Similar Triangles

🔺 Similar Triangles

Part 1 of 5 — What Similarity Means

Topics in This Part

🔑 Key Concept: Two triangles are similar when they have the same shape but not necessarily the same size. One is a scaled copy of the other — every length stretched (or shrunk) by the same factor, with all angles kept identical.

Congruent vs. Similar

It helps to compare the two big ideas side by side.

Congruence is the special case of similarity where the scale factor is exactly $1$. So every pair of congruent triangles is also similar, but most similar triangles are not congruent.

🔑 Two conditions must both hold for similarity:

1. All pairs of corresponding angles are equal, and
2. All pairs of corresponding sides are in the same ratio (proportional).

Concept Check 🎯

Reading a Similarity Statement

When we write $\triangle ABC \sim \triangle DEF$, the order of the letters matters. It tells you exactly which parts correspond:

$\triangle ABC \sim \triangle DEF$

The proportional sides give us the key chain of equal ratios:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

⚠️ Order is not optional. $\triangle ABC \sim \triangle DEF$ is a different claim from $\triangle ABC \sim \triangle EFD$. Always match the letters in the same position.

Match the Corresponding Parts 🔽

Given $\triangle PQR \sim \triangle XYZ$, pick the part that corresponds to each one.

The Scale Factor

The scale factor $k$ is the single number that every length is multiplied by to go from one triangle to its similar partner:

$k = \frac{\text{a side of the new triangle}}{\text{the matching side of the original}}$

If $\triangle ABC \sim \triangle DEF$ and $k = \dfrac{DE}{AB}$, then every side of $\triangle DEF$ is $k$ times its match in $\triangle ABC$.

Example

$\triangle ABC \sim \triangle DEF$ with $AB = 4$ and $DE = 12$. The scale factor from $ABC$ to $DEF$ is

$k = \frac{DE}{AB} = \frac{12}{4} = 3.$

So $DEF$ is three times as large. Going the other way, from $DEF$ to $ABC$, the scale factor is $\dfrac{1}{3}$.

💡 Reciprocal rule: the scale factor one direction is always the reciprocal of the scale factor the other direction. $k$ and $\dfrac{1}{k}$ are the two views of the same pair.

Find the Scale Factor 🧮

$\triangle ABC \sim \triangle DEF$. Use the matching sides given.

1) $AB = 5$, $DE = 15$. Scale factor from $ABC$ to $DEF$ $= \,?$ 2) $BC = 8$, $EF = 2$. Scale factor from $ABC$ to $DEF$ $= \,?$ (decimal or fraction) 3) $CA = 6$, $FD = 9$. Scale factor from $ABC$ to $DEF$ $= \,?$ (decimal or fraction)

🔺 Similar Triangles

Part 2 of 5 — The Three Similarity Shortcuts (AA, SSS, SAS)

🔑 The shortcut idea: You almost never have to check all six parts (3 angles + 3 sides). Just like congruence has SSS, SAS, ASA, similarity has its own three shortcuts: AA, SSS~, and SAS~. Any one of them is enough to prove two triangles similar.

AA — Angle-Angle (the most-used one)

AA Similarity: If two angles of one triangle equal two angles of another triangle, the triangles are similar.

Why only two? Because the angles of every triangle add to $180°$. If two pairs match, the third pair is forced to match too:

$\angle C = 180° - \angle A - \angle B.$

Example

In $\triangle ABC$, $\angle A = 50°$ and $\angle B = 60°$. In $\triangle DEF$, $\angle D = 50°$ and $\angle E = 60°$. Then the third angles are both

$180° - 50° - 60° = 70°,$

so all three angles match and $\triangle ABC \sim \triangle DEF$ by AA.

💡 AA is the workhorse of geometry. Parallel lines, shadows, and nested triangles almost always give you two equal angles for free.

SSS~ and SAS~ (the side shortcuts)

SSS Similarity: If all three pairs of corresponding sides are in the same ratio, the triangles are similar.

