Solve similar-triangle proportions

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Lesson progressPractice problems 0/3

Difficulty: Intermediate
Estimated time: 30 minutes
Domains: Geometry and Trigonometry
Techniques: Similar-trianglesCorrespondenceAngle-angle-similarityProportionsScale-factor

What you’ll learn

See that similar triangles are the same shape, possibly at different sizes.
Match the vertices, angles, and sides of two similar triangles, from the order of their letters or from their angles.
Use one scale factor for every pair of matching sides.
Show that two triangles are similar by AA, using shared angles, vertical angles, or parallel lines.
Find a missing length, or an angle that stays the same.
Solve a proportion with awkward numbers as one equation in Desmos, once the parts are matched.

Prerequisites

Why this matters on the SAT

Use one triangle to unlock another

SAT diagrams often show the same triangle shape twice: at two sizes, turned around, one tucked inside the other, or linked by parallel lines. When two triangles are similar, a part you know in one tells you the matching part in the other. The tricky part is matching the right parts. Once they’re matched, remember four words: sides scale, angles stay. Every side grows or shrinks by the same factor, and matching angles stay exactly equal.

Match the parts first, then pick your tool. Matching the vertices and setting up the proportion are your job, by hand. If the scale factor is friendly, or an angle follows straight from the match, finish by hand too. If the proportion has awkward decimals or fractions, type the whole equation into Desmos and let it do the arithmetic.

Solution to the example

The order of the letters tells you which parts match. $J$ , $K$ , and $L$ line up with $M$ , $N$ , and $P$ , so $L\leftrightarrow P$ . Matching angles in similar triangles are equal, so

m\angle P=m\angle L=37^\circ.

The answer is B. The $2.5$ in $MN=2.5(JK)$ tells you how the sides grow. Dividing $37^\circ$ by $2.5$ (choice A) or multiplying by it (choice C) puts a length factor on an angle. Choice D is $180^\circ-37^\circ$ , the supplement of $\angle L$ . The question asks for the matching angle, not one that makes a straight line with it.

SAT example

Match the vertices in the order the question gives, then decide whether the side-length scale factor changes angle P.

Triangle $JKL$ is similar to triangle $MNP$ , where $J$ , $K$ , and $L$ correspond to $M$ , $N$ , and $P$ , respectively. The measure of $\angle L$ is $37^\circ$ , and $MN=2.5(JK)$ . What is the measure of $\angle P$ ?

A
$14.8^\circ$
B
$37^\circ$
C
$92.5^\circ$
D
$143^\circ$

Match the vertices, then write one ratio

Similar triangles have the same shape, though maybe not the same size. Corresponding parts are the ones that play the same role in both triangles. A vertex matches a vertex, an angle matches an angle, and a corresponding side joins two matching vertices.

The statement

\triangle ABC\sim\triangle DEF

lists the matches in order: first letter with first letter, second with second, third with third.

First triangle	Second triangle
$A$	$D$
$B$	$E$
$C$	$F$

The second triangle is turned, but the order of the letters still tells you which vertices match.

Match the endpoints, and you get the corresponding sides:

AB\leftrightarrow DE,\qquad BC\leftrightarrow EF,\qquad AC\leftrightarrow DF.

You also get the corresponding angles:

\angle A\leftrightarrow\angle D,\qquad \angle B\leftrightarrow\angle E,\qquad \angle C\leftrightarrow\angle F.

Pick one direction for the scale factor and stick with it. Going from $\triangle ABC$ to $\triangle DEF$ ,

k =\frac{\text{side in }DEF}{\text{matching side in }ABC} =\frac{DE}{AB} =\frac{EF}{BC} =\frac{DF}{AC}.

Every pair of matching sides gives the same $k$ . The angles don't use $k$ at all. They stay equal:

m\angle A=m\angle D,\qquad m\angle B=m\angle E,\qquad m\angle C=m\angle F.

Check your understanding:

Triangle $ABC$ is similar to triangle $DEF$ , in that order. If $AB=8$ , $DE=12$ , $BC=10$ , and $m\angle B=64^\circ$ , what are $EF$ and $m\angle E$ ?

Common mistake:

Multiplying an angle by the scale factor treats a turn as if it were a distance. Before you calculate, ask whether you’re finding an angle or a length. Only a length gets the scale factor. To find a matching angle, trace it through the vertex order. An angle-sum check helps only when you already know the other two angles.

Find AA in a figure

Sometimes the question tells you two triangles are similar. Other times, the figure gives you two pairs of equal angles and you have to spot it. AA, or angle-angle similarity, says that two pairs of equal angles are enough to make two triangles the same shape. You don’t need the third pair: each triangle’s angles add up to $180^\circ$ , so the third pair has to match too.

Only count two angles as equal when you can point to one of these reasons in the figure:

A shared angle belongs to both triangles because it’s formed by the same two rays. Sharing just a vertex isn’t enough.
Vertical angles are the opposite angles formed where two lines cross.
A line crossing parallel lines makes equal corresponding angles and equal alternate interior angles.

Three ways to get the equal angles AA needs: a shared angle, vertical angles, and a line crossing parallel lines.

Don’t trust how the picture looks. A figure may not be drawn to scale, so the way a triangle is turned or how big it looks proves nothing. Mark the equal angles first, then write the triangles’ names in matching order.

Check your understanding:

Two triangles share a pair of vertical angles where their sides cross. What else would you need to show they’re similar by AA?

