Reason with congruence and sufficient information

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Difficulty: Intermediate
Estimated time: 30 minutes
Domains: Geometry and Trigonometry
Techniques: Triangle-congruenceSufficient-informationCorrespondenceTriangle-similarity

What you’ll learn

Explain what congruent means: two triangles with the same size and the same shape.
Match up the vertices, sides, and angles of two triangles, even when one is turned.
Recognize SSS, SAS, ASA, and AAS as enough information for congruence.
Tell a real shortcut from patterns that fall short, like AAA or SSA.
Decide whether the facts you’re given are enough for congruence or for similarity.
Use the matching parts once you know two triangles are congruent.

Prerequisites

Why this matters on the SAT

Decide whether the facts fix one triangle

An SAT question might give you a few matching sides and angles, then ask what must be true, or which extra fact would be enough. The figure may be turned or stretched on the page, so how it looks proves nothing. Your job is to match up the parts and decide whether the facts pin down one triangle.

Solution to the example

In the first triangle, the equal sides $AB$ and $BC$ meet at $B$ . In the second, their partners $DE$ and $EF$ meet at $E$ . So each equal angle sits right between the two sides you’re comparing. An angle in that spot is called an included angle.

Two pairs of equal sides plus the equal angle between them is the SAS pattern, and it’s enough to show the triangles are congruent:

\triangle ABC\cong\triangle DEF.

The answer is B. The letters are written in matching order: $A\leftrightarrow D$ , $B\leftrightarrow E$ , and $C\leftrightarrow F$ . Choice C gets the name wrong. SSA would put the angle outside the two sides, and here it sits between them.

SAT example

The equal angle in each triangle lies between the two equal side pairs.

In the figure, $AB=DE=7$ , $BC=EF=11$ , and $m\angle B=m\angle E=46^\circ$ . Which statement is true?

A
The triangles are similar, but they cannot be congruent.
B
$\triangle ABC\cong\triangle DEF$ by SAS.
C
$\triangle ABC\cong\triangle DEF$ by SSA.
D
There is not enough information to relate the triangles.

Congruence locks both size and shape

Congruent triangles have the same size and the same shape. One might be turned or flipped over, but every side and every angle in one triangle has an equal partner in the other.

The statement

\triangle ABC\cong\triangle DEF

also tells you which parts are partners. Read the letters in order, first with first, second with second, third with third:

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

Each side pairs with the side between the matching endpoints:

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

Once you know two triangles are congruent, every pair of matching parts is equal. That’s more than similarity gives you. Similar triangles can be different sizes, so their matching sides can differ by a scale factor. For congruent triangles, the scale factor is $1$ .

Check your understanding:

Triangle $JKL$ is congruent to triangle $MNP$ , in that order. Which side matches $\overline{JL}$ , and which angle matches $\angle K$ ?

Common mistake:

When one triangle is turned, its top vertex may not match the other triangle’s top vertex. So don’t pair vertices by where they sit on the page. Use the stated order, the matching marks, or the equal angles. Then pair each side by its endpoint letters, and check your matches against every fact you’re given.

Use the four congruence shortcuts

You don’t need all six pairs, three sides and three angles, before you can say two triangles are congruent. Certain combinations of three facts pin the triangle down completely. Those combinations are sufficient, which means they give you enough information.

Each set of marks pins down one triangle’s size and shape.

SSS (side-side-side): all three pairs of matching sides are equal.
SAS (side-angle-side): two pairs of matching sides are equal, and so is the angle between them.
ASA (angle-side-angle): two pairs of matching angles are equal, and so is the side between them.
AAS (angle-angle-side): two pairs of matching angles are equal, and so is one other pair of matching sides.

Why do ASA and AAS both work? Once two angles match, the third has to match too, because all three add up to $180^\circ$ . The equal side then fixes the size. Without it, even three pairs of equal angles only give you similarity.

The middle letter in SAS matters: the angle has to be sandwiched between the two sides. If it isn’t, the pattern is SSA (side-side-angle). In general, SSA can fit more than one triangle, so it isn’t a congruence shortcut.

Check your understanding:

Two triangles have three pairs of equal matching angles, and you know nothing about their sides. Is that enough for congruence, similarity, both, or neither?

Try it yourself:

Put a finger on each of the two sides in an SAS claim, then slide along them to the vertex where they meet. That’s where the equal angle has to be. If the marked angle is at a different vertex, it isn’t SAS.

Is it enough? Test one choice at a time

When a question asks which fact would be enough, don’t judge an answer choice on its own. Add it to everything the question already gave you, then see what you have.

Here’s the order that keeps it straight:

First, check what the question wants you to show: congruence or similarity.
Match up the vertices, so you know which sides and angles are partners.
Take stock of the givens, such as one pair of angles and one pair of sides.
Add one answer choice to that list, and test it together with the givens.
Look for a pattern you can name: SSS, SAS, ASA, or AAS for congruence, and AA, proportional SSS, or proportional SAS for similarity.

The word proportional is where the two goals split. Congruence needs matching sides to be equal. Similarity only needs them in the same ratio, like sides $3$ and $4$ in one triangle against $6$ and $8$ in the other.

