Step by step
The question
Question: find x.
Triangles ABC and DEF 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: A with D, B with E, C with F.
DEF is turned over.
That does not matter.
Angle A and angle D are both 45°: a pair of equal angles.
A for angle.
Angle B and angle E are both 70°: a pair of equal angles.
A for angle.
BC and EF are both 8: a pair of equal sides.
S for side.
2 Find the rule
BC is not between the two angles: it does not touch corner A.
Go round: angle, angle, side: A A S.
Two angles fix the third one too, because all three add up to 180°.
The angles fix the shape.
The side fixes the size.
So A A S shows the triangles are congruent.
3 Use the matching parts
DF matches AC: D with A, F with C.
In congruent triangles, matching parts are equal.
So 2x + 4 = 10.
We want x by itself.
Two things are stuck to x: times 2 and + 4.
The + 4 is on the outside, so it comes off first.
Then the times 2.
4 Remove the + 4
To remove + 4, we take away 4.
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.
+ 4 and − 4 make zero.
Cross them out.
10 − 4 = 6.
Now we have 2x = 6.
5 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
6 divided by 2 is 3.
So x = 3!
6 Check
Let's check: put x = 3 in 2x + 4.
We get 10, the same as AC.
So x = 3 is right!
Let's look back.
First: match the parts.
Then: find the rule.
Then: use the matching parts.
Then: remove the + 4.
Then: remove the 2.
Last: check.