You already know that two polygons are similar polygons if and only if the corresponding angles are congruent and corresponding sides are proportional. On this web page you will investigate some shortcuts for triangle similarity. The three activities that follow can be used to replace or extend the investigations in Lesson 11.2 of Discovering Geometry: An Investigative Approach.
Sketch
Is AA a similarity shortcut? If two angles of one triangle are congruent to two angles of another triangle, must the two triangles be similar? You'll answer this question with the sketch. Here angles A and D are congruent and fixed. Drag the other vertices—B, C, E, and F—to change the shape and size of the triangles. Press Start Over if you accidentally drag a vertex off the screen.
Investigate
- Make angle E congruent to angle B by dragging vertex F or by pressing Adjust Angle E. You now have two pairs of congruent angles: A and D, and B and E.
- What is true about angles C and F? Why?
- Compare the ratios of the corresponding sides. Is AB/DE = BC/EF = CA/FD?
- Based on your results in steps 1, 2, and 3, are the triangles similar?
- Formulate the AA Similarity Conjecture: If _____ angles of one triangle are congruent to _____ angles of another triangle, then _____.
- Explain how you could use this sketch to show that A (one pair of corresponding angles congruent) is not a similarity shortcut.
- Explain why there is no need to investigate AAA as a similarity shortcut.
- Explain why there is no need to investigate ASA or SAA.
Sketch
Is SSS a similarity shortcut? If three sides of one triangle are proportional to the three sides of another triangle, must the two triangles be similar? You'll answer this question with the next sketch. Triangles ABC and DEF have two pairs of sides that remain proportional: AB and DE, and BC and EF. You can change the shape and size of the triangles by dragging vertices B, C, E, or F. Press Start Over if you drag a vertex off the screen.
Investigate
- Make all three sides proportional by dragging vertex F or by pressing Adjust Side FD.
- Compare the measures of the corresponding angles of the two triangles when all three sides are proportional. What do you notice?
- Formulate the SSS Similarity Conjecture: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are _____.
- Based on this sketch, is SS a similarity shortcut? Why or why not?
Sketch
Is SAS a similarity shortcut? Use the next sketch to determine whether two triangles are similar if they have two pairs of sides proportional and the pair of included angles equal in measure. Triangles ABC and DEF have two pairs of sides that remain proportional: AB and DE, and AC and DF. You can change the shape and size of the triangles by dragging vertices B, C, E, and F. Press Start Over if you drag a vertex off the screen.
Investigate
- Make included angle D congruent to included angle B by dragging vertex F or by pressing Adjust Angle D. Are the two triangles similar?
- Does this relationship hold if you change the shape or size of the triangles?
- Formulate the SAS Similarity Conjecture: If two sides of one triangle are proportional to two sides of another triangle and _____, then the _____.
One question remains: Is SSA a similarity shortcut? You may recall that side SSA did not work for congruence, so you might guess that it does not work for similarity either.
- Consider angles C and F as the non-included angles for sides AB and DE, and AC and DF. Can you adjust the triangles such that angle C is congruent to angle F, but the triangles are not similar?
- Based on your results from question 4, is SSA a similarity shortcut?