Worked examples
The question
Question: which rule shows that triangles ABC and DEF are congruent?
Angle B = angle E = 90°.
AC = DF = 10.
AB = DE = 6.
1 Match the parts
Congruent means the same shape and the same size.
One fits exactly on top of the other.
The order of the letters shows which corners match: A with D, B with E, C with F.
DEF is turned over.
That does not matter.
Angle B and angle E are both 90°: a pair of equal angles.
A for angle.
angle B = angle E = 90° A
AC and DF are both 10: a pair of equal sides.
S for side.
AC = DF = 10 S
AB and DE are both 6: a pair of equal sides.
S for side.
AB = DE = 6 S
2 Find the rule
Angle B is 90°: a right angle.
AC is across from it: the hypotenuse.
Right angle, hypotenuse, side: R H S.
angle B → AC → AB
Why?
Pythagoras: BC² is the hypotenuse squared take away the other side squared: 64.
BC2 = 10² − 6² = 100 − 36 = 64
So BC is 8.
EF is the same, 8.
All three sides match, like S S S.
BC = EF = 8
Careful!
S S A is not a rule.
But with a right angle and the hypotenuse, it is: R H S.
Careful: S S A is not a rule, but R H S is
Pythagoras fixes the third side, so all three sides match.
There is only one way to draw it.
So R H S shows the triangles are congruent.
It is also called H L: hypotenuse, leg.
RHS → congruent
Let's look back.
First: match the parts.
Last: find the rule.
▶ Watch it solved
The question
Question: find DE.
Triangles ABC and DEF have the marked parts equal.
Angle B = angle E = 90°.
AC = DF = 10.
BC = EF = 8.
1 Match the parts
Congruent means the same shape and the same size.
One fits exactly on top of the other.
The order of the letters shows which corners match: A with D, B with E, C with F.
DEF is turned over.
That does not matter.
Angle B and angle E are both 90°: a pair of equal angles.
A for angle.
angle B = angle E = 90° A
AC and DF are both 10: a pair of equal sides.
S for side.
AC = DF = 10 S
BC and EF are both 8: a pair of equal sides.
S for side.
BC = EF = 8 S
2 Find the rule
Angle B is 90°: a right angle.
AC is across from it: the hypotenuse.
Right angle, hypotenuse, side: R H S.
angle B → AC → BC
Why?
Pythagoras: AB² is the hypotenuse squared take away the other side squared: 36.
AB2 = 10² − 8² = 100 − 64 = 36
So AB is 6.
DE is the same, 6.
All three sides match, like S S S.
AB = DE = 6
Careful!
S S A is not a rule.
But with a right angle and the hypotenuse, it is: R H S.
Careful: S S A is not a rule, but R H S is
Pythagoras fixes the third side, so all three sides match.
There is only one way to draw it.
So R H S shows the triangles are congruent.
It is also called H L: hypotenuse, leg.
RHS → congruent
3 Use the matching parts
DE matches AB: D with A, E with B.
In congruent triangles, matching parts are equal.
So DE = AB.
DE = AB
So DE = 6.
DE = 6
Let's look back.
First: match the parts.
Then: find the rule.
Last: use the matching parts.
▶ Watch it solved
The question
Question: find x.
Triangles ABC and DEF have the marked parts equal.
Angle A = angle D = 90°.
BC = EF = 41.
AC = DF = 9.
1 Match the parts
Congruent means the same shape and the same size.
One fits exactly on top of the other.
The order of the letters shows which corners match: A with D, B with E, C with F.
DEF is turned over.
That does not matter.
Angle A and angle D are both 90°: a pair of equal angles.
A for angle.
angle A = angle D = 90° A
BC and EF are both 41: a pair of equal sides.
S for side.
BC = EF = 41 S
AC and DF are both 9: a pair of equal sides.
S for side.
AC = DF = 9 S
2 Find the rule
Angle A is 90°: a right angle.
BC is across from it: the hypotenuse.
Right angle, hypotenuse, side: R H S.
angle A → BC → AC
Why?
Pythagoras: AB² is the hypotenuse squared take away the other side squared: 1600.
AB2 = 41² − 9² = 1681 − 81 = 1600
So AB is 40.
DE is the same, 40.
All three sides match, like S S S.
AB = DE = 40
Careful!
S S A is not a rule.
But with a right angle and the hypotenuse, it is: R H S.
Careful: S S A is not a rule, but R H S is
Pythagoras fixes the third side, so all three sides match.
There is only one way to draw it.
So R H S shows the triangles are congruent.
It is also called H L: hypotenuse, leg.
RHS → congruent
3 Use the matching parts
AB matches DE: A with D, B with E.
In congruent triangles, matching parts are equal.
So 3x + 1 = 40.
3x + 1 = 40
We want x by itself.
Two things are stuck to x: times 3 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 3.
4 Remove the + 1
To remove + 1, we take away 1.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
3x + 1 − 1 = 40 − 1
+ 1 and − 1 make zero.
Cross them out.
40 − 1 = 39.
40 − 1 = 39
Now we have 3x = 39.
3x = 39
5 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3x3 = 393
3 divided by 3 is 1.
So just x is left.
39 divided by 3 is 13.
393 = 13
So x = 13!
x = 13
6 Check
Let's check: put x = 13 in 3x + 1.
3x + 1 = 3 × 13 + 1 = 40
We get 40, the same as DE.
So x = 13 is right!
Let's look back.
First: match the parts.
Then: find the rule.
Then: use the matching parts.
Then: remove the + 1.
Then: remove the 3.
Last: check.
▶ Watch it solved
