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Almost-Twin Triangles (Triangle Similarity)

ForfatterAlexis Sparks

Introduction

Welcome, student-learner! In this activity, you will deal with the learning competency of proving the conditions for similarity of triangles, which is appropriate for 9th graders. Knowledge of ratios and proportions is required for this activity. You will be playing with triangles and changing their measurements. Be sure to follow the instructions in every section and answer the questions that follow honestly.

Triangle Similiarity

Definition "Triangles are similar if they have the same shape, but can be different sizes (Math Open Reference, 2011, para. 1)." (They are still similar even if one is rotated, or one is a mirror image of the other (Math Open Reference, 2011, para. 1)). Properties - The corresponding angles of similar triangles are "CONGRUENT (Math Open Reference, 2011, para. 5)." - The corresponding sides of similar triangles are "ALL IN THE SAME PROPORTION (Math Open Reference, 2011, para. 6)." However, you do not need to know all the measurements of the triangles to know whether or not they are similar. Three combinations of select measurements can be used to identify similarity.

Combination 1

In Plane 1, you will see two triangles, ABC and DEF, and sliders labelled AB, ABC, BAC, and s. Directions Move the sliders to any value you want. This will transform the triangles in the plane. You may drag the triangles and labels around to arrange them and avoid overlap and scroll your mouse to zoom in and out. It would help to have segments AB and DE be in the same orientation.

Plane 1

Assessment for Combination 1

1. Is BAC congruent to EDF?

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2. Is ABC congruent to DEF?

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3. Segment AB corresponds to segment DE. Segment BC corresponds to segment EF. Segment AC corresponds to segment DF. Are the corresponding segments in the same proportion?

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4. ABC corresponds to DEF. ACB corresponds to DFE. BAC corresponds to EDF. Are the corresponding angles congruent?

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5. Repeat questions 1-4 but, this time, change the value of the sliders. (You do not need to erase your answers in questions 1-4.) Did your answers to the Yes or No questions change?

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AA Similarity Theorem

The triangles you have made in Plane 1 exhibit AA similarity. The AA (angle-angle) similarity theorem states that for any pair of triangles: if two pairs of corresponding angles are congruent, then the pair of triangles are similar (SparkNotes, 2020). This means that with just the conditions said in the previous statement, the shapes already exhibit the properties of total side-proportionality and total angle-congruence.

Combination 2

In Plane 2, you will see two triangles, ABC and DEF, with points B and E split into two points each, B, B_{1}, E, and E_{1}. You will also see sliders labelled AB, AC, BC, and s. Directions Move the sliders to any value you want. This will transform the triangles in the plane. You may drag the triangles and labels around to arrange them and avoid overlap and scroll your mouse to zoom in and out. It would help to have segments AC and DF be in the same orientation. Then, drag points B and B_{1} so that they are in the same position and together act as the third vertex of ABC. Do the same for points E and E_{1} of DEF. There are certain limitations to GeoGebra that may make it difficult to accomplish this step but just try to make the points as close to each other as possible.

Plane 2

Assessment for Combination 2

1. Divide the length of segment DE by that of segment AB. Do the same for segments EF and BC. Are the two quotients equal?

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2. Divide the length of segment DF by that of segment AC. Is the quotient equal to the quotient of segments DE and AB?

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3. Is the quotient of segments DF and AC equal to the quotient of segments DE and AB?

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4. Are the quotients in past questions equal to the value of slider s?

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5. Segment AB corresponds to segment DE. Segment BC corresponds to segment EF. Segment AC corresponds to segment DF. Are the corresponding segments in the same proportion?

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6. ABC corresponds to DEF. ACB corresponds to DFE. BAC corresponds to EDF. Are the corresponding angles congruent?

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7. Repeat questions 1-6 but, this time, change the value of the sliders. Do not forget to make sure that points B and B_{1} are again as close to each other as possible, and same thing for points E and E_{1}. (You do not need to erase your answers in questions 1-6.) Did the quotients, ratios, and proportions change?

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8. Did your answer to the Yes or No questions change when you manipulated the sliders?

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SSS Similarity Theorem

The triangles you have made in Plane 2 exhibit SSS similarity. The SSS (side-side-side) similarity theorem states that for any pair of triangles: if all three pairs of corresponding sides are in proportion, then the pair of triangles are similar (SparkNotes, 2020). This means that with just the conditions said in the previous statement, the shapes already exhibit the properties of total side-proportionality and total angle-congruence.

Combination 3

In Plane 3, you will see two triangles, ABC and DEF, and sliders labelled AB, BC, and s. Directions Move the sliders to any value you want. This will transform the triangles in the plane. You may drag the triangles and labels around to arrange them and avoid overlap and scroll your mouse to zoom in and out. It would help to have segments AC and DF be in the same orientation. Then, drag the points of the triangles around so that the angles ABC and DEF have the equal degree. There are certain limitations to GeoGebra that may make it difficult to accomplish this step but just try to make the angles as close to equal as possible.

Plane 3

Assessment for Combination 3

1. Divide the length of segment DE by that of segment AB. Do the same for segments EF and BC. Are the two quotients equal?

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2. Divide the length of segment DF by that of segment AC. Is the quotient equal to the quotients in the previous question?

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3. Are the quotients in past questions equal to the value of slider s?

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4. Find the ratios for the lengths of the following segments: AB:BC (AB/BC), DE:EF (DE/EF). Are these ratios proportional?

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5. Segment AB corresponds to segment DE. Segment BC corresponds to segment EF. Segment AC corresponds to segment DF. Are the corresponding segments in the same proportion?

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6. ABC corresponds to DEF. ACB corresponds to DFE. BAC corresponds to EDF. Are the corresponding angles congruent?

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7. Repeat questions 1-6 but, this time, change the valueof the sliders. Do not forget to make sure that angles ABC and DEF are again as close to equal as possible. (You do not need to erase your answers in questions 1-6.) Did the quotients, ratios, and proportions change?

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  • B
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8. Did your answer to the Yes or No questions change when you manipulated the sliders?

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SAS Similarity Theorem

The triangles you have made in Plane 3 exhibit SAS similarity. The SAS (side-angle-side) similarity theorem states that for any pair of triangles: if two pairs of corresponding sides are in proportion and the corresponding included angles are congruent, then the pair of triangles are similar (SparkNotes, 2020). This means that with just the conditions said in the previous statement, the shapes already exhibit the properties of total side-proportionality and total angle-congruence.

Summary

Triangles are similar if a) their corresponding angles are "CONGRUENT (Math Open Reference, 2011, para. 5);" and b) their corresponding sides are "ALL IN THE SAME PROPORTION (Math Open Reference, 2011, para. 6)." Such triangles have the same shape but may differ in size. There are three conditions or theorems for triangle similarity. In triangles with AA similarity, two pairs of corresponding angles are congruent (SparkNotes, 2020). In triangles with SSS similarity, all three pairs of corresponding sides are in proportion (SparkNotes, 2020). In triangles with SAS similarity, two pairs of corresponding sides are in proportion and the corresponding included angles are congruent (SparkNotes, 2020).

References

Math Open Reference. (2011). Similar Triangles. Retrieved September 25, 2020, from https://www.mathopenref.com/similartriangles.html SparkNotes. (2020). Proving Similarity of Triangles. Retrieved September 25, 2020, from https://www.sparknotes.com/math/geometry2/congruence/section5/

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