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \;\Rightarrow\; \triangle ABC \sim \triangle DEF$

SAS Similarity: If two pairs of corresponding sides are in the same ratio and the included angles (the angles between those sides) are equal, the triangles are similar.

⚠️ For SAS~ the equal angle must be the one between the two proportional sides. An angle somewhere else does not count.

Concept Check 🎯

Test the SSS~ Ratios 🧮

$\triangle ABC$ has sides $6, 9, 12$. $\triangle DEF$ has the corresponding sides $8, 12, 16$.

1) Ratio of the first pair $\dfrac{6}{8}$ as a decimal $= \,?$ 2) Ratio of the second pair $\dfrac{9}{12}$ as a decimal $= \,?$ 3) Ratio of the third pair $\dfrac{12}{16}$ as a decimal $= \,?$

Pick the Right Shortcut 🔽

For each situation, choose the similarity criterion that fits.

🔺 Similar Triangles

Part 3 of 5 — Solving for Missing Sides

🔑 Why we care: The real power of similarity is that proportional sides let you solve for an unknown length. Set up a proportion, cross-multiply, and solve. This is the single most-tested skill in this topic.

The Proportion Method

When $\triangle ABC \sim \triangle DEF$, set the matching sides equal as ratios, then cross-multiply.

Worked Example

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

Step 1 — Write the proportion (matching sides): $\frac{AB}{DE} = \frac{BC}{EF} \;\Rightarrow\; \frac{6}{9} = \frac{8}{EF}$

Step 2 — Cross-multiply: $6 \cdot EF = 9 \cdot 8 = 72$

Step 3 — Solve: $EF = \frac{72}{6} = 12$

✅ Check: $\dfrac{6}{9} = \dfrac{2}{3}$ and $\dfrac{8}{12} = \dfrac{2}{3}$ ✓ — both ratios match, so $EF = 12$ is correct.

Worked Example: scale factor first

Sometimes it is fastest to find the scale factor $k$, then multiply.

$\triangle PQR \sim \triangle STU$ with $PQ = 10$ and $ST = 25$. Find $TU$ if $QR = 4$.

Scale factor (from $PQR$ to $STU$): $k = \frac{ST}{PQ} = \frac{25}{10} = 2.5$

Every side of $STU$ is $2.5$ times its match in $PQR$, so: $TU = k \cdot QR = 2.5 \cdot 4 = 10$

💡 Two valid routes, same answer. A proportion and the scale-factor method always agree — use whichever feels cleaner for the numbers in front of you.

Concept Check 🎯

Solve for the Missing Side 🧮

In each, $\triangle ABC \sim \triangle DEF$.

1) $AB = 4$, $DE = 6$, $BC = 10$. Find $EF$. 2) $AB = 8$, $DE = 12$, $AC = 6$. Find $DF$. 3) Scale factor from $ABC$ to $DEF$ is $0.5$; $CA = 14$. Find $FD$.

Order the Solve 🔽

You're finding $EF$ given $\triangle ABC \sim \triangle DEF$, $AB = 5$, $DE = 20$, $BC = 3$. Choose each step.

🔺 Similar Triangles

Part 4 of 5 — Real Applications: Shadows, Mirrors & the Side-Splitter

🔑 Similarity in the wild: Before lasers, people measured the height of trees, buildings, and pyramids using nothing but shadows and similar triangles. The same idea powers indirect measurement today.

Shadow Problems (Indirect Measurement)

At the same time of day, the sun hits everything at the same angle. So a person and a tree each cast a shadow, forming two similar right triangles (AA: both have a right angle and the same sun angle).

$\frac{\text{person height}}{\text{person's shadow}} = \frac{\text{tree height}}{\text{tree's shadow}}$

Worked 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?

$\frac{6}{4} = \frac{h}{20}$

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

✅ Check: $\dfrac{6}{4} = 1.5$ and $\dfrac{30}{20} = 1.5$ ✓ — the tree is $30$ feet tall.

