triangle in the above activity?[/color][/b]

Translation Reflection Dilation Rotation

[b][color=#0000ff]4) Therefore, two triangles with the same sides will always be congruent. [/color][/b]

FALSE; a non-rigid motion is needed to map one triangle onto the other. TRUE; since one can be mapped onto the other using rigid motions.

[size=200][b][color=#ff0000][u]SAS (Side-Angle-Side) Criteria[br][/u][/color][/b][/size][b][color=#ff0000][u][br][/u][/color][/b]

Slide the bar to watch the animation and transformations

[b][color=#0000ff]5) What transformations are observed in the above activity ?[/color][/b]

RIGID transformation NON-RIGID transformations Both RIGID and NON-RIGID transformations

[b][color=#0000ff]6) Therefore, two triangles with the same 2 sides and same included angle (SAS) will always be congruent. [/color][/b]

TRUE FALSE

[size=200][b][color=#ff0000][u]SSA (Side-Side-Angle) Criteria[br][/u][/color][/b][/size][b][color=#ff0000][u][br][/u][/color][/b]

SSA Criteria Part A: Slide the bar to watch the animation and transformations

SSA Criteria Part B: Slide the bar for the animations. Then, determine if the triangle on the right can be mapped on to the left one. (One vertex will rotate it while the other will move it ).

[b][color=#0000ff]7) If two triangles have the same SSA conditions, then one triangle [u]CAN ALWAYS[/u] be mapped onto the other using rigid motions only. [/color][/b]

TRUE; as see in both cases above. FALSE; sometimes, the triangles will not overlap despite attempting several rigid motions.

[size=200][b][color=#ff0000][u]ASA (Angle-Side-Angle) Criteria[br][/u][/color][/b][/size][b][color=#ff0000][u][br][/u][/color][/b]

Slide the bar to watch the animation. Map one triangle to the other (Blue dot will rotate while pink will move the triangles)

[b][color=#0000ff]7) If 2 angles and the included side of one triangle are fixed, which of these is TRUE? [/color][/b]

There's [b][u]only one way[/u][/b] to complete the triangle. ASA [b][u]doesn't guarantee[/u] [/b]congruence. There's [b][u]more than one way[/u][/b] to complete the triangle. A series of rigid motions [b][u]can map[/u][/b] one triangle to the other. ASA [u][b]guarantees[/b] [/u]congruence. A series of rigid motions [b][u]may or may not[/u][/b] be sufficient to map one triangle to the other.

[size=200][b][u][color=#ff0000]AAS (Angle-Angle-Side) Criteria;[br][/color][/u][/b][/size][b][u][size=200][color=#ff0000][br][/color][/size][/u][/b]

Use a series of transformations to map the triangle on the right to the one on the left. (One white vertex will move the triangle and the other will rotate it)

[b][color=#0000ff]8) Therefore, are the triangles above congruent?[/color][/b]

YES; one triangle can be mapped onto the other using rigid motions NO; triangles do not overlap

Take a minute to interact with this activity. Be very observant. Then, answer the two questions that follow.

[b][color=#0000ff]9) If 2 angles of a triangle are congruent to 2 angles of another triangle, what can we conclude about the third pair of angles? [/color][/b]

They are congruent by the Third Angle Theorem Nothing Deductively. The third pair of angles may or may not be congruent. They are congruent by the Vertical Angle Theorem

[b][color=#0000ff]10) Therefore, the [/color][color=#ff0000][i]Angle-Angle-Side[/i][/color][color=#0000ff] Triangle Congruence Theorem should be no surprise to us. Which other triangle congruence theorem is disguised here? [br][/color][/b][b][color=#0000ff][br][/color][/b]

SSS ASA SAS

[b][color=#ff0000][u][size=200]HL(Hypotenuse-Leg) Criteria[/size][/u][/color][/b]

Use a series of rigid motions to map the triangle on the right to the one on the left. (One white vertex will allow you move it, the other will rotate it)

[b][color=#0000ff]11) Therefore, are the triangles above congruent?[/color][/b]

No; triangles do not overlap despite attempting several rigid motions Yes; a series of rigid motions can be used to show that HL guarantees congruence.

[size=200][b][u][color=#ff0000]REFLECT & SUMMARIZE[/color][/u][/b][/size]

[color=#0000ff][b]12) What have you learned in this, and yesterday's lesson regarding triangle congruence criteria[/b]. ([/color][i]State at least two ideas)[/i]