Step by step
The question
Question: find DF.
Triangles ABC and DEF have the marked parts equal.
Angle A = angle D = 90°.
BC = EF = 25.
AB = DE = 7.
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.
BC and EF are both 25: a pair of equal sides.
S for side.
AB and DE are both 7: a pair of equal sides.
S for side.
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.
Why?
Pythagoras: AC² is the hypotenuse squared take away the other side squared: 576.
So AC is 24.
DF is the same, 24.
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
DF matches AC: D with A, F with C.
In congruent triangles, matching parts are equal.
So DF = AC.
So DF = 24.
Let's look back.
First: match the parts.
Then: find the rule.
Last: use the matching parts.