Congruence and Similarity

📐 Congruence and Similarity

Part 1 of 5 — Congruent Figures

Topics in This Part

🔑 Key Concept: Two figures are congruent when they have exactly the same size and shape. You could slide, flip, or turn one to land perfectly on top of the other. Same size, same shape — that's congruence.

What Does "Congruent" Mean?

Two figures are congruent ($\cong$) if one can be placed exactly on top of the other so that every part matches.

When two figures are congruent:

• Corresponding sides are equal in length.
• Corresponding angles are equal in measure.

Think of two copies of the same key, or two cut-outs from the same cookie cutter — they fit perfectly.

⚠️ Careful with symbols: We say the figures are congruent ($\triangle ABC \cong \triangle DEF$), but their measurements are equal ($AB = DE$). Use $\cong$ for shapes and $=$ for numbers.

Corresponding Sides and Angles

"Corresponding" parts are the parts that match up when the figures are lined up.

Suppose $\triangle ABC \cong \triangle DEF$. The order of the letters tells you what matches what:

$A \leftrightarrow D, \quad B \leftrightarrow E, \quad C \leftrightarrow F$

So the corresponding parts are:

🔑 Read the statement in order! In $\triangle ABC \cong \triangle DEF$, the first letters ($A$ and $D$) correspond, the second letters ($B$ and $E$) correspond, and the third letters ($C$ and $F$) correspond.

Match the Parts 🔽

Given $\triangle ABC \cong \triangle DEF$, choose the part that corresponds to each one.

Why the Order Matters

Notice that you never had to look at a picture to match those parts — the order of the letters told you everything. That's the power of a congruence statement: it carries the matching information inside it.

So whenever you see a statement like $\triangle PQR \cong \triangle STU$, you can read off corresponding parts directly:

$P \leftrightarrow S, \quad Q \leftrightarrow T, \quad R \leftrightarrow U$

💡 Same idea for sides: pair the letters position-by-position. $\overline{PR}$ (positions 1 and 3) matches $\overline{SU}$ (positions 1 and 3).

Concept Check 🎯

Using Corresponding Parts to Find Measures

Here's where congruence becomes useful: if two figures are congruent, knowing a measurement in one instantly gives you the matching measurement in the other.

If $\triangle ABC \cong \triangle DEF$ and you measure $AB = 7$, then you automatically know $DE = 7$ — no need to measure it. The same goes for every corresponding side and angle.

🔑 Corresponding Parts Rule: In congruent figures, corresponding parts are equal. Match the letters, copy the measurement.

Use Corresponding Parts 🧮

Given $\triangle ABC \cong \triangle DEF$ with $AB = 7$, $BC = 10$, $AC = 5$, and $\angle A = 40^\circ$, find each measurement.

1) $DE = \,?$ 2) $EF = \,?$ 3) $\angle D = \,?$ (in degrees, just the number)

Writing a Congruence Statement

When you write a congruence statement, list the vertices in matching order. This is not optional — the order is the information.

If $\angle X = \angle M$, $\angle Y = \angle N$, and $\angle Z = \angle O$, then you must write:

$\triangle XYZ \cong \triangle MNO$

Writing $\triangle XYZ \cong \triangle NMO$ would be wrong, because it claims $X$ matches $N$ — which isn't true.

💡 A good habit: figure out which vertices match first, then write both triangles with their letters in the same matching order. You now know what congruence means — in Part 2 we'll see how you can move a figure without changing it.

📐 Congruence and Similarity

Part 2 of 5 — Rigid Motions (Transformations)

🔑 The Idea: A rigid motion is a move that slides, flips, or turns a figure without changing its size or shape. Two figures are congruent exactly when one can be carried onto the other by a sequence of rigid motions.

The Three Rigid Motions

All three are rigid — they preserve:

• every side length,
• every angle measure,
• and therefore the figure stays congruent to the original.

🔑 Key Fact: Rigid motions never stretch, shrink, or distort. The image is always congruent to the original — only its position (and maybe its facing direction) changes.

Translations on the Coordinate Plane

A translation slides a figure. We describe it by how far it moves right/left ($x$) and up/down ($y$).

The rule "$(x, y) \to (x + 3,\, y - 2)$" means: move every point 3 right and 2 down.

Example

Translate the point $A(1, 4)$ by the rule $(x, y) \to (x + 3,\, y - 2)$:

$A'(1 + 3,\; 4 - 2) = A'(4, 2)$

The image $A'$ is read "A prime." Do the same arithmetic to every vertex, and the whole figure slides as one.

