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.

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 =

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!!

What do you think SSS stands for?Side-side-side

What do you think SAS stands for?Side-angle-side

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

OUR FIRST PROOF is here!!!!!

yay

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”

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

Example 1: Using SSS to Prove Triangle CongruenceProve:∆ABC ∆DBC.

1. Given2. Given3. Reflexive Property4. SSS

SS

S

USE: SSS

Statements Reasons

✔ ✔ ✔

The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

Caution

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

An example of a SAS impersonator

S

S

A

Yes, the impersonator forms a bad word. We will be discussing this one later…

Example 2: Engineering Application

Prove: ∆XYZ ∆VWZ.

1. Given2. Vertical angles are congruent3. Given4. SAS

SA

S

USE: SAS

Statements Reasons

✔ ✔ ✔

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

Check It Out! Example 4

Given: QP bisects RQS

Prove: ∆RQP ∆SQP

ReasonsStatements

R

Q

SP

Not enough info!