SSS and SAS Similarity

In **AA Similarity** you proved triangles similar with angles only. Now you will do it with **sides**. Remember: similar triangles have the same shape, their corresponding sides are in the same ratio, the **scale factor** $k$. **SSS Similarity (Side-Side-Side):** if all three pairs of corresponding sides are in the same ratio, the triangles are similar. $$\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF$$ **SAS Similarity (Side-Angle-Side):** if two pairs of corresponding sides are in the same ratio and the **included angle** (the angle between those two sides) is congruent, the triangles are similar. $$\frac{DE}{AB} = \frac{EF}{BC} = k \text{ and } \angle B = \angle E \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF$$ **How to match the sides:** when you only know the lengths, sort the sides of each triangle from **shortest to longest** and pair them in that order: shortest with shortest, middle with middle, longest with longest. **Once you know the triangles are similar:** - every side of the image is $k$ times the matching side - the **perimeter ratio** is $k$ - the **area ratio** is $k^2$ Compare with congruence: SSS and SAS congruence need **equal** sides ($k = 1$), SSS and SAS similarity only need **proportional** sides.

Checking SSS Similarity

$\triangle ABC$ has sides $AB = 4$, $BC = 6$, $CA = 8$. $\triangle DEF$ has sides $DE = 6$, $EF = 9$, $FD = 12$. Are the triangles similar?

Sort each triangle's sides from shortest to longest: $\triangle ABC$: $4, 6, 8$. $\triangle DEF$: $6, 9, 12$ = Pairs: $4 \leftrightarrow 6$, $6 \leftrightarrow 9$, $8 \leftrightarrow 12$

Divide each side of $\triangle DEF$ by the matching side of $\triangle ABC$: $\frac{6}{4} = 1.5$, $\frac{9}{6} = 1.5$, $\frac{12}{8} = 1.5$ = All three ratios are $1.5$

Apply SSS Similarity: Three equal ratios, so the sides are proportional = $\triangle ABC \sim \triangle DEF$, $k = 1.5$

Answer: Yes, $\triangle ABC \sim \triangle DEF$ by SSS Similarity with scale factor $k = 1.5$.

When SSS Similarity Fails

One triangle has sides $5$, $7$, $9$. Another has sides $10$, $14$, $17$. Are they similar?

Pair the sides in order: $5 \leftrightarrow 10$, $7 \leftrightarrow 14$, $9 \leftrightarrow 17$ = Sides sorted from shortest to longest

Compute the three ratios: $\frac{10}{5} = 2$, $\frac{14}{7} = 2$, $\frac{17}{9} \approx 1.89$ = The third ratio is different

Decide: SSS needs ALL three ratios equal. Two out of three is not enough. = Not similar

Answer: No. The ratios are $2$, $2$ and about $1.89$, so the triangles are not similar. For a similar triangle the longest side would have to be $9 \times 2 = 18$.

SAS Similarity and a Missing Side

In $\triangle PQR$: $PQ = 5$, $QR = 8$, $\angle Q = 60°$, $PR = 7$. In $\triangle XYZ$: $XY = 10$, $YZ = 16$, $\angle Y = 60°$. Are the triangles similar? If so, find $XZ$.

Compare the two pairs of sides: $\frac{XY}{PQ} = \frac{10}{5} = 2$, $\frac{YZ}{QR} = \frac{16}{8} = 2$ = Same ratio $k = 2$

Check that the angle is the included angle: $\angle Q$ lies between $PQ$ and $QR$, $\angle Y$ lies between $XY$ and $YZ$, and $\angle Q = \angle Y = 60°$ = Included angles are congruent

Apply SAS Similarity: Two proportional sides and the included angle = $\triangle PQR \sim \triangle XYZ$, $k = 2$

Use the scale factor for the missing side: $XZ = k \cdot PR = 2 \cdot 7$ = $XZ = 14$

Answer: Yes, $\triangle PQR \sim \triangle XYZ$ by SAS Similarity with $k = 2$, and $XZ = 14$.

Nested Triangles with a Shared Angle

In $\triangle ABC$, point $D$ lies on $AB$ and point $E$ lies on $AC$. $AD = 4$, $AB = 12$, $AE = 5$, $AC = 15$ and $DE = 6$. Show that $\triangle ADE \sim \triangle ABC$ and find $BC$.

Find the shared angle: Both triangles contain the angle at $A$ = $\angle DAE = \angle BAC$

Compare the sides next to angle $A$: $\frac{AB}{AD} = \frac{12}{4} = 3$, $\frac{AC}{AE} = \frac{15}{5} = 3$ = Same ratio $k = 3$

Apply SAS Similarity: The shared angle is the included angle of both pairs of sides = $\triangle ADE \sim \triangle ABC$

Find the third side: $BC = k \cdot DE = 3 \cdot 6$ = $BC = 18$

Answer: $\triangle ADE \sim \triangle ABC$ by SAS Similarity (shared angle $A$, $k = 3$), so $BC = 18$. Use the whole sides $AB$ and $AC$, not the pieces $DB$ and $EC$.

Perimeter Ratio and Area Ratio

A right triangle has sides $3$, $4$, $5$. A second triangle has sides $9$, $12$, $15$. Compare their perimeters and areas.