💡 A translation changes a point's location but never its size — so the image is always congruent to the original.

Translate the Point 🧮

Apply the translation rule $(x, y) \to (x - 4,\, y + 5)$ to each point. Enter the new coordinates.

1) $B(6, 1) \to B'(\,?\,,\ ?\,)$ — enter the new $x$, then the new $y$. 2) $C(-2, 3) \to C'(\,?\,,\ ?\,)$ — enter the new $x$, then the new $y$.

Reflections Across the Axes

A reflection flips a figure across a line (the "mirror"). Two common mirrors are the $x$-axis and the $y$-axis.

Example: Reflect $P(3, 5)$ across the $x$-axis

Keep $x$, flip the sign of $y$:

$P(3, 5) \to P'(3, -5)$

⚠️ Don't flip both signs! Reflecting across the $x$-axis changes only the $y$-coordinate's sign. Flipping both signs at once is actually a $180^\circ$ rotation, not a reflection.

Name That Motion 🔽

Choose the correct rule or result for each transformation.

What All Three Have in Common

Slides, flips, and turns look different, but they share the one property that matters most for this lesson:

🔑 Rigid motions preserve congruence. No matter how you translate, reflect, or rotate a figure, the image is the same size and shape as the original — so it is always congruent to it.

This is the link back to Part 1: two figures are congruent exactly when a sequence of these rigid motions carries one onto the other.

Concept Check 🎯

📐 Congruence and Similarity

Part 3 of 5 — Similar Figures & Scale Factor

🔑 The Idea: Two figures are similar ($\sim$) when they have the same shape but (possibly) a different size. Think of a photo and a smaller print of it — same picture, scaled down.

What Does "Similar" Mean?

Two figures are similar if:

• all pairs of corresponding angles are equal, and
• all pairs of corresponding sides are proportional (in the same ratio).

🔑 Key Idea: Congruent figures are a special case of similar figures — they're similar with a size ratio of $1$. Every congruent pair is similar, but not every similar pair is congruent.

Scale Factor

The scale factor is the number you multiply the original's side lengths by to get the new figure's side lengths.

$\text{scale factor} = \frac{\text{new (image) length}}{\text{original length}}$

• Scale factor $> 1$ → the figure gets bigger (enlargement).
• Scale factor $< 1$ → the figure gets smaller (reduction).
• Scale factor $= 1$ → the figures are congruent.

Example

A triangle has a side of length $4$. A similar triangle has the corresponding side of length $12$.

$\text{scale factor} = \frac{12}{4} = 3$

Every side of the bigger triangle is $3$ times the matching side of the smaller one.

💡 To go backward (big → small), use the reciprocal. Here the scale factor from the big triangle to the small one is $\dfrac{4}{12} = \dfrac{1}{3}$.

Concept Check 🎯

Dilations on the Coordinate Plane

A dilation is the transformation that resizes a figure. A dilation centered at the origin with scale factor $k$ uses the rule:

$(x, y) \to (kx,\, ky)$

You multiply both coordinates by $k$.

Example: Dilate $A(2, 3)$ by scale factor $k = 2$ (center at origin)

$A(2, 3) \to A'(2 \cdot 2,\; 2 \cdot 3) = A'(4, 6)$

The image $A'$ is twice as far from the origin as $A$ was.

⚠️ A dilation is NOT a rigid motion (unless $k = 1$). It changes size, so the image is similar to the original — same shape, different size — but generally not congruent.

Dilate the Point 🧮

Each dilation is centered at the origin. Apply the rule $(x, y) \to (kx,\, ky)$.

1) Dilate $P(3, 5)$ by $k = 2$: enter the new $x$, then the new $y$. 2) Dilate $Q(10, 6)$ by $k = \tfrac{1}{2}$: enter the new $x$, then the new $y$.

Congruent or Just Similar?

Now you have two families of transformations, and they produce different relationships:

💡 A handy test: if a transformation can change the size, the result is only similar. If the size must stay the same, the result is congruent.

Congruent or Similar? 🔽

Decide what relationship each transformation produces.

📐 Congruence and Similarity

Part 4 of 5 — Finding Missing Lengths with Proportions

🔑 Big Payoff: Because similar figures have proportional sides, you can set up a proportion to find any missing length — including real-world heights and distances you could never measure directly.

