Similarity Secrets Slides Similarity Secrets
Mastering AA, SAS, and SSS
Essential Question
How can we measure objects that are too large or distant to reach?
The "Cousin" Challenge
Are these two triangles related (similar) or just strangers? What evidence would you need to prove they are "family"?
What is Similarity?
Corresponding Angles are congruent (\(\cong\)).
Corresponding Sides are proportional (same ratio).
"Same shape, different size."
A B C
Criterion 1: AA
Angle-Angle Similarity
The Shortest Path
If two angles of one triangle are congruent to two angles of another, the triangles are similar.
\(\angle A \cong \angle D\)
\(\angle B \cong \angle E\)
\(\triangle ABC \sim \triangle DEF\)
Criterion 2: SAS
Side-Angle-Side Similarity
The Sandwich Rule
If two sides are proportional AND the included angle is congruent.
Conditions:
1. \(\frac{AB}{DE} = \frac{AC}{DF}\)
2. \(\angle A \cong \angle D\)
Angle MUST be between the sides!
Criterion 3: SSS
Side-Side-Side Similarity
If all three sides of two triangles are proportional...
\(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\)
Check every ratio. They must be IDENTICAL.
Similarity Scrutiny
CASE #1
Triangle A has sides 4, 6, 8. Triangle B has sides 10, 15, 20.
Are they similar?
CASE #2
Two right triangles each have a 35° angle.
Which theorem proves it?
Think before you ink. Evidence required.
Triangle Cousins Worksheet Triangle Cousins Worksheet
Topic: AA, SAS, and SSS Similarity Criteria
Name:
Date:
The "Cousin" Rule
In geometry, triangles are like family. We call them similar (or "cousins") if they have the same shape but different sizes. To prove they are related, we use three main pieces of evidence: AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side).
1
Are they related?
Determine if the following pairs of triangles are similar. If they are, state the criterion (AA, SAS, or SSS) and write the similarity statement. If not, explain why.
A B C 60° 70° D E F 60° 70°
Similar? (Yes/No):
Criterion:
Statement: \(\triangle ABC \sim \triangle\)
8 10 12 4 5 6
Similar? (Yes/No):
Criterion:
Statement: \(\triangle\) \(\sim \triangle\)
15 20 6 8
Similar? (Yes/No):
Criterion:
A B E C D
Similar? (Yes/No):
Criterion:
2
Solving for Cousins
The triangles in each pair are similar. Set up a proportion and solve for the missing variable.
24 x 10 15
Assume parallel horizontal lines.
Show your work:
\(x =\) ________
12 9 x + 2 15
Show your work:
\(x =\) ________
The "Why" Behind the Math
If you only know that two sides are proportional (\(1:2\)), why isn't that enough to say the triangles are similar? What else must you know to confirm the "family relationship"?
Side Splitter Slides Side Splitter Logic
Proportional Parts in Triangles
The Ladder Mystery
Imagine a ladder leaning against a wall. The rungs are perfectly parallel to the ground.
If you know the distance between rungs on one side, can you predict the distance on the other side?
"Parallel lines create proportional segments."
A B C
The Side-Splitter Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
\(\frac{AD}{DB} = \frac{AE}{EC}\)
B C A D E
Warning: This only applies to the segments of the split sides, not the parallel bases themselves!
Expanding the Rule
If three or more parallel lines intersect two transversals, they cut the transversals proportionally.
The Ratio Rule:
\(\frac{a}{b} = \frac{c}{d}\)
a b c d
Split Decision
If a triangle's side is split into segments of 5 and 10, and the other side has a segment of 7...
What is the last segment length?
Setup your proportion:
\(\frac{5}{10} = \frac{7}{x}\)
x = ?
Split the Side Worksheet Split the Side Worksheet
Topic: Triangle Proportionality Theorem & Parallel Lines
Name:
Date:
The Side-Splitter Rule
When a line is parallel to one side of a triangle, it cuts the other two sides into segments that are proportional. Formula: \(\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}}\)
1
Basic Proportions
In each figure below, the internal line is parallel to the base. Solve for \(x\).
8 12 10 x
\(x =\) ________
9 6 x + 3 4
\(x =\) ________
2
Transversal Tussle
15 25 18 y
Set up the proportion for the three parallel lines shown:
\(y =\) ________
3
The Rung Calculation
A custom ladder is built where the rungs are parallel. On the left side, the distance between the first two rungs is 12 inches, and the distance between the next two is 16 inches. If the distance between the first two rungs on the right side is 15 inches, what is the distance between the next two rungs on the right side?
Diagram & Calculation:
"Parallel rungs preserve the ratio!"
Mean Geometry Slides Mean Geometry
Altitudes & Geometric Means
The Altitude Split
When you draw an altitude from the right angle to the hypotenuse...
Magic Happens!
You create three triangles that are all similar to each other.
A C B D
\(\triangle ABC \sim \triangle ADB \sim \triangle BDC\)
The Geometric Mean
The geometric mean of two positive numbers \(a\) and \(b\) is the positive number \(x\) such that:
\(\frac{a}{x} = \frac{x}{b}\)
\(x = \sqrt{ab}\)
It's the "middle" value in a proportional sequence.
Geometric Mean Altitude
Theorem 1
The altitude is the geometric mean of the two segments of the hypotenuse.
\(\frac{s_1}{h} = \frac{h}{s_2}\)
s₁ s₂ h
Geometric Mean Leg
Theorem 2
Each leg is the geometric mean of the adjacent segment and the entire hypotenuse.
\(\frac{\text{segment}}{\text{leg}} = \frac{\text{leg}}{\text{hypotenuse}}\)
\(leg = \sqrt{seg \cdot hyp}\)
The Mean Calculation
Find the altitude:
Hypotenuse segments are 4 and 9.