Check SSS Similarity: $\frac{9}{3} = \frac{12}{4} = \frac{15}{5} = 3$ = Similar, $k = 3$

Compare the perimeters: $3 + 4 + 5 = 12$ and $9 + 12 + 15 = 36$, $\frac{36}{12} = 3$ = Perimeter ratio $= k = 3$

Compare the areas (legs are base and height): $\frac{3 \cdot 4}{2} = 6$ and $\frac{9 \cdot 12}{2} = 54$, $\frac{54}{6} = 9$ = Area ratio $= 9$

Connect the area ratio to $k$: $k^2 = 3^2 = 9$ = Area ratio $= k^2$

Answer: The perimeters are $12$ and $36$ (ratio $3 = k$), the areas are $6$ and $54$ (ratio $9 = k^2$).

Often you can measure lengths more easily than angles, and then SSS and SAS similarity are the tools you need: - **Surveying**: to find the width of a lake you cannot cross, surveyors stake out a small triangle that shares an angle with a big one and use SAS similarity - **Engineering**: a model of a roof truss or a bridge is built with all lengths scaled by the same factor, so it has exactly the same shape as the real one - **Printing and design**: when a logo is enlarged, every length is multiplied by $k$ but the ink or material needed grows by $k^2$ - **Checking a drawing**: a carpenter can check that a triangular frame matches a plan by comparing three length ratios, no protractor needed Knowing that area grows with $k^2$ (not $k$) also prevents costly mistakes, like ordering twice the paint for a wall that is twice as wide and twice as tall.

Mistake: Pairing the sides in the order they are listed, for example $4, 6, 8$ with $12, 9, 6$, and getting ratios $3$, $1.5$, $0.75$

Why: The order in which side lengths are written says nothing about which sides correspond. Wrong pairs give different ratios even for similar triangles.

Correct: Sort both triangles from shortest to longest first: $4, 6, 8$ and $6, 9, 12$. Then all ratios are $1.5$.

Mistake: Using SAS with an angle that is not between the two sides

Why: Two proportional sides and a non-included angle (SSA) do not fix the shape: two different triangles can have the same two sides and the same angle opposite one of them.

Correct: Check that the congruent angle is formed by exactly the two sides you compared. Otherwise SAS Similarity does not apply.

Mistake: Thinking the area ratio is also $k$

Why: Area is length times length, so both factors grow by $k$: the area grows by $k \cdot k = k^2$.

Correct: Perimeter ratio $= k$, area ratio $= k^2$. With $k = 3$, the area is $9$ times as large, not $3$ times.

Mistake: Dividing big by small for one pair and small by big for another, for example $\frac{6}{4} = 1.5$ but $\frac{6}{9} \approx 0.67$

Why: The ratios then look different although the sides are proportional.

Correct: Always divide in the same direction: image side over original side (or the other way round for every pair).

Enlarging a Logo

Designers enlarge logos so that every length grows by the same scale factor. The enlarged triangle is similar to the original by SSS Similarity, so it keeps exactly the same shape.

A triangular logo with sides $2$ cm, $3$ cm and $4$ cm is printed with $k = 5$ on a poster: the sides become $10$ cm, $15$ cm and $20$ cm.

Measuring Across a Lake

Surveyors cannot stretch a tape measure across water. They mark two points on lines leading away from a corner point and use the small triangle that shares the corner angle with the big one.

From point $A$ two straight paths lead past the lake. A small triangle with sides $\frac{1}{4}$ as long as the big one along both paths shares the angle at $A$, so the width of the lake is $4$ times the short distance measured on land.

Fabric for a Bigger Sail

Sailmakers scale a proven sail design up or down. The new sail is similar to the old one, but the fabric needed grows with the area ratio $k^2$, not with $k$.

A sail that is $1.5$ times as long on every side needs $1.5^2 = 2.25$ times as much fabric.

SSS Similarity: if all three pairs of corresponding sides have the same ratio $k$, the triangles are similar

SAS Similarity: two pairs of sides with the same ratio $k$ and a congruent included angle make the triangles similar

When only lengths are given, sort the sides from shortest to longest and pair them in that order

The scale factor $k$ is image side divided by original side; every missing side is $k$ times its partner

Perimeter ratio $= k$, area ratio $= k^2$

Choose the criterion from what you know: two angles means AA, three sides means SSS, two sides and the angle between them means SAS

Q: How is SSS Similarity different from SSS Congruence?

A: SSS Congruence needs three pairs of **equal** sides. SSS Similarity only needs three pairs of sides in the **same ratio**. Congruence is the special case $k = 1$.

Q: Why must the angle in SAS be the included angle?

A: The two sides and the angle between them fix the shape completely. If the angle is somewhere else (SSA), two different triangles can fit the same data, so you cannot conclude that the triangles are similar.

Q: When should I use AA, SSS or SAS?

A: Look at what is given. Two pairs of congruent angles: AA. Three pairs of side lengths: SSS. Two pairs of side lengths and the angle between them: SAS. In nested triangles a shared angle often gives you the angle for AA or SAS for free.

Q: Does it matter which triangle I divide by?

A: Only for the value of $k$. Big over small gives $k > 1$ (enlargement), small over big gives $k < 1$ (reduction). Both work, as long as you divide in the same direction for every pair.

Exit Presentation

SSS and SAS Similarity

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Geometry / Congruence & Similarity

SSS and SAS Similarity

Prove triangles similar from side ratios (SSS) or two side ratios and the included angle (SAS), then use the scale factor for sides, perimeter and area.

Advanced30 min

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