Middle Years

Lesson

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.

In two congruent triangles, any sides or angles that match up are referred to as corresponding.

Corresponding sides and angles

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).

We can use the congruence test SSS to establish that $\triangle CDE\equiv\triangle LMN$△`C``D``E`≡△`L``M``N`.

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 $\angle DEC$∠`D``E``C`, we replace $D\rightarrow M$`D`→`M`, $E\rightarrow N$`E`→`N`, and $C\rightarrow L$`C`→`L`, which gives us $\angle MNL$∠`M``N``L`. Since these two angles have the same position, they are corresponding (and must be equal).

Doing this for each angle in $\triangle CDE$△`C``D``E` matches them with an angle in $\triangle LMN$△`L``M``N`:

$\angle DEC=\angle MNL$∠`D``E``C`=∠`M``N``L`

$\angle CDE=\angle LMN$∠`C``D``E`=∠`L``M``N`

$\angle ECD=\angle NLM$∠`E``C``D`=∠`N``L``M`

We can use the congruence test AAS to establish that $\triangle PRQ\equiv\triangle SUT$△`P``R``Q`≡△`S``U``T`.

Which side in $\triangle STU$△`S``T``U` is equal in length to $PQ$`P``Q`?

**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 $PQ$`P``Q`, we replace $P\rightarrow S$`P`→`S` and $Q\rightarrow T$`Q`→`T` to find the corresponding side $ST$`S``T`. 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: $\triangle CDE\equiv\triangle LMN$△`C``D``E`≡△`L``M``N` means $C\leftrightarrow L$`C`↔`L`, $D\leftrightarrow M$`D`↔`M`, $E\leftrightarrow N$`E`↔`N`.

Example 2: $\triangle PRQ\equiv\triangle SUT$△`P``R``Q`≡△`S``U``T` 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.

It is known that $\triangle STU\equiv\triangle ABC$△`S``T``U`≡△`A``B``C`.

- Which of the following equalities do we know to be true? Select all correct options.
$TU=BC$

`T``U`=`B``C`A$US=AB$

`U``S`=`A``B`B$TU=CA$

`T``U`=`C``A`C$ST=AB$

`S``T`=`A``B`D$TU=BC$

`T``U`=`B``C`A$US=AB$

`U``S`=`A``B`B$TU=CA$

`T``U`=`C``A`C$ST=AB$

`S``T`=`A``B`D

It is known that $\triangle STU\equiv\triangle PQR$△`S``T``U`≡△`P``Q``R`.

- Which of the following equalities do we know to be true? Select all correct options.
$\angle STU=\angle PQR$∠

`S``T``U`=∠`P``Q``R`A$\angle STU=\angle QPR$∠

`S``T``U`=∠`Q``P``R`B$SU=PQ$

`S``U`=`P``Q`C$ST=PQ$

`S``T`=`P``Q`D$\angle STU=\angle PQR$∠

`S``T``U`=∠`P``Q``R`A$\angle STU=\angle QPR$∠

`S``T``U`=∠`Q``P``R`B$SU=PQ$

`S``U`=`P``Q`C$ST=PQ$

`S``T`=`P``Q`D

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.

We can use the congruence test AAS to establish that $\triangle GHI\equiv\triangle LMN$△`G``H``I`≡△`L``M``N`.

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 $\triangle GHI$△`G``H``I` must also have a length of $y$`y`.

**Do:** In $\triangle GHI$△`G``H``I`, the side $HI$`H``I` is opposite the angle of size $28^\circ$28°, and it has length $6$6. This means $y=6$`y`=6 as well.

These two triangles are congruent. Find the value of $y$`y`.

Consider the two triangles below:

Together with the given information, which other condition would make sure that these two triangles are congruent?

$\angle FGE=\angle JKH$∠

`F``G``E`=∠`J``K``H`A$EG=HJ$

`E``G`=`H``J`B$EF=HJ$

`E``F`=`H``J`C$EF=HK$

`E``F`=`H``K`D$\angle FGE=\angle JKH$∠

`F``G``E`=∠`J``K``H`A$EG=HJ$

`E``G`=`H``J`B$EF=HJ$

`E``F`=`H``J`C$EF=HK$

`E``F`=`H``K`DGiven that $EF=HJ$

`E``F`=`H``J`, and $EG=7$`E``G`=7, find the value of $m$`m`.