Step by step
The question
Question: find TU.
Triangles PQR and STU have the marked parts equal.
Angle Q = angle T = 90°.
PR = SU = 29.
PQ = ST = 20.
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 S, Q with T, R with U.
STU is turned over.
That does not matter.
Angle Q and angle T are both 90°: a pair of equal angles.
A for angle.
PR and SU are both 29: a pair of equal sides.
S for side.
PQ and ST are both 20: a pair of equal sides.
S for side.
2 Find the rule
Angle Q is 90°: a right angle.
PR is across from it: the hypotenuse.
Right angle, hypotenuse, side: R H S.
Why?
Pythagoras: QR² is the hypotenuse squared take away the other side squared: 441.
So QR is 21.
TU is the same, 21.
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
TU matches QR: T with Q, U with R.
In congruent triangles, matching parts are equal.
So TU = QR.
So TU = 21.
Let's look back.
First: match the parts.
Then: find the rule.
Last: use the matching parts.