7.03 Using triangle congruence
Once we have shown two triangles are congruent using some of the side and angle information, we know the other sides and angles must match up as well.
Corresponding sides and angles in congruent triangles
In two congruent triangles, any sides or angles that match up are referred to as corresponding.
If two triangles are congruent, then:
- The sides in the same relative position are equal, and are called corresponding sides (of congruent triangles).
- The angles in the same relative position are equal, and are called corresponding angles (of congruent triangles).
Worked examples
Example 1
We can use the congruence test SSS to establish that $\triangle CDE\equiv\triangle LMN$△CDE≡△LMN.
What are the three corresponding angle pairs?
Think: If we arranged the two triangles so they were lying on top of each other, then $C$C and $L$L would be the same point. Similarly, $D$D and $M$M would be the same point, and $E$E and $N$N as well.
Do: Starting with $\angle DEC$∠DEC, we replace $D\rightarrow M$D→M, $E\rightarrow N$E→N, and $C\rightarrow L$C→L, which gives us $\angle MNL$∠MNL. Since these two angles have the same position, they are corresponding (and must be equal).
Doing this for each angle in $\triangle CDE$△CDE matches them with an angle in $\triangle LMN$△LMN:
$\angle DEC=\angle MNL$∠DEC=∠MNL
$\angle CDE=\angle LMN$∠CDE=∠LMN
$\angle ECD=\angle NLM$∠ECD=∠NLM
Example 2
We can use the congruence test AAS to establish that $\triangle PRQ\equiv\triangle SUT$△PRQ≡△SUT.
Which side in $\triangle STU$△STU is equal in length to $PQ$PQ?
Think: If we arranged the two triangles so they were lying on top of each other, then $P$P and $S$S would be the same point, and $Q$Q and $T$T would be the same as well.
Do: Starting with the side $PQ$PQ, we replace $P\rightarrow S$P→S and $Q\rightarrow T$Q→T to find the corresponding side $ST$ST. Since the triangles are congruent, these sides must be equal.
In both of these examples the order of the points making up each triangle was listed in the same order.
Example 1: $\triangle CDE\equiv\triangle LMN$△CDE≡△LMN means $C\leftrightarrow L$C↔L, $D\leftrightarrow M$D↔M, $E\leftrightarrow N$E↔N.
Example 2: $\triangle PRQ\equiv\triangle SUT$△PRQ≡△SUT means $P\leftrightarrow S$P↔S, $R\leftrightarrow U$R↔U, $Q\leftrightarrow T$Q↔T.
This is a very useful habit to develop when you are writing your own congruence statements, since it makes identifying corresponding sides and angles much easier.
Practice questions
Question 1
It is known that $\triangle STU\equiv\triangle ABC$△STU≡△ABC.
- Which two of the following equalities do we know to be true?
$TU=BC$TU=BC
A$US=AB$US=AB
B$TU=CA$TU=CA
C$ST=AB$ST=AB
D
Question 2
It is known that $\triangle STU\equiv\triangle PQR$△STU≡△PQR.
- Which two of the following equalities do we know to be true?
$\angle STU=\angle PQR$∠STU=∠PQR
A$\angle STU=\angle QPR$∠STU=∠QPR
B$SU=PQ$SU=PQ
C$ST=PQ$ST=PQ
D
Matching information across congruent triangles
If two corresponding sides or angles must be equal in congruent triangles then knowing the value of one gives us the value of the other.
Worked example
Example 3
We can use the congruence test AAS to establish that $\triangle GHI\equiv\triangle LMN$△GHI≡△LMN.
Find the value of $y$y.
Think: The side of length $y$y is opposite the angle of size $28^\circ$28°. The corresponding side in $\triangle GHI$△GHI must also have a length of $y$y.
Do: In $\triangle GHI$△GHI, the side $HI$HI is opposite the angle of size $28^\circ$28°, and it has length $6$6. This means $y=6$y=6 as well.
Practice questions
Question 3
These two triangles are congruent. Find the value of $y$y.
$\triangle SUT$△SUT is congruent to $\triangle ACB$△ACB.
In the top triangle, the vertices are labeled $S$S, $U$U, and $T$T. The angle at $U$U is a right angle and shaded purple, while the angle at $T$T is $44^\circ$44°. The side $SU$SU is marked with a single tick to indicate congruence with side $AC$AC in the second triangle. The side $ST$ST is marked with a double tick to indicate congruence with side $AB$AB.
In the bottom triangle, the vertices are labeled $A$A, $C$C, and $B$B. The angle at $A$A is labeled $\left(y\right)$(y), corresponding to angle at $S$S, the angle at $C$C is a right angle and shaded purple, corresponding to the angle at $U$U in the top triangle. The angle at $B$B is corresponding to the angle at $T$T. The side $AC$AC is congruent to side $SU$SU, the side $AB$AB is congruent to side $ST$ST, and the side $CB$CB is congruent to side $UT$UT.
Question 4
Consider the two triangles below:
Together with the given information, which other condition would make sure that these two triangles are congruent?
$\angle FGE=\angle JKH$∠FGE=∠JKH
A$EG=HJ$EG=HJ
B$EF=HJ$EF=HJ
C$EF=HK$EF=HK
DGiven that $EF=HJ$EF=HJ, and $EG=7$EG=7, find the value of $m$m.