Example: Solve an intersecting-triangle proportion

Worked example

The parallel sides and intersecting diagonals create two pairs of equal angles. Figure not drawn to scale.

In the figure, $\overline{AB}\parallel\overline{CD}$ . Segments $\overline{AD}$ and $\overline{BC}$ intersect at $E$ . Also, $AE=15$ , $DE=10$ , and $BE=18$ . What is the length of $\overline{CE}$ ?

Step 1

Show AA

$\angle AEB$ and $\angle DEC$ are vertical angles, so they’re equal.

$\overline{AD}$ crosses the parallel segments $\overline{AB}$ and $\overline{CD}$ , so $\angle BAE$ and $\angle CDE$ are alternate interior angles. They’re equal too.

That’s two equal pairs, so by AA,

\triangle ABE\sim\triangle DCE.

The order records the match: $A\leftrightarrow D$ , $B\leftrightarrow C$ , and $E\leftrightarrow E$ . Notice it’s $DCE$ , not $CDE$ : $A$ matches $D$ because their angles are the equal alternate interior pair.

Step 2

Match sides by their endpoints

Use the vertex map to pair the sides:

AE\leftrightarrow DE,\qquad BE\leftrightarrow CE,\qquad AB\leftrightarrow DC.

The scale factor from the left triangle to the right triangle is

\frac{DE}{AE} =\frac{10}{15} =\frac23.

It’s less than $1$ , so the right triangle is the smaller one. Each of its sides should be shorter than the matching side on the left.

Step 3

Scale the side you want

Keep the right-triangle side over the matching left-triangle side in both ratios:

\frac{DE}{AE}=\frac{CE}{BE}.

Put in the lengths you know:

\begin{aligned} \frac{10}{15}&=\frac{CE}{18}\\[1.4em] CE&=18\left(\frac{10}{15}\right)\\[1.4em] &=18\left(\frac23\right)\\[1.4em] &=12. \end{aligned}

\boxed{CE=12}.

That passes the size check: $12<18$ , just as a scale factor of $\frac23$ predicts.

Try it yourself:

Cover up the proportion in the last step. Starting from $A\leftrightarrow D$ , $B\leftrightarrow C$ , and $E\leftrightarrow E$ , rebuild the two side pairs you need to find $CE$ . Then check that both ratios go from the left triangle to the right one.

Match by role, not by picture position

A side that’s on the left in one triangle can be on the right in the other. In a nested figure, it can be one piece of a longer side. So left, right, top, and bottom don’t tell you which sides match.

Work in this order instead:

Find the equal angles, or use the vertex order you’re given.
Write the vertex map.
Match each side by its two endpoints.
Pick one direction for the scale factor.
Use that same direction in every ratio.

For example, if $A\leftrightarrow D$ and $B\leftrightarrow E$ , then $AB\leftrightarrow DE$ , however either triangle is turned. You can also match a side by the angle across from it: the side opposite $\angle A$ matches the side opposite $\angle D$ .

In a nested figure, compare whole sides. If the smaller triangle uses $AD$ and the larger one uses $AB$ , then $AD$ matches the whole length $AB$ , not the leftover piece $DB$ .

Common mistake:

If two side pairs give you two different scale factors, the triangles didn’t change size halfway through. Your match or your ratio direction flipped somewhere. Go back to the equal angles, write the vertex map, and pair sides by their endpoints. Then keep the same direction, second over first or first over second, in every ratio. Last, check sizes: a scale factor greater than $1$ should give a longer side, and one less than $1$ a shorter side.

Practice problems

Match the vertices before you calculate. The numbers then tell you whether to finish by hand or in Desmos.

Use named correspondence

Practice problem

Triangle $GHI$ is similar to triangle $KLM$ , where $G$ , $H$ , and $I$ correspond to $K$ , $L$ , and $M$ , respectively. If $GH=13.2$ , $HI=17.5$ , and $KL=18.48$ , what is the length of $LM$ ?

Calculator loads as you approach

Match the sides first, then type the whole proportion as one equation.

Use complete sides in a nested figure

Practice problem

The smaller triangle

ADE

shares vertex

A

with the larger triangle

ABC

. Figure not drawn to scale.

In the figure, $D$ lies on $\overline{AB}$ , $E$ lies on $\overline{AC}$ , and $\overline{DE}\parallel\overline{BC}$ . If $AD=8$ , $DB=4$ , and $AE=10$ , what is the length of $\overline{AC}$ ?

Calculator loads as you approach

Use Desmos if it helps you solve or check.

Transfer similarity to a shadow

Practice problem

At the same time on level ground, a vertical $4$ -foot reference pole casts a $3$ -foot shadow. A second vertical structure consists of a $5$ -foot support with a flagpole directly above it. The entire structure casts an $18$ -foot shadow.

What is the height, in feet, of the flagpole?

Your answer

Calculator loads as you approach

Use Desmos if it helps you solve or check.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Sides scale, angles stay: matching angles are equal, and every pair of matching sides uses one scale factor.
Use AA only with angle pairs you can name, such as shared, vertical, or parallel-line angles.
Match sides by their endpoints from the vertex map, and keep every ratio in one direction.
Once the parts are matched, put an awkward proportion into Desmos as one equation.
In a nested figure, use whole sides, and check that your answer’s size fits the scale factor.

Related lessons

When a question asks how perimeter, area, surface area, or volume changes, see Geometric scale factors. Later, Similarity inside right triangles uses similarity on the altitude drawn to a right triangle’s hypotenuse.

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156 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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