Step by step
The question
Question: find x.
Triangles ABC and DEF have the marked parts equal.
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.
AC and DF are both 10: a pair of equal sides.
S for side.
AB and DE are both 6: a pair of equal sides.
S for side.
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.
Why?
Pythagoras: BC² is the hypotenuse squared take away the other side squared: 64.
So BC is 8.
EF is the same, 8.
All three sides match, like S S S.
Careful!
S S A is not a rule.
But with a right angle and the hypotenuse, it is: R H S.
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.
3 Use the matching parts
EF matches BC: E with B, F with C.
In congruent triangles, matching parts are equal.
So x + 3 = 8.
4 Remove the + 3
We want x by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
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.
+ 3 and − 3 make zero.
Cross them out.
8 − 3 = 5.
So x = 5!
5 Check
Let's check: put x = 5 in x + 3.
We get 8, the same as BC.
So x = 5 is right!
Let's look back.
First: match the parts.
Then: find the rule.
Then: use the matching parts.
Then: remove the + 3.
Last: check.