1. 2-110. SIDE SPLITTER Hot Potato; e together
Janelle & Kamraan were working on their hw together.
For the diagram at right, Kamraan wrote the equation 4 6 = 5 5+𝑥 .  Is his equation correct?  Explain.  If needed, revise his equation, and then solve for x.
4(5 + x) = 6(5) → x = 2.5
Janelle wrote the equation  4 2 = 5 𝑥 .  Janelle said, “Oops, I wrote the wrong equation, but I still got the same answer!”  How did Janelle come up with her equation?
She just used the parts of the corresponding sides.
Kamraan asked Janelle, “Will that always work?  Can we prove it?”  He drew the figure at right.  What information should he add to his diagram?  What does he need to prove?
Must have parallel lines so that it can be AA~.
With your team, write an argument that Janelle’s method will always work, or provide a counterexample.  First, use the definition of similar triangles to write a proportion comparing the corresponding side lengths.
Since they are similar, 𝑎 𝑎+𝑏 = 𝑐 𝑐+𝑑
Use algebra to rewrite the proportion you wrote in part (d) to show that . The relationship that you proved is sometimes called the Side Splitter Theorem.  Write the relationship as a conditional statement.
𝑎 𝑎+𝑏 = 𝑐 𝑐+𝑑 and 𝑎 𝑏 = 𝑐 𝑑 both yield ad = bc
If a segment intersects two sides of a triangle and is parallel to the third side of the triangle, then the lengths of the parts are proportional.
Hot Potato; e together
Hot Potato; e together

2. 2.3.4 Similar Triangle Proofs
HW: to 2-119
September 20, 2018

3. Objectives
CO: SWBAT write proofs using similar triangles and their properties. LO: SWBAT explain their reasoning using proofs.

4. 2-109.  DOES AA ~ ALWAYS WORK?
Scott knows that if two angles of a triangle are congruent to two angles of another triangle, then the triangles will always be similar.  But now his teacher has asked his team to use transformations to explain why AA ~ always works.
Eliana says, “Let’s start with a diagram.  I’m going to name my triangles ΔPQR and ΔTUV.  I know that two pairs of angles are congruent, so I can mark ∠P ≅ ∠T and ∠Q ≅ ∠U on the diagram.  Do we know anything else?” Copy Eliana’s diagram.  If you know anything else about the triangles, mark that information and justify your reasoning.  
Scott says, “I think it will help if we can match up two of the congruent angles.” Eliana says, “Well, if the angles are congruent, that means there are rigid transformations that will move one onto the other, right?  Let’s map ∠P onto ∠T.”  Is Eliana correct?  Justify your answer and draw a diagram that shows ∆P'Q'R' and ∆TUV after the sequence of transformations.
What do the angle relationships in your diagram tell you about  𝑄 ′ 𝑅′  and  𝑈𝑉 ?  Explain.
Eliana says, “Hey, that looks like those problems we did with rubber bands.  We can just dilate ∆P'Q'R' and then we’ll have ∆TUV!” Scott says, “But we don’t know the side lengths, so how do we dilate so that the vertices match up?” Help them write a scale factor that will take point Q' to point U if point T is the center of dilation.  (Use the letters of the sides like we did in the flow proofs)
Kendall is confused.  “Okay, I know that parallel lines stay parallel after a dilation.  So, if   𝑄 ′ 𝑅′  and  𝑈𝑉 are parallel and point Q“ coincides with point U, then point R" will coincide with point V.  But my diagram doesn’t look like Eliana’s diagram.  Will it still work?” See Kendall’s diagram at right.  Will the dilation you described in part (d) still work?  Explain.
Progress Chart

5. Progress Chart a-b c-e Purple Stripes Blue Green Pink Orange Yellow
Red

6. 2-109.  DOES AA ~ ALWAYS WORK?
Scott knows that if two angles of a triangle are congruent to two angles of another triangle, then the triangles will always be similar.  But now his teacher has asked his team to use transformations to explain why AA ~ always works.
Eliana says, “Let’s start with a diagram.  I’m going to name my triangles ΔPQR and ΔTUV.  I know that two pairs of angles are congruent, so I can mark ∠P ≅ ∠T and ∠Q ≅ ∠U on the diagram.  Do we know anything else?” Copy Eliana’s diagram.  If you know anything else about the triangles, mark that information and justify your reasoning.  
Scott says, “I think it will help if we can match up two of the congruent angles.” Eliana says, “Well, if the angles are congruent, that means there are rigid transformations that will move one onto the other, right?  Let’s map ∠P onto ∠T.”  Is Eliana correct?  Justify your answer and draw a diagram that shows ∆P'Q'R' and ∆TUV after the sequence of transformations.
What do the angle relationships in your diagram tell you about  𝑄 ′ 𝑅′  and  𝑈𝑉 ?  Explain.
Eliana says, “Hey, that looks like those problems we did with rubber bands.  We can just dilate ∆P'Q'R' and then we’ll have ∆TUV!” Scott says, “But we don’t know the side lengths, so how do we dilate so that the vertices match up?” Help them write a scale factor that will take point Q' to point U if point T is the center of dilation.  (Use the letters of the sides like we did in the flow proofs)
Kendall is confused.  “Okay, I know that parallel lines stay parallel after a dilation.  So, if   𝑄 ′ 𝑅′  and  𝑈𝑉 are parallel and point Q“ coincides with point U, then point R" will coincide with point V.  But my diagram doesn’t look like Eliana’s diagram.  Will it still work?” See Kendall’s diagram at right.  Will the dilation you described in part (d) still work?  Explain.
∠𝑅≅∠𝑉 because of triangle angles sum theorem
Any transformation makes angles congruent. We can use rotation.
They must be parallel because they are corresponding angles and they are congruent.
Progress Chart
𝑈𝑉 𝑄 ′ 𝑅′
We technically do not know which triangle must be bigger since it doesn’t say they are drawn to scale.

7. 2-111. SIDE SPLITTER CONVERSE
In problem 2-110, you proved that if a segment intersects two sides of a triangle and is parallel to the third side of the triangle, then the lengths of the parts are proportional.  Now you and your team will prove the converse of the Side Splitter Theorem. 
Write the converse of the Side Splitter Theorem as a conditional statement or as an arrow diagram.
If a segment intersects two sides of a triangle so that the lengths of the parts are proportional, then the segment is parallel to the third side of the triangle.
The hypothesis, or the “if” part of your if-then statement, tells what you can use as given information in your proof.  Using the diagram at right, what equation can you write from the given information?
𝑎 𝑏 = 𝑐 𝑑
Are there similar triangles in the figure at right?  If so, write a similarity statement and justify your conclusion using a similar triangles condition.
If 𝑎 𝑏 = 𝑐 𝑑 , 𝑡ℎ𝑒𝑛 𝑎 𝑎+𝑏 = 𝑐 𝑐+𝑑 . They share <M, which means they are similar by SAS~.
Using your findings from part (c), prove that  𝑇𝑉  is parallel to  𝑆𝑁 .
If they are similar, then it is a dilation, which makes the lines parallel.
a & d = partners; b & c = together
a & d = partners; b & c = together