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7. Geometry
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Middle Years

7.03 Congruent triangle tests

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Two triangles are congruent if one can be moved (by translation, rotation, and/or reflection) to lie exactly on top of the other. It is a more precise way of saying that two triangles are "the same". 

Here is an example of two congruent triangles:

Three pairs of matching equal sides, and three pairs of matching equal angles.

If we reflected one of these triangles, rotated it, and translated it, we could place it directly on top of the other.

When deciding whether or not two triangles are congruent, we don't need to know six pieces of information (three sides and three angles), we need only three pieces in one of these combinations:


Side-side-side congruence (SSS)

If two triangles have three equal side lengths, then the triangles must be congruent. Try this yourself with three straight objects - once you put them together, you can rotate, translate, and reflect the triangle to make every other possible combination:

Each triangle is made from the same three sides, so they are all congruent.

This kind of congruence is called side-side-side, or SSS.


Side-angle-side congruence (SAS)

If two triangles have a pair of matching sides and the angles between them are equal, then the triangles must be congruent. Try this yourself with two straight objects - if you hold them together at one end and form an angle, there is only one triangle you can form by joining the ends together:

After fixing the two given sides about the given angle, there is only one possible triangle.

This kind of congruence is called side-angle-side, or SAS. We write this test with the "A" in between the two "S"s, because the angle must be between the matching sides - the long name for this kind of congruence is "two sides and the included angle".

There is no SSA!

It is possible for triangles to have two pairs of equal sides and a pair of matching angles, yet not be congruent overall. Here is an example:

Two obviously non-congruent triangles!

This can only happen when the pair of equal angles is not included between the sides.

Try using this applet to find the two different triangles that have two matching angles and a matching non-included angle, just like the picture above:

Created with Geogebra
Lydia Kenney, Ben Richards, nbianculli


Right angle-hypotenuse-side congruence (RHS)

If two right-angled triangles have equal lengths hypotenuses and another pair of equal sides, then the triangles must be congruent:

There is only one possible triangle that lines up the hypotenuse with the given side at a right angle.

Notice that the right angle is not included between the sides - this is the only exception to the general rule, which is why we mention it as a separate case. This congruence test is called right angle-hypotenuse-side, or RHS.


Angle-angle-side congruence (AAS)

What if we are only given one pair of equal sides? In this case we need two pairs of equal angles. Here is the construction if the two angles are made with the given side:

The projected lines meet at exactly one point, so we can only build one triangle from this information.

If one of the given angles is opposite the given side, we can always find the third one by using the angle sum of a triangle:

This kind of congruence is called angle-angle-side congruence, or AAS.

We need a matching side for AAS

To use AAS to show that two triangles are congruent, the matching sides must have the same position relative to the matching angles. 

The three triangles below have two pairs of equal angles and a pair of equal sides, but because the side of length $7$7 is in a different position, none of them are congruent:

 

Summary of the four congruence tests

Proving congruence in triangles

These are the four congruence tests:

  • SSS: Three pairs of equal sides
  • SAS: Two pairs of equal sides with an equal included angle
  • AAS: Two pairs of equal angles and one pair of equal sides
  • RHS: Both have right angles, equal hypotenuses, and another equal side

 

Insufficient information

If two triangles don't satisfy these tests, one of two things could be true:

  • The two triangles are definitely not congruent, or
  • We don't have enough information to know whether or not they're congruent

Two triangles are definitely not congruent if there is a pair of sides or angles that are in the same relative position but are not equal to each other.

Worked examples

Example 1

Are these triangles congruent?

Think: Both have matching sides with lengths $9$9 and $5$5. We can use the angle sum of a triangle to find the missing angle in the first triangle.

Do: In the first triangle, we subtract the known angles to find the missing angle: $180^\circ-56^\circ-44^\circ=80^\circ$180°−56°−44°=80°. We now know that the first triangle doesn't have a right angle, so it cannot be congruent to the second triangle.

Example 2

Are these triangles congruent?

Think: Both have matching sides with lengths $7$7 and $6$6. We don't know anything about the angles in the second triangle. They may look congruent, but that isn't enough - either we know for sure, or we don't.

Do: There is no way to calculate any additional side or angle information in the second triangle, and we need at least three matching pairs to make conclusions about congruence. We don't know whether or not they are congruent.

 

Practice questions

QUESTION 1

Consider the two triangles in the diagram below:

Two triangles, $\triangle PQR$△PQR and $\triangle STU$△STU, have their vertices marked with a solid dot.

