Now using what you learned about Angle-Angle-Angle(AAA) Similarity, we will explore if there are any other ways to tell if triangles are similar to each other.
First, let's see if these triangles are similar checking AAA similarity.[br]We will measure each of the angles to see if the triangles are similar in terms of AAA similarity. To measure the angles of the triangle select [img]data:image/png;base64,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[/img] then click the center of the triangle. All three interior angles of the triangle will be revealed.
[b]Q1: For triangle ABC what is the interior angle A and for triangle A’B’C’ what is the [b]interior [/b]angle A’? Are angle A, and angle A’ the same?[/b]
[b]Q2: For triangle ABC what is [b]interior[/b] angle B and for triangle A’B’C’ what is [b]interior [/b]angle B’ ? Are angle B and angle B’ the same?[/b]
[b]Q3: For triangle ABC what is [b]interior [/b]angle C ? And for triangle A’B’C’ what is the [b]interior [/b]angle C’? Are angle C and angle C’ the same?[/b]
[b]Q4: If 2 or more angles are the same in both triangles are they similar triangles? (Hint think back to AAA similar triangles).[/b]
Please press the undo button[img]data:image/png;base64,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[/img] on your GeoGebra worksheet to reset your worksheet. After resting back to the initial worksheet please explore how the two triangles move, when you drag/move[br]around the points of triangle ABC.
[b]Q5: What happens to triangle A’B’C’ when you stretch triangle ABC?[/b]
[b]Q6: What happens to triangle A’B’C' when you shrink triangle ABC?[/b]
[b]Q7: Drag point A so that it is on top of point A’, then manipulate points B and C. What do you notice? What happens to triangle ABC and triangle A’B’C’? Does the relationship between the two triangles change?[/b]
[b]Q8: Drag point B so that it is top of point B’, then manipulate points A and C. What do you notice? What happens to triangle ABC and triangle A’B’C’? Does the relationship between the two triangles change?[/b]
[b]Q9: Drag point C so that it is top of point C’, then manipulate points A and C. What do you notice? What happens to triangle ABC and triangle A’B’C’? Does the relationship between the two triangles change?[/b]
Now take the opportunity to explore around the GeoGebra worksheet to see if you can discover anything new about these triangles and see if there is another property that makes them similar triangles. [b]Please take a few minutes to explore.[/b]
By now you may have noticed that when you manipulate the points on triangle ABC the side lengths of triangle A’B’C’ change accordingly. Try and now move points A and B to have the side length of AB=5 or another whole number.
[b]Q10: When AB=5 or another whole number, what does side length A’B’ equal? Is there a relationship between AB and A’B’?[/b]
Now please move points A and C to have the side length of AC= 5 or another whole number. Now please move points B and C to have a side length of BC=5 or another whole number.
[b]Q11: When AC=5 or another whole number, what does side length A’C’ equal? Is there a relationship between AC and A’C'?[/b]
Now please move points B and C to have a side length of BC=5 or another whole number.[br][br][br]
[b]Q12: When BC=5 or another whole number, what does side length B’C’ equal? Is there a relationship between BC and B’C’?[/b]
Perhaps now we may have noticed that triangle ABC is [b]x [/b]times larger than A’B’C’. This means that A’B’=[b]x [/b]AB, B’C’=[b]x[/b] BC, A’C’=[b]x[/b] AC.
[b]Q13: What could we conclude from this? Can you think of a similar triangle rule that could be created regarding Side-Side-Side (SSS)? [Please answer this question before moving on.][/b]
Now that we have recognized previously that these triangles are similar due to AAA similarity. We have explored and witnessed that triangle ABC had a relationship with A’B’C’ in regard to side length we can conclude that when triangles have the same ratios between side lengths they are similar triangles by Side-Side-Side (SSS) if this holds true: [math]\frac{A'B'}{AB}=\frac{A'C'}{AC}=\frac{B'C'}{BC}[/math]
[math][/math][br]To conclude this exploration we know that triangles are similar if they share two of the[br]same angles, AAA similarity, or if all three side ratios equal each other[br] [ [math]\frac{A'B'}{AB}=\frac{A'C'}{AC}=\frac{B'C'}{BC}[/math]]SSS similarity.
Stretch or shrink your triangles and please measure all six side lengths to see if [br][math]\frac{A'B'}{AB}=\frac{A'C'}{AC}=\frac{B'C'}{BC}[/math] holds true. Please record your ratios below.
Thank you for taking your time to complete this exploration good luck with the rest of[br]your similar triangle exploration. Please comment below if you have any questions or if something was confusing to you.