Example: Using Corresponding Parts of Congruent Triangles Given: ∆ABC  ∆DBC. Find the value of x.  BCA and  BCD are rt.  s.  BCA   BCD m  BCA = m  BCD (2x – 16)° = 90° 2x = 106 x = 53 Def. of  lines. Rt.   Thm. Def. of   s Substitute values for m  BCA and m  BCD. Add 16 to both sides. Divide both sides by 2.

Example: Using Corresponding Parts of Congruent Triangles Given: ∆ABC  ∆DBC. Find m  DBC. m  ABC + m  BCA + m  A = 180° m  ABC = 180 m  ABC = 180 m  ABC = 40.7  DBC   ABC m  DBC = m  ABC ∆ Sum Thm. Substitute values for m  BCA and m  A. Simplify. Subtract from both sides. Corr.  s of  ∆s are . Def. of   s. m  DBC  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!!

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

A two-column proof has…surprise… TWO columns… StatementsReasons “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 using Step 3: ATTACK! Check off the use Step 4: Get to the end goal, the PROVE

Example 1: Using SSS to Prove Triangle Congruence Prove:∆ABC  ∆DBC. 1. Given 2. Given 3. Reflexive Property 4. SSS S S S USE: SSS StatementsReasons ✔✔ ✔

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 particular An example of SAS S S A

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. Given 2. Vertical angles are congruent 3. Given 4. SAS S A S USE: SAS StatementsReasons ✔✔ ✔

Example 3: Proving Triangles Congruent Given: BC ║ AD, BC  AD Prove: ∆ABD  ∆CDB ReasonsStatements 5. SAS 5. ∆ 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 S P Not enough info!