Altitude = ?
Think deeply:
Why do we use the square root? How is this different from a normal average (arithmetic mean)?
Mean Geometry Practice Worksheet Mean Geometry Practice
Topic: Altitudes and Geometric Means in Right Triangles
Name:
Date:
The Altitude Rule
\(h = \sqrt{s_1 \cdot s_2}\)
Altitude is the geometric mean of the two hypotenuse segments.
The Leg Rule
\(leg = \sqrt{seg \cdot hyp}\)
Leg is the geometric mean of its adjacent segment and the whole hypotenuse.
1
Calculate the Mean
Find the geometric mean of the following pairs of numbers. Leave answers in simplest radical form if necessary.
A) 4 and 9
B) 2 and 32
C) 5 and 20
2
Solve for Variables
3 12 x
Setup & Solve:
x = ________
4 5 y
Setup & Solve:
y = ________
3
The Bridge Support
An architect is designing a bridge support shaped like a right triangle. A vertical support beam (the altitude) is placed such that it meets the horizontal base (the hypotenuse) at a right angle. The beam divides the base into two segments: 8 meters and 18 meters.
How tall is the support beam?
Height: ________ m
What is the length of the shorter slanted side (leg) of the bridge?
Length: ________ m
Mastery Challenge
Prove why the Altitude Rule works using similar triangles. If \(\triangle ABC \sim \triangle BDC\), set up the proportion that leads to \(h^2 = s_1 \cdot s_2\).
Tall Task Slides Tall Task
Mastering Indirect Measurement
The Problem of Scale
"How do you measure a skyscraper without a tape measure that long?"
Indirect Measurement
Using known lengths and similar triangles to calculate heights or distances that are impossible to measure directly.
Method 1: The Shadow
Sunny Days
The sun's rays hit the earth at the same angle in a local area.
\(\frac{\text{Object Height}}{\text{Shadow Length}} = \frac{\text{Person Height}}{\text{Shadow Length}}\)
This is AA Similarity in action!
Height? Shadow A Shadow B
Method 2: The Mirror
Any Time
Angle of Reflection
Light reflects off a mirror at the same angle it arrives.
You must measure:
1. Your eye height
2. Your distance to mirror
3. Mirror distance to building
Accuracy Matters
Verticality
The object AND the person must be 90° to the ground for the right triangles to work.
Eye Level
In the mirror method, use your eye height, not your total body height!
Flat Ground
If the ground is sloped, your similar triangles will be "broken." Find a level spot!
Ready to head outside?
Tall Task Lab Sheet Tall Task Lab Sheet
Indirect Measurement: Shadows & Mirrors
Team:
Date:
The Mission
Your team must calculate the height of ____________________ (e.g., flagpole, basketball hoop, tree) using two different methods. Compare your results for accuracy.
1
Method 1: The Shadow
Data Collection
Your Height (H₁): in.
Your Shadow (S₁): in.
Object's Shadow (S₂): in.
The Proportion Setup:
Ratio Table
Show your algebraic steps to find H₂ (Object Height):
Calculated Height: ________________ in.
2
Method 2: The Mirror
Place a mirror on the ground. Walk backward until you can see the very top of the object in the center of the mirror.
Data Collection
Your Eye Level Height: in.
Dist. (You to Mirror): in.
Dist. (Mirror to Object): in.
Mirror Diagram
Show your work:
Calculated Height: ________________ in.
Analysis & Error
Compare your two results. Why might they be different? What were the potential sources of measurement error (human or environmental)?
If you were doing this on a cloudy day, which method would still be useful? Why?
Modeling Scale Slides Presentation Design by Scale
Similarity in Engineering & Film
Forced Perspective
"How did they make the hobbits look so small next to Gandalf?"
The Visual Trick
By placing objects at specific distances, the eye is tricked into seeing different sizes. This is pure similarity and scale at work.
Actor A Actor B
From the camera's view, they look the same size!
The Scale Factor (\(k\))
The ratio of any two corresponding lengths in similar figures.
Enlargement
\(k > 1\)
Reduction
\(0 < k < 1\)
\(Scale = \frac{\text{Model Length}}{\text{Actual Length}}\)
Precision in Design
Architecture
In architecture, a scale of 1:48 means every 1 inch on the model represents 48 inches (4 feet) on the real building.
Check for Understanding:
If a wall is 12 feet long, how many inches should it be on a 1:48 model?
1:48 Scale
Project Brief
You are the lead Miniature Effects Supervisor for a new sci-fi film.
Your Task:
Create a detailed scale model of a futuristic vehicle or structure. Every measurement must follow a strict scale factor!
Accuracy is non-negotiable
Scale Designer Project Guide Scale Designer Project Guide
Application: Engineering, Cinema, & Architecture
Name:
The Brief
You are the lead design engineer for Vector Dynamics. Your team has been tasked with creating a precise scale model of a landmark, vehicle, or set piece. Every dimension must be calculated using a consistent scale factor to ensure the final product is a "similar figure" to the real-world original.
01
Selection & Scale Factor
A) Choose your object:
B) Select your Scale Factor (k):
1 :
Example: 1:48 (1 inch on model = 4 feet in reality)
Why this scale?
02
Dimension Blueprint
Part Description Actual Size (in/ft) Model Size (in/cm)
03
The Similarity Check
Pick two corresponding parts from your data table and prove they are proportional. Set up the ratio and show the calculation.
Calculated Scale Factor: ________________
The Film Studio Extension
If a film director wants to use your model in a "Forced Perspective" shot to look 10 times larger than it actually is, how much farther away should the actor stand compared to the model? Use your knowledge of similar triangles to explain.
Vector Dynamics Design Lab • Educational Prototype • Rev 2026
