Verifying Similarity and the Big Picture

Lesson 2 of 2: Testing and Comparing

In this lesson:

• Test similarity by checking angles and side ratios
• Place congruence and similarity in one framework

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Verify similarity by checking corresponding angles and side ratios
2. Connect those measurements back to the transformation sequence
3. Classify a pair as congruent, similar but not congruent, or neither
4. Explain why congruence is similarity with scale factor

Be the Similarity Detective Here

You're handed two figures with their measurements — but no sequence.

How could you decide whether they're similar, without trying to find the sequence?

Measure the Corresponding Side Ratios

All four corresponding ratios equal .

Now Check the Corresponding Angles

In both rectangles, every angle is .

• Corresponding angles match
• Sides are proportional

So far, both conditions hold.

The Two Conditions for Similarity

Two figures are similar when both hold:

1. All corresponding angles are equal
2. All corresponding sides are proportional

Both conditions are required.

Why Both Conditions Are Needed

• Equal ratios detect the dilation — it scales every length the same
• Equal angles detect the angle-preservers — every move keeps angles fixed

The measurements are evidence of a hidden sequence.

A Rectangle Pair That Passes

rectangle and rectangle:

Both conditions hold, so the rectangles are similar.

Equal Angles Is Not Enough

Short sides: . Long sides: . Not similar.

A Catch That Trips Up Quadrilaterals

• For quadrilaterals, equal angles alone do not force similarity
• For triangles, equal angles alone do force proportional sides

So always check both — unless you already know it's a triangle.

Quick Check With Two Triangles

Triangle 1: sides , angles
Triangle 2: sides , same angles

Similar? Decide on your own.

Congruence Lives Inside the Bigger Idea

What if the scale factor turns out to be exactly ?

• Sides aren't just proportional — they're equal
• The figures are congruent

Congruence is the case of similarity.

Three Pairs and Three Verdicts

Congruent · Similar (not congruent) · Neither

The Congruence and Similarity Table

Think Each One Through Carefully

• If two figures are congruent, are they similar?
• Are all circles similar?
• Can a square be similar to a non-square rectangle?

Answer each in your head, with a reason.

Classify the Pair and Describe It

: , ,
: , ,

Are they similar? If so, give the scale factor and a full sequence.

The Key Takeaways to Remember

✓ Similarity is a measurable test: equal angles and proportional sides
✓ Both conditions required — equal angles alone can fail
✓ Congruence is similarity with

Watch out: use corresponding sides; orientation may differ

What Is Coming Up Next

You can now define, build, and test similarity.

Next you'll use it: informal arguments about triangle angles and the AA shortcut (8.G.A.5) — and later, why every line has one slope (8.EE.B.6).