Chapter 2 Justification and Similarity 2.3 Students will determine conditions that can be used to identify similar triangles and support this by writing proofs. Students will solve mathematical and everyday problems using similar triangles

Similar Triangles Can you prove triangles similar Same I proof ideas as before but you state sides proportional instead of congruent AA SAS SSS Why not ASA or AAS

Are the triangles similar Are the triangles similar. If yes then state the conjecture as to why and state the similarity statement (can you support the why what work or statements would you need), if no state why not.

Example

Prove these 2 Triangles Similar

2.3.3 Applying Similarity This uses proportions and setting up similar triangles Using a Mirror and Using shadows

Shadows These are the same proportion, but give you a different way to set them up

Example A tree casts a 13ft shadow, while a 5 ft tall person casts a shadow that’s is 1.5 ft. Sketch a picture and label it Set up a proportion and solve for the height of the tree. A flagpole casts a 20 m shadow while a 1.5 m person casts a shadow that is 1.2 m long. Sketch a picture and label it Set up a proportion and solve for the height of the flagpole.

Mirrors Instead of your total height it is only the height to your eyes and instead of the shadow length it is the distance to mirror. You place a mirror between you and the object. Move backwards until you see the top of the object in the middle of the mirror.

Mirror Cont.

Example A person places a mirror between herself and a building. When she is 3 m away from the mirror and the mirror is 17.5 m away from the building, she can see the top of the building in the center of the mirror. She is 2 m tall. Sketch and label a picture Set up proportion and find the height of the building

Example 2 A mirror is placed between a person who is 5.5 ft tall and a statue of an unknown height. When the person is 6 ft away from the mirror and the mirror is 20 ft away from the statue, he can see the top of the statue in the center of the mirror. Sketch and label a picture Set up a proportion and solve for the height of the statue

2.3.4 Similar Triangle Proofs Similar Triangles and Parallel lines

Parallel Lines and Transversal Alternate interior angles are congruent if the lines are parallel Corresponding angles are congruent if the lines are parallel Could use SSI to prove lines parallel Think of the triangle inside of a triangle and how you might prove them similar Why are angles 1 and 2 congruent

Parallel/Proportionality Conjecture If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. (This ratio is not the scale factor between the triangles) The sides the parallel line intersects creates 2 segments on each side, those 4 segments are proportional Assume line is parallel to base of triangle

Converse If the segments are proportional then the lines are parallel. You need to know these conjectures forwards and backwards

Example You can use either the triangles or the proportional segments Using triangles – ab cancels

Example Find AB ___ if EC=9

Extended Parallel/Proportionality Conjecture If two or more lines pass through two sides of a triangle parallel to the third side, then they divided the two sides proportionally. This just means you can set up multiple proportions depending on the segments you are using on one side of the figure

What proportions can be set up

Example Show that sides are divided proportionally, set up similar triangles and just proportional sides.

Answers to Examples 1. 2. 3. Proportional segments x = 24 Here you need to set up similar triangles AB=27 X= 36