Shadow Practice 🧮

Set up height-over-shadow proportions.

1) A $5$-ft person casts a $3$-ft shadow. A flagpole casts a $21$-ft shadow at the same time. Find the flagpole's height (ft). 2) A $2$-ft sign casts a $1.5$-ft shadow. A nearby lamppost is $12$ ft tall. How long is its shadow (ft)?

The Side-Splitter Theorem

Side-Splitter (Triangle Proportionality) Theorem: A line drawn parallel to one side of a triangle, cutting the other two sides, divides those sides proportionally.

If $\overline{DE} \parallel \overline{BC}$ in $\triangle ABC$ (with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$):

$\frac{AD}{DB} = \frac{AE}{EC}$

This works because $\overline{DE} \parallel \overline{BC}$ creates equal corresponding angles, so $\triangle ADE \sim \triangle ABC$ by AA.

Worked Example

In $\triangle ABC$, $\overline{DE} \parallel \overline{BC}$ with $AD = 4$, $DB = 6$, and $AE = 8$. Find $EC$.

$\frac{AD}{DB} = \frac{AE}{EC} \;\Rightarrow\; \frac{4}{6} = \frac{8}{EC}$ $4 \cdot EC = 48 \;\Rightarrow\; EC = 12$

⚠️ The side-splitter ratio compares the two pieces of the same side ($AD$ to $DB$), not a piece to the whole. Don't accidentally use $AB$.

Concept Check 🎯

Bonus: the Mirror Method

Place a mirror flat on the ground between you and a tall object. Step back until you see the object's top in the mirror. The light reflects at equal angles, so your "eye-height-and-distance" triangle is similar to the "object-height-and-distance" triangle:

$\frac{\text{your eye height}}{\text{your distance to mirror}} = \frac{\text{object height}}{\text{object's distance to mirror}}$

💡 Shadows, mirrors, and side-splitters are three faces of the same tool. Spot the two equal angles, declare the triangles similar, write a proportion, and solve.

Side-Splitter & Mirror Practice 🧮

1) $\overline{DE} \parallel \overline{BC}$, $AD = 5$, $DB = 10$, $AE = 4$. Find $EC$. 2) Mirror method: your eye is $5$ ft up, you stand $2$ ft from the mirror, and the tree is $14$ ft from the mirror. Find the tree's height (ft).

🔺 Similar Triangles

Part 5 of 5 — Perimeter, Area & Mastery Check

You can now (1) recognize similarity, (2) prove it with AA / SSS~ / SAS~, (3) solve for missing sides, and (4) apply it to shadows and the side-splitter. One last idea ties it together: how similarity scales perimeter and area.

How Perimeter and Area Scale

If two triangles are similar with scale factor $k$:

Perimeter is just a sum of sides, so it scales by the same factor $k$. But area depends on two dimensions (base $\times$ height), so it scales by $k^2$.

Example

$\triangle ABC \sim \triangle DEF$ with scale factor $k = 3$. If $\triangle ABC$ has area $5$, then

$\text{area of } \triangle DEF = k^2 \cdot 5 = 9 \cdot 5 = 45.$

⚠️ Classic trap: if the sides triple, the area does not triple — it grows $3^2 = 9$ times. Doubling the sides quadruples the area.

Scale Perimeter & Area 🧮

$\triangle ABC \sim \triangle DEF$ with scale factor $k$ (from $ABC$ to $DEF$).

1) $k = 4$. $\triangle ABC$ has perimeter $7$. Find the perimeter of $\triangle DEF$. 2) $k = 4$. $\triangle ABC$ has area $3$. Find the area of $\triangle DEF$. 3) $k = 2$. $\triangle ABC$ has area $10$. Find the area of $\triangle DEF$.

Quick Reference

🔑 The whole topic reduces to one habit: find the two equal angles, declare similarity, then write and solve a proportion.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.