Step by step
The question
Question: find x.
Triangles PQR and XYZ have the marked parts equal.
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: P with X, Q with Y, R with Z.
XYZ is turned over.
That does not matter.
PQ and XY are both 6: a pair of equal sides.
S for side.
QR and YZ are both 8: a pair of equal sides.
S for side.
Angle Q and angle Y are both 90°: a pair of equal angles.
A for angle.
2 Find the rule
Angle Q is between the two sides: PQ and QR both touch corner Q.
Go round: side, angle, side: S A S.
Two sides and the angle between them fix the triangle.
There is only one way to draw it.
So S A S shows the triangles are congruent.
3 Use the matching parts
XZ matches PR: X with P, Z with R.
In congruent triangles, matching parts are equal.
So 5x + 10 = 10.
We want x by itself.
Two things are stuck to x: times 5 and + 10.
The + 10 is on the outside, so it comes off first.
Then the times 5.
4 Remove the + 10
To remove + 10, we take away 10.
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.
+ 10 and − 10 make zero.
Cross them out.
10 − 10 = 0.
Now we have 5x = 0.
5 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
0 divided by 5 is 0.
So x = 0!
6 Check
Let's check: put x = 0 in 5x + 10.
We get 10, the same as PR.
So x = 0 is right!
Let's look back.
First: match the parts.
Then: find the rule.
Then: use the matching parts.
Then: remove the + 10.
Then: remove the 5.
Last: check.