Download 9-6 Proving Triangles Similar Day 2

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Geometry Per ____
Lesson 9-6 Notes
Date: _________________
8-9 Proving Similar Triangles Day 2
Learning Goals: How can we prove two triangles are similar? How can we use similarity to prove proportions? How can
we use similarity to prove cross product equivalency?
Today we are going to continue with similarity proofs. From
yesterday, we learned how to prove triangles are similar.
Now, we are going to extend that and prove proportions and
equal cross products.
We use 3 tiers to guide us through proofs of and/or involving similarity:
*We can say that these two triangles are similar by the
____________ shortcut.
*Since they are similar we can set up the following proportion:
𝐴𝐵 𝐴𝐶
=
𝑆𝑅
*Take it one step further! What cross products do we get when we
cross multiply?
Let’s try one:
Given
is an isosceles triangle with base AC and BD is perpendicular to AC, prove AD ∗ BC = BA ∗ CD
Where should we start?
What tier do we need to use?
What triangle are we going to prove to be similar?
What’s our plan? Justify at all steps!
*What is given to us?
*What can we infer from the givens?
Statement
1.
is isosceles triangle with base AC and BD is perpendicular to AC.
Reason
1.
𝟐. ∠BDA and ∠BDC are right angles.
2.
𝟑. ∠BDA ≅ ∠BDC
3. All right angles are congruent.
𝟒. ∠A ≅ ∠𝑩𝑪𝑨
𝟒.
5. ∆𝑩𝑫𝑨 and ∆𝑩𝑫𝑪 are ________________.
5.
Perfect Practice Makes Perfect!
You are going to make a fruit salad! Your salad needs to consist of at least two apples,
2 pineapples, and 2 watermelons! You get to pick which one’s you want to do!
Apples - you need at least 2!
1. Given: AB//DE Prove: ∆𝐷𝐶𝐸 ~ ∆𝐵𝐶𝐴
Statement
Reason
Pre-proof work
Do I need to re-draw my
triangles separately?
Did I mark my diagram?
Do I have a plan?
2. Given: ∆ABC with BA ≅ BC and BD is an angle bisector of<B. Prove: ∆ABD ~ ∆CBD
*Careful! What does it mean for BD to be an angle bisector?
3. Given: ∆ABC is isosceles with base AC; BD and EC are altitudes. Prove: ∆𝐵𝐷𝐶 ~ ∆𝐶𝐸𝐴.
(*Hint! What do you remember about the altitude of an isosceles triangle? Does it have any other special properties?)
Statements
Reasons
4. Givens: ∆FGH is an isosceles triangle with legs FG and HG. GK bisects ∠FGH.
Prove: ∆FGK ~ HGK
PINEAPPLES - you need at least 2!
1. Given: WA // CH and WH and AC intersect at point T.
𝑾𝑻
𝑾𝑨
Prove:
=
𝑯𝑻
𝑯𝑪
2. Given: HW//TA and HY//AX, prove
𝑨𝑿
𝑯𝒀
=
𝑨𝑻
𝑯𝑾
hH lines!
Hint: extend
Pre-Proof work:
3. Given: Trapezoid ABCD with bases BC and AD
𝐵𝐶
Prove:
𝐴𝐷
=
𝐸𝐶
𝐸𝐴
a) What tier type of question is this?
b) Based on the given information mark your diagram
appropriately (In a trapezoid the bases are _______________)
c) Determine the 2 triangles we are looking to prove similar based
on the sides we are working with (redraw the triangles below).
WATERMELON - you need at least 2!
1. Given AC ⊥ BD and DE ⊥ AB. Prove: AC * BD = DE * AB
Where should we start?
What tier do we need to use?
What triangle are we going to prove to be similar?
Redraw Triangles!
What’s our plan? Justify at all steps!
*What is given to us?
*What can we infer from the givens?
Statement
1.
AC ⊥ BD and DE ⊥ AB.
Reason
1.
2.
𝟐. ∠________and ∠________ are right angles.
𝟑. ∠______ ≅ ∠______
3. All right angles are congruent.
4.
4.
𝟓. ∆
and ∆
6.
=
are ________________.
7. AC * BD = DE * AB
2.
̅̅̅̅ ⊥ 𝐴𝐵
̅̅̅̅̅ and 𝐶𝐵
̅̅̅̅ ⊥ 𝐶𝐴
̅̅̅̅̅
Given: 𝐶𝐷
5.
𝟔.
7.
Prove: BC * DB = AC * CD
Pre-proof work
Do I need to re-draw my triangles separately?
Did I mark my diagram?
Do I have a plan? (What tier should I use?)
3. Given CB ⊥ BA, CD ⊥ DE, prove AB * CD = DE * CB