Logic Link Slides LOGIC BLUEPRINTS
Mastering Congruence & Similarity Criteria
The Detective's Dilemma
Before we build a proof, we must ask:
Are they Twin or Mini-Me?
Congruent: Identical size and shape.Similar: Same shape, scaled size.
Standard HS.G-SRT.B.5: Using criteria to solve and prove relationships.
Congruent (Same) Similar (Scaled)
Congruence Toolbox (CPCTC Ready)
SSS
Side-Side-Side
SAS
Side-Angle-Side
ASA
Angle-Side-Angle
AAS
Angle-Angle-Side
HL
Hypotenuse-Leg
DANGER: "ASS" or "AAA" do NOT prove congruence!
Similarity Toolbox
AA ~
Angle-Angle
If two angles match, the third MUST match. The shape is locked!
SSS ~
Side-Side-Side
All three pairs of sides are in the same ratio .
SAS ~
Side-Angle-Side
Two pairs of sides are proportional AND the included angle is equal.
Decision Tree Strategy
Start: Look at the Diagram
Identify Givens (Sides? Angles?)
Equal or Proportional?
Equal → Congruence
Ratio → Similarity
Pick Your Tool
Match your givens to the criterion (e.g., ASA).
Team Blueprint Check
Diagram Situation:
Triangles $ABC$ and $DEF$ have $\angle A \cong \angle D$.
The ratio of sides $AB/DE = 2$.
The ratio of sides $AC/DF = 2$.
Discuss: Which similarity criterion proves they are similar?
The Logic Check:
S? A? S?
SAS ~
Criteria Compass Reference CRITERIA COMPASS
Decision Support for Geometric Proofs
Name: ________________________________
Date: ________________________________
The Proof Path: Where do I start?
Follow the arrows based on what you see in the diagram or read in the "Givens".
Step 1 Are the side lengths EQUAL or PROPORTIONAL?
1
Congruence ($\cong$)
Use these if the triangles are the exact same size .
SSS
3 Sides: All corresponding sides are equal.
SAS
Side-Angle-Side: Angle is between the two sides.
ASA
Angle-Side-Angle: Side is between the two angles.
AAS
Angle-Angle-Side: Side is not between the angles.
HL
Hypotenuse-Leg: Only for Right Triangles .
2
Similarity ($\sim$)
Use these if the triangles are different sizes but same shape.
AA~
Angle-Angle: 2 matching angles lock the shape.
SSS~
3 Sides: All sides have the same ratio ($\frac{a}{A} = \frac{b}{B} = \frac{c}{C}$).
SAS~
Side-Angle-Side: Angle equal, surrounding sides proportional.
PRO TIP:
If you find parallel lines , look for Alternate Interior Angles or Corresponding Angles to use AA~ !
Detective's Secret Tools (Find hidden information)
Reflexive Property
Side is shared ($AB = AB$)
Vertical Angles
Angles across from the X
Parallel Line Angles
Alternate Interior or Corresponding
Criteria Compass - Geometry Intervention Resource Standard: HS.G-SRT.B.5
Proof Blueprint Worksheet PROOF BLUEPRINTS
Building Logical Foundations
NAME:
PARTNER:
DIRECTIONS: Use your Criteria Compass to analyze each pair of triangles. Identify the given information, determine if they are Congruent ($\cong$) or Similar ($\sim$), and select the correct criterion.
1
Case Study: Shared Walls
A B C D
Given: $C$ is the midpoint of $AD$. $BC \perp AD$.
1. List the Givens (Equal Parts):
2. Hidden Property Found (Reflexive/Vertical):
3. Final Criterion:
SSS
SAS
ASA
HL
2
Case Study: Proportional Profiles
5 8 10 16 35° 35°
1. Check the Ratio (Small/Large):
$\frac{5}{10} = $ _______ $\frac{8}{16} = $ _______
2. Is the Angle "Included"?
Yes
No
3. Conclusion:
These triangles are similar by:
FIELD CHALLENGES
Challenge A: The Bowtie
Given: Left and Right vertical lines are parallel.
1. Find TWO pairs of equal angles:
Pair 1: ____________ because ____________
Pair 2: ____________ because ____________
2. Are they similar or congruent? Why?
Challenge B: The Missing Piece
You want to prove $\triangle ABC \cong \triangle XYZ$ using **ASA**. You already know:
$\angle A \cong \angle X$
$\angle B \cong \angle Y$
What is the third piece of information you need?
$AC \cong XZ$
$AB \cong XY$
$BC \cong YZ$
Explain your choice:
Reflect: When you are stuck, what is the FIRST tool you check in your Criteria Compass?
Bridge Builder Exit Ticket Bridge Builder
EXIT TICKET
NAME:
DATE:
Task 1: The Blueprint Match
8 8 15 15
Which congruence criterion proves these triangles are identical?
SSS
SAS
ASA
HL
Task 2: Scaling Up
Triangle $PQR$ has sides $6, 8, 10$. Triangle $STU$ has sides $9, 12, 15$.
1. Calculate the ratio for each side pair:
$\frac{6}{9} = $ ________
$\frac{8}{12} = $ ________
$\frac{10}{15} = $ ________
2. Are they similar? If so, by what criterion?
Confidence Check
Struggling Getting It Mastered
Intervention Intel Teacher Guide INTERVENTION INTEL
Teacher Facilitation Guide
Targeted Skill
HS.G-SRT.B.5
Learning Objective
Students will correctly identify and apply the minimum criteria required to prove triangle congruence and similarity in varied geometric configurations. Focus is on moving from "seeing" to "proving" using mathematical language.
Group Profile
Tier 2 Intervention: Small group (3-5 students). Students may struggle with spatial reasoning or logical sequencing.
High-Alert Misconceptions
The "ASS" Trap: Students often assume Side-Side-Angle works. Remind them that if the angle isn't "sandwiched" (included), the triangle can swing and change shape.Similarity vs. Congruence: Students confuse ratio with equality. Use the "Twin vs. Mini-Me" analogy from the slides.
Instructional Flow
1
The Hook & Visual Scan (10 mins)
Use the Logic Link Slides . Don't let students guess; force them to point at the diagram and say "I see an angle" or "I see a shared side."
Prompt: "If you were a detective, what evidence is already marked on the wall?"
2
The Criteria Compass (10 mins)
Introduce the Criteria Compass Reference . This is their permanent scaffold. Practice navigating the flowchart with 1 simple example before moving to the worksheet.
3
Collaborative Blueprinting (20 mins)
Students work in pairs on the Proof Blueprint Worksheet .Intervention Move: If a student is stuck, ask "What does the middle letter in SAS stand for?" to redirect their focus to the 'included' angle.
4
Progress Check (5 mins)
Administer the Bridge Builder Exit Ticket independently. Use the results to group students for the next session's deeper proof writing.
Effective Questioning Prompts
"Can you find a side that both triangles use at the same time?" (Reflexive)
"If these were pieces of a puzzle, would they fit perfectly, or would one be a zoomed-in version of the other?"
"Why isn't AA enough for congruence, but it's enough for similarity?"
