1. Draw two different acute angles with long rays each on a separate piece of patty paper. Trace the rays onto two more pieces of patty paper. Use each set to make a triangle being sure that the triangles are non-congruent. Tape paper paper so that each triangle stays consistent. Verify that the third angles are congruent. Measure the corresponding sides. What do you notice about the ratios of each set of corresponding sides?
2. Are the triangles you drew in Exercise 1 similar? Explain.
3. Why is it that you only need to construct triangles where two pairs of angles are equal but not three?
4. Why were the ratios of the corresponding sides proportional?
5. Do you think that what you observed will be true when you construct any pair of triangles with two pairs of equal angles? Explain.
6. Make another set of two triangles of different sizes with two pairs of equal angles. Then, measure the lengths of the corresponding sides to verify that the ratio of their lengths is proportional.
7. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and side-length measures.
3. Why is it that you only need to construct triangles where two pairs of angles are equal but not three?
4. Why were the ratios of the corresponding sides proportional?
5. Do you think that what you observed will be true when you construct any pair of triangles with two pairs of equal angles? Explain.
6. Make another set of two triangles of different sizes with two pairs of equal angles. Then, measure the lengths of the corresponding sides to verify that the ratio of their lengths is proportional.
7. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and side-length measures.
8. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and side-length measures.
9. The triangles shown below are similar. Use what you know about similar triangles to find the missing side lengths x and y.
10. The triangles shown below are similar. Write an explanation to a student, Claudia, of how to find the lengths of x and y
At a certain time of day, a 12m flagpole casts an shadow. Write an equation that would allow you to find the height, h, of the tree that uses the length, s, of the tree’s shadow.
Example 2
In the diagram above, a large flagpole stands outside of an office building. Marquis realizes that when he looks up from the ground 60 m away from the flagpole, the top of the flagpole and the top of the building line up. If the flagpole is 35 m tall and Marquis is 170 m from the building, how tall is the building?
- Are the triangles in the diagram similar? Explain.
- Determine the height of the building using what you know about scale factors.
- Determine the height of the building using ratios between similar figures.
- Determine the height of the building using ratios within similar figures.