The triangle above is $\triangle PQR$△PQR labeled with vertices $P$P, $Q$Q, and $R$R. One of the sides of $\triangle PQR$△PQR is labeled $4$4 units indicating its length. The angle at vertex $R$R, $\angle PRQ$∠PRQ, is marked with a yellow-shaded double arc and is opposite the $4$4-unit side. The angle at vertex $P$P, $\angle QPR$∠QPR, is marked with a blue-shaded single arc and is adjacent the $4$4-unit side.

The triangle below is $\triangle STU$△STU labeled with vertices $S$S, $T$T, and $U$U. One of the sides of $\triangle STU$△STU is labeled $4$4 units indicating its length. The angle at vertex $S$S, $\angle TSU$∠TSU, is marked with a blue-shaded single arc and is opposite the $4$4-unit side. The angle at vertex $U$U, $\angle SUT$∠SUT, is marked with a yellow-shaded double arc and is adjacent the $4$4-unit side.

 

  1. Are $\triangle PQR$△PQR and $\triangle STU$△STU congruent?

    Yes, they satisfy SSS.

    A

    Yes, they satisfy SAS.

    B

    Yes, they satisfy AAS.

    C

    Yes, they satisfy RHS.

    D

    No, they are definitely not congruent.

    E

    Unknown, there is not enough information.

    F

QUESTION 2

Consider the following:

  1. Which two of the following triangles are congruent?

    A triangle has vertices marked with solid dots. The first angle, marked with a red-shaded double arc, is labeled $44^\circ$44°. The leg opposite to this angle measures $5$5 units as labeled. The second angle, marked with a blue-shaded arc, is labeled $56^\circ$56°. The leg opposite to this angle is not labeled. The third angle is not marked, and the leg opposite to the unmarked angle measures $9$9 units as labeled.
    A
    A triangle has vertices marked with solid dots. The right angle is indicated by a yellow-shaded square. The hypotenuse, which lies opposite the right angle, is labeled $9$9 units. The horizontal leg is labeled $5$5 units.
    B
    A triangle has vertices marked with solid dots. The first angle, marked with a green-shaded double arc, is labeled $34^\circ$34°. The leg opposite to this angle measures $5$5 units as labeled. The second angle, marked with a blue-shaded arc, is labeled $56^\circ$56°. The leg opposite this angle is not labeled. The third angle is not marked, and the leg opposite it measures $9$9 units as labeled.
    C
    A triangle has vertices marked with solid dots. The right angle is indicated by a yellow-shaded square. The vertical leg is labeled $9$9 units. The horizontal leg is labeled $5$5 units.
    D

  2. What congruence test does this pair satisfy?

    SSS

    A

    SAS

    B

    AAS

    C

    HL

    D

QUESTION 3

Consider the following diagram:

A quadrilateral has vertices labeled $A$A, $B$B, $C$C, and $D$D arranged in a clockwise direction. Vertex $B$B is positioned at the top-left, vertex $C$C at the top-right, vertex $D$D at the bottom-right, and vertex $A$A at the bottom-left. A diagonal line segment $BD$BD runs from vertex $B$B to vertex $D$D, slanting downward and to the right, dividing the quadrilateral into two triangles: $\triangle ABD$△ABD on the lower-left and $\triangle CDB$△CDB on the upper-right. In $\triangle ABD$△ABD, line segment $AB$AB is marked with single arrowhead pointing from vertex $A$A to vertex $B$B and a single tick mark. In $\triangle CDB$△CDB, line segment $DC$DC is similarly marked with single arrowhead pointing from vertex $D$D to vertex $C$C and a single tick mark. The marked segments and the diagonal form two angles: $\angle ABD$∠ABD in $\triangle ABD$△ABD and $\angle DBC$∠DBC in $\triangle CDB$△CDB. These two angles are positioned on opposite sides of the diagonal $BD$BD.

  1. Are the triangles $\triangle ABD$△ABD and $\triangle CDB$△CDB definitely congruent?

    Yes

    A

    No

    B

  2. What congruence test does this pair satisfy?

    SSS

    A

    SAS

    B

    AAS

    C

    RHS

    D

  3. Select the three statements that, when put together, establish congruence for this test.

    Make sure each reason is correct as well.

    $\angle ADB=\angle CBD$∠ADB=∠CBD
    Alternate angles on parallel lines

    A

    $AD=CB$AD=CB
    Corresponding sides on parallel lines

    B

    $\angle ABD=\angle CDB$∠ABD=∠CDB
    Corresponding angles on parallel lines

    C

    $AB=CD$AB=CD
    Given

    D

    $BD$BD is common

    E

    $\angle ABD=\angle CDB$∠ABD=∠CDB
    Alternate angles on parallel lines

    F

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