Setting Up a Proportion

In similar figures, the ratio of any pair of corresponding sides equals the scale factor — and all the ratios are equal:

$\frac{\text{side of figure 1}}{\text{matching side of figure 2}} = \frac{\text{another side of figure 1}}{\text{its match in figure 2}}$

Example

$\triangle ABC \sim \triangle DEF$. We know $AB = 6$, $DE = 9$, and $BC = 8$. Find $EF$.

Set up matching ratios and cross-multiply:

$\frac{AB}{DE} = \frac{BC}{EF} \;\Rightarrow\; \frac{6}{9} = \frac{8}{EF}$

$6 \cdot EF = 9 \cdot 8 = 72 \;\Rightarrow\; EF = \frac{72}{6} = 12$

✅ Check: The scale factor from $\triangle ABC$ to $\triangle DEF$ is $\dfrac{9}{6} = 1.5$, and $8 \times 1.5 = 12$ ✓.

Concept Check 🎯

The Cross-Multiply Shortcut

When a proportion has the unknown by itself in one denominator (or numerator), cross-multiplying is the fastest way to solve it.

For a proportion $\dfrac{a}{b} = \dfrac{c}{d}$, cross-multiplying gives:

$a \cdot d = b \cdot c$

Quick Example

$\frac{2}{x} = \frac{6}{18} \;\Rightarrow\; 6x = 2 \cdot 18 = 36 \;\Rightarrow\; x = 6$

💡 Always multiply along the diagonals, then divide to isolate the unknown. Practice it on the drill below.

Solve the Proportion 🧮

In each pair of similar figures, find the missing length $x$.

1) $\dfrac{3}{x} = \dfrac{9}{15}$, so $x = \,?$ 2) $\dfrac{x}{12} = \dfrac{10}{8}$, so $x = \,?$ 3) Two similar triangles have a scale factor of $4$. A side of $2.5$ in the small triangle corresponds to what length in the large one? $x = \,?$

Real-World Application: Indirect Measurement

Similar triangles let you measure things you can't reach — like the height of a tree using its shadow.

At the same time of day, a tree and a person both cast shadows. The triangles (object + shadow + sun ray) are similar because the sun's rays hit at the same angle.

Example

A $6$-ft person casts a $4$-ft shadow. At the same moment, a tree casts a $20$-ft shadow. How tall is the tree?

$\frac{\text{person height}}{\text{person shadow}} = \frac{\text{tree height}}{\text{tree shadow}} \;\Rightarrow\; \frac{6}{4} = \frac{h}{20}$

$4h = 6 \cdot 20 = 120 \;\Rightarrow\; h = \frac{120}{4} = 30 \text{ ft}$

💡 The tree is $30$ feet tall — found without a ladder, just a proportion and a shadow.

Indirect Measurement 🧮

1) A $5$-ft student casts a $3$-ft shadow while a flagpole casts a $18$-ft shadow. How tall is the flagpole, in feet? $h = \,?$ 2) A model car is built to scale. The real car is $15$ ft long and the model is $1$ ft long, so the scale factor (real → model) is $\tfrac{1}{15}$. The real car is $6$ ft wide. How wide is the model, in feet? Enter a fraction or decimal. $w = \,?$

📐 Congruence and Similarity

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) identify congruent figures and their corresponding parts, (2) use rigid motions, (3) recognize similar figures and scale factor, and (4) find missing lengths with proportions. Let's put it all together.

Quick Reference

⚠️ Remember: Translations, reflections, and rotations keep figures congruent. A dilation (with $k \ne 1$) makes figures similar but not congruent.

Mixed Practice 🎯

Blending the Skills

The next check mixes all four ideas — corresponding parts, dilations, the congruent-vs-similar distinction — into one set of dropdowns. Slow down on each one and ask yourself:

• Are the letters in matching order? (corresponding parts)
• Does the size change? (congruent vs. similar)
• Am I multiplying both coordinates? (dilation)

💡 You have everything you need — these just ask you to pick the right tool for each line.

Putting It Together 🔽

You're Ready

Take stock of what you can now do:

• ✅ Spot congruent figures and read off corresponding sides and angles.
• ✅ Slide, flip, and turn figures with the right coordinate rules.
• ✅ Tell congruent figures from merely similar ones, using scale factor.
• ✅ Solve a proportion to find any missing length — even a tree's height.

One last quiz to lock it in.

Exit Quiz ✅

Answer all three to finish the lesson.