Monday, September 23 rd. 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
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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
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
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 _________
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!