Investigation: Constructing Similar Triangles
Tools Needed: pencil, paper, protractor, ruler
- Draw a
45∘angle. Extend the horizontal side and then draw a 60∘angle on the other side of this side. Extend the other side of the 45∘angle and the 60∘angle so that they intersect to form a triangle. What is the measure of the third angle? Measure the length of each side.
- Repeat Step 1 and make the horizontal side between the
45∘and 60∘angle at least 1 inch longer than in Step 1. This will make the entire triangle larger. Find the measure of the third angle and measure the length of each side. Find the ratio of the sides. Put the sides opposite the 45∘angles over each other, the sides opposite the 60∘angles over each other, and the sides opposite the third angles over each other. What happens?
Watch this video beginning at the 2:09 mark.
AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar.
The AA Similarity Postulate is a shortcut for showing that two triangles are similar. If you know that two angles in one triangle are congruent to two angles in another, which is now enough information to show that the two triangles are similar. Then, you can use the similarity to find the lengths of the sides.
Determining if Two Triangles are Similar
1. Determine if the following two triangles are similar. If so, write the similarity statement.
2. Determine if the following two triangles are similar. If so, write the similarity statement.
3. Are the following triangles similar? If so, write the similarity statement.
Are the following triangles similar? If so, write a similarity statement.
Use the diagram to complete each statement.
Name two similar triangles. How do you know they are similar?
Write a true proportion.
Name two other triangles that might not be similar.
Writing How many angles need to be congruent to show that two triangles are similar? Why?
Writing How do congruent triangles and similar triangles differ? How are they the same?
Use the triangles below for questions 12-15.
Are the two triangles similar? How do you know?
Fill in the blanks: If an acute angle of a _______ triangle is congruent to an acute angle in another ________ triangle, then the two triangles are _______.
Use the diagram below to answer questions 16-20.
Draw the three separate triangles in the diagram.
Complete the following proportionality statements.
To view the Review answers, open this PDF file and look for section 7.4.