**Monday, September 23rd** Warm Up S is the centroid. If NP=7, what is PL? If SQ=4, what is SN?

**Week at a Glance** Monday: Review-Centroid and Proofs Tuesday: parallelogram proofs Wednesday: Similar Triangles Thursday: Similarity Triangle Proofs Friday: Similiarity Triangle Proofs CW Test #2-Next Wednesday

**Part 1: Centroids**

**A median of a triangleis a segment whose endpoints are a** vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent.

**The point of concurrency of the medians of a triangle is the** centroid of the triangle. The centroid is always inside the triangle.

**Vertex vs. Mid-Segment ** • Vertex to Centroid = 2/3 distance of the median • Mid-segment to centroid=1/3 distance of median

**Centroid** midsegment Vertex Median

**#1** In ∆LMN, RL = 21. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify.

**#2** In ∆LMN, SQ=4. Find NQ. SQ= mid-segment to centroid

**Total: 4 + 4 + 4 = 12 ** Centroid 4 4 4 midsegment Vertex Median

**Algebra ** Midsegment to centroid= 1/3 distance 4=1/3NQ *Use division (multiply by reciprocal)

**In ∆JKL, ZW =8 find ZK** #3 ZW=midsegment to centroid

**Total: 8 + 8 =16 ** Centroid 8 8 8 midsegment Vertex Median

**#4** In ∆JKL, LX= 8.1. Find LZ. Centroid Thm. Substitute 8.1 for LX. LZ = 5.4 Simplify.

**You Try! ** Z is the centroid. YZ = 5, JX= 10, KW = 15 KX ZJ YJ KZ

**Part 2: Proofs **

**Don’t Forget! ** 2 markings you can add if they aren’t marked already

**Share a side** Reason: reflexive property Vertical Angles Reason: Vertical Angles are congruent

**Congruence Statement** In a congruence statement ORDER MATTERS!!!! Everything matches up.

**Congruence Statement** Angles are written with either 1 letter or three letters! Sides are written with two letters

**1. Write down two congruent sides 2. Write two congruent** angles3. GAU is congruent to _________

**There are 5 ways to prove triangles congruent.**

**Remember ** • Go around clockwise • Can’t skip sides and angles

**Side-Side-Side (SSS) Congruence Postulate** All Three sides in one triangle are congruent to all three sides in the other triangle

**Side-Angle-Side (SAS) Congruence Postulate** Two sides and the INCLUDED angle (the angle is in between the 2 marked sides)

**A** A A A S S Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included

**Angle-Side-Angle (ASA) Congruence Postulate** A A S S A A Two angles and the INCLUDED side (the side is in between the 2 marked angles)

**There is one more way to prove triangles congruent, but** it’s only for RIGHT TRIANGLES…Hypotenuse Leg HL

**SSS** SAS ASA AAS HL NO BAD WORDS The ONLY Ways To Prove Triangles Are Congruent

**Key Tips ** • First statement is always GIVEN. • Other statements may also be GIVEN • Last statement should be what you are trying to prove and be using one of our 5 triangle Postulates (SSS, ASA, Etc.) • Draw in your dashes and angles • Try to remember all the rules/theorems/postulates we have covered since the beginning of the year!

**CW- More Practice with Proofs ** 1-9

**HW: Proof Scramble **