Intervention Roadmap Teacher Guide Intervention Roadmap
Teacher Facilitation Guide: Rigid Motion Congruence
Standard
HS.G-CO.B.8 (Colorado)
Lesson Objective
Students will explain the sufficiency of SSS, SAS, and ASA criteria for triangle congruence by describing the sequence of rigid motions (translations, rotations, reflections) that maps one triangle onto another.
Facilitation Steps
1
The "Anchor" Vertex (5 mins)
Start by explaining that for two triangles to be congruent, we must be able to "stack" them perfectly. The first step is always moving one vertex to match its corresponding vertex using translation .
2
Aligning the Side (10 mins)
Once a vertex is anchored, use rotation to align a side. Emphasize that in SSS, SAS, and ASA, we are given that this side is equal, so the second vertex MUST now land on the corresponding vertex.
3
The "Fixed" Logic (15 mins)
This is the heart of the lesson. Explain that once two points are fixed, the third point is "locked" into place by the remaining given information:
SSS: The lengths from the two anchored points determine the third point's exact location (intersection of two circles).SAS: The angle and the side length determine exactly where the third vertex must go.ASA: The two angles "point" toward each other, intersecting at a unique third point.
Tier 2 Scaffolding
Visual Cues
Use patty paper or transparency film for students to physically translate and rotate triangles before writing.
Sentence Starters
"First, I will translate point A to point D because..."
Misconception Alert
The "Over-Rotation"
Students often think rotation changes the triangle. Remind them rigid motions preserve distance and angle measure .
Ambiguity of SSA
Briefly show that with SSA, the third vertex isn't "locked" — it could be in two different places. This reinforces why we need SAS/ASA.
Progress Monitoring
Look for these indicators during the small group work:
Can identify which rigid motion aligns a vertex.
Uses the term "preserve" when talking about side lengths.
Can explain why SAS "locks" the third side.
Motion Match Slides Motion Match
Proving Congruence with Rigid Motions
Translate
Rotate
Reflect
What is the Goal?
To show two triangles are congruent , we must prove that one can be exactly mapped onto the other using only rigid motions.
"If they stack, they match!"
1 Universal First Steps
A: Translate
Pick a vertex on Triangle 1 and translate it to its partner on Triangle 2.
Result: 1 point matched
B: Rotate
Using the matched vertex as a pivot, rotate the side to align with the partner side.
Result: 1 side aligned
The SAS Logic
Side-Angle-Side
Since the Angle is identical, the side points in the right direction.
Since the Side Length is identical, the vertex lands on its partner.
Locked In:
Distance and Direction are fixed.
Side Angle Side
The ASA Logic
Angle-Side-Angle
The two Angles act like two laser beams pointing at each other.
They must intersect at exactly one unique point.
Locked In:
The intersection is unique.
Side Angle Angle
Proof Architect Worksheet Proof Architect Worksheet
Name: _________________________________
Date: __________________________________
Objective: Use rigid motions (translation, rotation, reflection) to explain why triangles are congruent.
1
Blueprint: SSS Logic
Given that all three sides of \(\triangle ABC\) match the three sides of \(\triangle DEF\).
A B C
Construction Steps
Step 1: ____________________ point A to point D.
(Movement: Move the entire triangle to match a vertex)
Step 2: ____________________ \(\overline{AB}\) to align with \(\overline{DE}\).
(Movement: Pivot around the matched vertex)
The Result: Since side lengths match, point B lands on point _____. Because the third side length is fixed, point C must land on point _____.
2
Blueprint: SAS Logic
Why is one angle and two sides enough to "lock" the triangle?
Side (Given) Side (Given) Angle
Once we align vertex A and side AB...
1. Because the Angle is the same, side AC is pointing in the ____________________ direction as side DF.
2. Because the Side Length AC is congruent to DF, point C must land ____________________ on point F.
3
Blueprint: ASA Logic
The Unique Intersection
"If you have a fixed base and two fixed angles, the lines will only meet at one specific point."
Final Argument:
Imagine you are explaining this to a classmate. Why does knowing the Side and the Two Angles at the ends of that side mean the whole triangle is "locked" into place?
Snapshot Check Exit Ticket Snapshot Check
Rigid Motion Reasoning
Quick Progress Check
Name: ________________________
Date: _________________________
1 Which rigid motion is best described by each action?
Slide a triangle to align a vertex ____________________
Spin a triangle around a point ____________________
Flip a triangle over a line ____________________
Rigid motions always preserve: Side length & _________
2 The "Logic Lock"
You have mapped vertex A to vertex D and aligned side AB with side DE. Explain why knowing SAS means the third vertex C must land on point F.
3 Check for Understanding
Why is knowing SSS sufficient for triangle congruence?
A) Because it is a shorter way to say the triangles look the same.
B) Because the three side lengths fix the positions of the vertices.
C) Because rigid motions can change side lengths but not angles.
D) Because translation and rotation are only used for SSS.
How confident do you feel explaining triangle congruence logic?
🤔 😐 😊
Congruence Logic Answer Key ANSWER KEY
Proof Architect Worksheet
Part 1: SSS Logic
Step 1: Translate point A to point D.
Step 2: Rotate \(\overline{AB}\) to align with \(\overline{DE}\).
Conclusion: Since the lengths are equal, point B must land on point E. Since the third side lengths are fixed, point C must land on point F.
Part 2: SAS Logic
First, map vertex A to vertex D.
Part 3: ASA Logic
Exemplar Explanation:
"Knowing the side length fixes points A and B onto D and E. Because we know both angles, we know the direction of the lines from A and B. Since lines with these specific angles can only intersect at one unique point, the third vertex C must land exactly where the third vertex F is located."
Snapshot Check (Exit Ticket) Key
Q1: Matching
Slide = Translation
Spin = Rotation
Flip = Reflection
Preserves = Angle measure
Q3: Multiple Choice
Correct Answer: B
"Because the three side lengths fix the positions of the vertices."
