Similar and Congruent Triangles, SAT math lesson

A photo enlarged on a copier keeps every angle and stretches every length by the same factor. A tree’s shadow and your own shadow, cast by the same sun, form two triangles with the same shape. Triangles with the same shape let you find a length you cannot measure from lengths you can. The hard part is rarely the arithmetic. It is matching the right sides to each other.

See also Similar Triangles and Changing Dimensions in Foundations for Calculus.

Key terms

Correspondence of vertices

Definition A correspondence between triangles $ABC$ and $DEF$ pairs each vertex of one with a vertex of the other. The statement $\triangle ABC \sim \triangle DEF$ uses the order of the letters to give the correspondence $A \leftrightarrow D$ , $B \leftrightarrow E$ and $C \leftrightarrow F$ . Corresponding angles are the angles at matched vertices, such as $\angle A$ and $\angle D$ . Corresponding sides join matched vertices, such as $\overline{AB}$ and $\overline{DE}$ . Equivalently, corresponding sides lie opposite corresponding angles.

In plain words Before you compare two triangles, decide which corner goes with which. The order of the letters in the name tells you.

Example If $\triangle PQR \sim \triangle XYZ$ , then $\angle Q$ corresponds to $\angle Y$ , side $\overline{QR}$ corresponds to $\overline{YZ}$ , and side $\overline{PR}$ corresponds to $\overline{XZ}$ . Side $\overline{PR}$ does not correspond to $\overline{XY}$ , even if they look alike in a figure.

Similarity

Definition Triangles $ABC$ and $DEF$ are similar, written $\triangle ABC \sim \triangle DEF$ , when corresponding angles are congruent and corresponding sides are proportional,

\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}.

Similar triangles have the same shape and may have different sizes. A similar copy can be obtained by a rotation, reflection or translation combined with a uniform scaling.

In plain words Similar triangles are enlargements or reductions of each other. Angles stay the same, and every side changes by the same factor.

Example Suppose $\triangle ABC \sim \triangle DEF$ with $AB = 6$ , $BC = 8$ , $AC = 10$ and $DE = 9$ . Each side of $DEF$ is $\frac{9}{6} = 1.5$ times the matching side of $ABC$ , so $EF = 1.5 \cdot 8 = 12$ and $DF = 1.5 \cdot 10 = 15$ .

Scale factor

Definition If $\triangle ABC \sim \triangle DEF$ , the scale factor from $ABC$ to $DEF$ is the common ratio

k = \frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}.

This $k$ is always positive. The scale factor from $DEF$ back to $ABC$ is $\frac{1}{k}$ . The perimeters have the same ratio $k$ , and the areas have ratio $k^2$ .

In plain words The scale factor is the one number you multiply every length by to go from one triangle to the other.

Example In the triangles above, $k = 1.5$ . The perimeter of $ABC$ is $24$ and the perimeter of $DEF$ is $1.5 \cdot 24 = 36$ . Triangle $ABC$ is a right triangle with legs 6 and 8, so its area is $24$ , and the area of $DEF$ is $1.5^2 \cdot 24 = 54$ .

Congruence

Definition Triangles $ABC$ and $DEF$ are congruent, written $\triangle ABC \cong \triangle DEF$ , when all corresponding sides and all corresponding angles are equal. Congruent triangles are similar with scale factor $k = 1$ . One can be moved onto the other by rotations, reflections and translations, with no scaling.

In plain words Congruent triangles are exact copies, possibly turned or flipped.

Example If $\triangle JKL \cong \triangle MNO$ with $JK = 5$ and $m\angle L = 40^\circ$ , then $MN = 5$ and $m\angle O = 40^\circ$ .

AA similarity

Definition If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. The third angles are then equal as well, because each triangle’s angles add up to $180^\circ$ . Matching all three angles (often called AAA) gives the same conclusion. It proves similarity but not congruence, since equal angles allow any size.

In plain words Two matching angles fix the shape. They say nothing about the size.

Example In triangle $ABC$ , point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ , with $\overline{DE}$ parallel to $\overline{BC}$ . The triangles share $\angle A$ , and $\angle ADE \cong \angle ABC$ because they are corresponding angles for parallel lines. So $\triangle ADE \sim \triangle ABC$ by AA. If $AD = 4$ , $DB = 6$ and $DE = 5$ , then $AB = 4 + 6 = 10$ , and

\begin{aligned} \frac{BC}{DE} &= \frac{AB}{AD} = \frac{10}{4}, \\[0.4em] BC &= \frac{10}{4} \cdot 5 = 12.5. \end{aligned}

Notice that the matching side of $\overline{AD}$ is the whole side $\overline{AB}$ , not the piece $\overline{DB}$ .

SSS criteria

Definition If all three sides of one triangle are proportional to the three sides of another, with the same ratio, the triangles are similar (SSS similarity). If the three sides of one triangle are equal to the three sides of another, the triangles are congruent (SSS congruence). To check, pair the shortest with the shortest, the middle with the middle and the longest with the longest.