For SAS similarity, the equal angle still has to be sandwiched between the two pairs of proportional sides. For SSS similarity, all three pairs of matching sides need the same ratio.

Example: Test each added statement

Worked example

Test each statement on its own, together with the angle pair and side pair already marked.

In the figure, $\angle A\cong\angle D$ and $AB=DE$ . Which additional statements are each sufficient to prove that $\triangle ABC\cong\triangle DEF$ ?

I. $\angle B\cong\angle E$

II. $BC=EF$

III. $\angle C\cong\angle F$

A
I only
B
II only
C
I and III only
D
I, II, and III

Step 1

Take stock of the givens

The givens tell you how the triangles match up:

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

So far, you know one pair of matching angles and one pair of matching sides:

\angle A\cong\angle D \quad\text{and}\quad AB=DE.

That isn’t one of the four shortcuts yet. Each statement is a chance to finish one.

Step 2

Test statement I

Statement I adds

\angle B\cong\angle E.

Now the known side $AB$ sits between the two known angles, $\angle A$ and $\angle B$ . Its partner $DE$ sits between $\angle D$ and $\angle E$ . That’s ASA, so statement I is enough.

Step 3

Test statement II

Statement II adds $BC=EF$ . That gives you two pairs of sides,

AB=DE \quad\text{and}\quad BC=EF,

plus $\angle A\cong\angle D$ .

This one looks close to SAS, so check where the angle is. $AB$ and $BC$ meet at $B$ , not at $A$ , so $\angle A$ isn’t between them. That’s the SSA setup, and statement II isn’t enough.

Step 4

Test statement III

Statement III adds

\angle C\cong\angle F.

Now you have two pairs of angles and the side pair $AB=DE$ . That side isn’t between $\angle A$ and $\angle C$ , so the pattern is AAS, which is enough.

Statements I and III each work, and statement II doesn’t. The answer is C.

Check your understanding:

Suppose statement II said $AC=DF$ instead. Would that be enough?

Common mistake:

It’s tempting to reject a choice because it adds only one fact. But one fact can finish a pattern, the way statements I and III did. Write the givens next to each choice before you judge it.

Keep congruence and similarity separate

SSS and SAS show up for both congruence and similarity, but they mean different things for each. So check which one the question wants before you name a pattern.

What you know	Congruent?	Similar?
Two pairs of equal angles	Not enough	Yes, by AA
Three pairs of equal sides	Yes, by SSS	Yes, with scale factor $1$
Three pairs of proportional sides	Only if the ratio is $1$	Yes, by SSS similarity
Two pairs of equal sides and the equal angle between them	Yes, by SAS	Yes, with scale factor $1$
Two pairs of proportional sides and the equal angle between them	Only if both ratios are $1$	Yes, by SAS similarity

Congruent triangles are always similar too, with a scale factor of $1$ . It doesn’t work the other way around, because similar triangles can be different sizes.

The SAT won’t ask you to write a proof. It wants the quick version: match the parts, name the pattern, and use what it tells you.

Choose the theorem before the tool

Do this work by hand. Desmos can check that two ratios match, like

\frac{6}{10}=\frac{9}{15},

but it can’t tell you whether the equal angle sits between those sides, or whether the question wants congruence or similarity. Those are the real decisions here, and they come from reading the figure and the matches. In the practice below, use the calculator only if you want to check some arithmetic.

Practice problems

Use SSS to match the angles

Practice problem

In triangles $JKL$ and $MNP$ ,

JK=MN=8,\qquad KL=NP=13,\qquad JL=MP=15.

Which angle must have the same measure as $\angle K$ ?

Calculator loads as you approach

Match the vertices by their side lengths. Use the graph only as scratch space.

Use congruence to find a segment

Practice problem

Triangles $ABC$ and $BDE$ are congruent, where $A$ , $B$ , and $C$ correspond to $B$ , $D$ , and $E$ , respectively. Point $C$ lies on $\overline{BD}$ . If $AB=42$ and $BC=19$ , what is the length of $\overline{CD}$ ?

Your answer

Calculator loads as you approach

Match the sides from the stated order. The subtraction after that is quick by hand.

Is it enough for similarity?

Practice problem

Compare the two side ratios that surround the marked equal angles.

In the figure, $\angle Q\cong\angle V$ , $PQ=6$ , $QR=9$ , $UV=10$ , and $VW=15$ . Which statement is best supported?

Calculator loads as you approach

Name the pattern by hand. Use Desmos only if you want to check the two ratios.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Congruent triangles have the same size and shape, so every pair of matching sides and angles is equal.
Match the parts first, from the letter order, the marks, or the equal parts, not from where they sit on the page.
SSS, SAS, ASA, and AAS are each enough for congruence.
In SAS, the angle has to be sandwiched between the two sides. SSA isn’t a congruence shortcut.
AA fixes the shape but not the size, so it shows similarity, not congruence.
Similarity also follows from three pairs of proportional sides, or from two pairs of proportional sides with the equal angle between them.
When a question asks what’s enough, add each choice to the givens and look for a complete pattern.

Next lesson

Apply the Pythagorean theorem

Use two known sides of a right triangle to find a missing distance.

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Practice

Practice this lesson

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

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