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3.3 – Similar Right Triangles
Objectives
 Determine if right triangles are similar when given only one acute angle measure in each triangle.
 Prove that if an altitude is drawn from the right angle vertex of a right triangle to its hypotenuse, then three similar triangles are formed.
 Calculate the missing sides of similar right triangles using proportions.
 Solve realworld problems by using the properties of similar right triangles.
Notes
Similar Right Triangles 
 Similar right triangles have the same shape, but not always the same size.
 Their corresponding angles are congruent.
 Their corresponding sides are proportional.
 If two triangles are similar, all of their angles are congruent.
 Transitive Property (like a Syllogism)
 If a = b and b = c, then a = c.
 If a ~ b and b ~ c, then a ~ c.
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Right Triangle Similarity Theorem 
 You have a right triangle.
 You draw an altitude from its right angle to its hypotenuse.
 This forms two smaller right triangles.
 Both smaller triangles are similar to the original!

 How to Use Proportions to Find a Side Length
(A proportion is an equation that shows two ratios are equal).
 Step 1: Write the proportion with side names.
 Step 2: Replace the names with numbers (side lengths you know) and a variable (unknown side length).
 Step 3: Cross multiply.
 Step 4: Solve the equation for the variable.

Examples 
 Distance from a Tree (Example)
 Leslie places a mirror on the ground 63 feet from the base of a tree. She walks backwards until she can see the top of the tree in the middle of the mirror. At that point, Val’s eyes are 5 feet above the ground and she is 9 feet from the image in the mirror. What is the height of the tree?
 Steps
 Step 1: Draw the diagram (like the one above, where the triangles are facing each other)
 Step 2: Label the side lengths
 Step 3: Set up the proportions
 Step 4: Cross multiply
 Step 5: Simplify & solve for x
 Solve
 : Set up the proportion
 : Cross multiply
 : Divide both sides by 10
 : The tree is 35 feet tall.

 Two right triangles are can be proven using the AA Similarity Postulate
 The sum of all three angles of a triangle add up to 180.
 Similar right triangles will both have a congruent 90 degree angle.
 You can find the missing angle of each triangle below using this postulate.
 Both triangles have a 90, 72, and 18 degree angles

Important!
Practice (Apex Study 3.3)
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