In plain words Three side lengths fix a triangle’s shape, and equal side lengths fix the triangle completely.

Example Triangles with sides 4, 6, 8 and 6, 9, 12 are similar, since

\frac{6}{4} = \frac{9}{6} = \frac{12}{8} = 1.5.

Triangles with sides 4, 6, 8 and 6, 9, 13 are not similar, because $\frac{13}{8} = 1.625$ differs from $1.5$ .

SAS criteria

Definition If two sides of one triangle are proportional to two sides of another and the angles included between those sides are congruent, the triangles are similar (SAS similarity). If the two pairs of sides are equal and the included angles are congruent, the triangles are congruent (SAS congruence). The angle must be the included angle, the one formed by the two sides.

In plain words Two sides and the angle between them pin the triangle down. The angle has to sit between the two sides you used.

Example In triangle $ABC$ , $AB = 3$ , $AC = 5$ and $m\angle A = 40^\circ$ . In triangle $DEF$ , $DE = 6$ , $DF = 10$ and $m\angle D = 40^\circ$ . The ratios are $\frac{6}{3} = \frac{10}{5} = 2$ and the included angles match, so $\triangle ABC \sim \triangle DEF$ with scale factor 2, and $EF = 2 \cdot BC$ .

Sufficient information

Definition Information is sufficient to prove two triangles congruent when it fits one of the criteria SSS, SAS, ASA (two angles and the included side) or AAS (two angles and a side that is not included). For similarity, AA, SSS similarity or SAS similarity is sufficient. Two other patterns are not sufficient for congruence in general. Three equal angles give only similarity. Two sides and an angle that is not between them (SSA) can fit two different triangles, so SSA is not a congruence criterion in general. It does work in special cases, for example right triangles with equal hypotenuses and one pair of equal legs (the hypotenuse leg case), or more generally when the given angle lies opposite the longer of the two given sides. Without such extra information, SSA does not guarantee congruence. On the SAT a figure is drawn to scale unless the question says otherwise, but a conclusion still has to rest on the given data and on one of these criteria, not on how the figure looks.

In plain words Before you solve, ask whether the facts you have actually force the triangles to match. Looking alike is not a reason.

Example Take $m\angle A = 30^\circ$ , $AB = 10$ and $BC = 6$ , which is an SSA pattern. Two different triangles fit these facts. In one, angle $C$ is about $56.4^\circ$ and $AC \approx 12.0$ . In the other, angle $C$ is about $123.6^\circ$ and $AC \approx 5.3$ . Since one set of data allows two shapes, SSA cannot prove congruence.

Check yourself

Triangles $PQR$ and $STU$ are similar, with $\triangle PQR \sim \triangle STU$ . The lengths are $PQ = 5$ , $QR = 7$ and $ST = 15$ . What is the length of $\overline{TU}$ ? (A) $\frac{7}{3}$ (B) $15$ (C) $17$ (D) $21$

Show the answer

The correct choice is (D). The order of the letters matches $\overline{PQ}$ with $\overline{ST}$ and $\overline{QR}$ with $\overline{TU}$ . The scale factor is $\frac{15}{5} = 3$ , so $TU = 3 \cdot 7 = 21$ . Choice (C) adds $10$ instead of multiplying by $3$ , choice (A) multiplies by $\frac{5}{15}$ , the ratio upside down, and choice (B) matches $\overline{TU}$ with $\overline{ST}$ .

In triangle $ABC$ , point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ , with $\overline{DE}$ parallel to $\overline{BC}$ . If $AD = 6$ , $AB = 15$ and $BC = 20$ , what is the length of $\overline{DE}$ ?

Show the answer

The answer is $8$ . Because $\overline{DE}$ is parallel to $\overline{BC}$ , $\triangle ADE \sim \triangle ABC$ by AA, so

\frac{DE}{BC} = \frac{AD}{AB} = \frac{6}{15}, \quad \text{so} \quad DE = 8.

Using the piece $DB = 9$ in place of $AB$ would give $\frac{40}{3}$ , which is wrong because $\overline{DB}$ is not a side of either triangle.

Triangles $ABC$ and $DEF$ have $AB = DE$ and $BC = EF$ . Which additional fact is enough to prove that $\triangle ABC \cong \triangle DEF$ ? (A) $m\angle A = m\angle D$ (B) $m\angle B = m\angle E$ (C) $m\angle C = m\angle F$ (D) $AB = BC$

Show the answer

The correct choice is (B). Angle $B$ is included between sides $\overline{AB}$ and $\overline{BC}$ , so the triangles are congruent by SAS. Choices (A) and (C) give an angle that is not between the two known sides, an SSA pattern. Since nothing is known about which side is longer or whether there is a right angle, SSA does not guarantee congruence here. Choice (D) only says each triangle has two equal sides, which says nothing about the angle between them.

Similar and Congruent Triangles

Theory, part 95

Practice, part 96

Key terms

Correspondence of vertices

Similarity

Scale factor

Congruence

AA similarity

SSS criteria

SAS criteria

Sufficient information

Check yourself

How this lesson connects

Builds on

Used later in