example: using corresponding parts of congruent triangles given: ∆abc ∆dbc. find the value of...
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In Lesson 4-5, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. Luckily for us! There is a short cut!!TRANSCRIPT
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Example: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC ∆DBC.
Find the value of x.
BCA and BCD are rt. s.
BCA BCD
mBCA = mBCD
(2x – 16)° = 90°
2x = 106
x = 53
Def. of lines.
Rt. Thm.
Def. of s
Substitute values for mBCA and mBCD.
Add 16 to both sides.
Divide both sides by 2.
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Example: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC ∆DBC.
Find mDBC.
mABC + mBCA + mA = 180°
mABC + 90 + 49.3 = 180
mABC + 139.3 = 180
mABC = 40.7
DBC ABC
mDBC = mABC
∆ Sum Thm.
Substitute values for mBCA and mA.
Simplify.
Subtract 139.3 from both sides.
Corr. s of ∆s are .
Def. of s.
mDBC 40.7° Trans. Prop. of =
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In Lesson 4-5, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.
Luckily for us! There is a short cut!!
Hurray!!
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What do you think SSS stands for?Side-side-side
What do you think SAS stands for?Side-angle-side
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Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!
Reflexive Property is your new best friend
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OUR FIRST PROOF is here!!!!!
yay
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A two-column proof has…surprise… TWO columns…
Statements Reasons
“Word” stuff“Math” stuff
You will always be given 1 or more “Givens” and you will always be given a “Prove”
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Step 1: MARK IT UP!!!Step 2: Decide what you are usingStep 3: ATTACK! Check off the useStep 4: Get to the end goal, the PROVE
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Example 1: Using SSS to Prove Triangle CongruenceProve:∆ABC ∆DBC.
1. Given2. Given3. Reflexive Property4. SSS
SS
S
USE: SSS
Statements Reasons
✔ ✔ ✔
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The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.
Caution
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An included angle is an angle formed by two adjacent sides of a polygon.B is the included angle between sides AB and BC.
SAS is sassy and particularAn example of SAS
S
SA
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An example of a SAS impersonator
S
S
A
Yes, the impersonator forms a bad word. We will be discussing this one later…
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Example 2: Engineering Application
Prove: ∆XYZ ∆VWZ.
1. Given2. Vertical angles are congruent3. Given4. SAS
SA
S
USE: SAS
Statements Reasons
✔ ✔ ✔
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Example 3: Proving Triangles Congruent
Given: BC ║ AD, BC AD
Prove: ∆ABD ∆CDB
ReasonsStatements
5. SAS5. ∆ABD ∆ CDB
4. Reflex. Property
1. Given
3. Alt. Int. s Thm.3. CBD ABD
2. Given2. BC || AD
1. BC AD
4. BD BD
USE: SAS
Step 1 MARK IT UP!
S
S
A
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Check It Out! Example 4
Given: QP bisects RQS
Prove: ∆RQP ∆SQP
ReasonsStatements
R
Q
SP
Not enough info!