What makes triangles congruent?
This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. Today you and your partner are going to investigate some triangles.
Table of Contents
- Congruent Figures
- Congruent Figures: Dynamic Illustration
- Three Sides
- Given three sides
- Two Sides and a Right Angle
- Given hypotenuse and leg
- Two Angles
- Given two angles
- Two Angles One Side
- Given Angle-Side-Angle
- Given Angle-Angle-Side
- Two Sides One Angle
- Given Side-Angle-Side
- Given Side-Side-Angle
- Practice
- Animation 136
- Animation 138
- Animation 137
- Animation 225
Congruent Figures: Dynamic Illustration
[color=#0000ff]Recall an RIGID MOTION is a transformation that preserves distance.[/color] [br][br]
CONGRUENT FIGURES
[b]Definition: [br][br]Any two figures are said to be CONGRUENT if and only if one can be mapped perfectly onto the other using [color=#0000ff]any 1 or composition of 2 (or more) RIGID MOTIONS.[/color][/b][br][br]The applet below dynamically illustrates, [b]by DEFINITION[/b], what it means for any 2 figures (in this case, triangles) to be [b]CONGRUENT.[/b] [br][br]Feel free to move the BIG WHITE VERTICES of either triangle anywhere you'd like at any time.
Given three sides
[color=#000000][b]Given: three side lengths[br][/b][br]Feel free to move the [b]BIG WHITE POINTS[/b] anywhere you'd like! [br]Be sure to do this each time after resetting and before re-sliding the sliders! [/color]
What geometry transformations did you observe in the applet above? List them. [br]
What common trait do all these transformations have?
[i][b]So....[/b][/i][br][br]If 3 sides of one triangle are congruent to 3 sides of another triangle, can we say that those 2 triangles are congruent? Fully explain why or why not.
Given hypotenuse and leg
[b]Given: Two Side Lengths, One Angle is 90[sup]o[/sup] opposite the larger side.[br][/b][br]Feel free to move the BIG WHITE POINTS anywhere you'd like! [br]Be sure to do this each time after resetting and before re-sliding the sliders!
What geometry transformations did you observe in the applet above? List them. [br][br][br]
What common trait do all these transformations have?
[i]So....[/i][br][br]If the [color=#9900ff]hypotenuse[/color] and leg of [color=#bf9000]one right triangle[/color] are congruent to the [color=#9900ff]hypotenuse[/color] and leg of another right triangle, [color=#bf9000]then those 2 triangles are congruent? [/color]Fully explain why or why not.
Given two angles
[b]Given: Two angle measures[/b][color=#980000] Â [/color][br][br][color=#000000]Feel free to move the locations of the [/color][color=#38761d][b]BIG GREEN VERTICES[/b][/color][color=#000000] of either triangle before slowly dragging the slider. [/color][b]Â [/b][i][color=#ff0000]Pay careful attention to what happens as you do.[/color][/i]
What geometry transformations did you observe in the applet above? List them. [br][br][br]
How are these transformations different from the previous examples?
[i]So....[/i][br][br]If 2 angles are congruent to 2 angles of another triangle, can we say that those 2 triangles are congruent? Fully explain why or why not.
Given Angle-Side-Angle
[color=#000000][b]Given: Two angles and one side, where the side is in between the two angles[br][/b][br]Suppose 2 triangles have 2 pairs of congruent angles. Suppose we also know that the side between each set of given angles (in one triangle) is congruent to the side between this same pair of angles in the other triangle. [br][br]Does knowing only this constitute sufficient evidence to prove the triangles congruent? If so, explain how/why with respect to the transformations and/or triangle congruence theorems you've previously learned. If not, clearly explain why not. [/color]
[i]So....[/i][br][br]If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, can we say that those 2 triangles are congruent? Fully explain why or why not.
Given Side-Angle-Side
[b][color=#000000]Given: 2 sides [/color][color=#000000]and their [/color][color=#ff00ff]included angle[/color][/b][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?
[i]So....[/i][br][br]If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, can we say those 2 triangles are congruent? Fully explain why or why not.
Animation 136
What geometry theorem is dynamically being illustrated below? [br](Feel free to move the white points anywhere you'd like.)