Congruent triangles
Two triangles are congruent if all their corresponding angles have the same measure and all their corresponding sides have the same length.
Four rules of proving that two triangles are congruent
Rule 1: The SSS rule: Side-Side-Side rule
The side-side-side rule states that if the three sides of a triangle are equal to the three sides of the other triangle then those two triangles are congruent.
AB = EF, BC = FG and CA = GE
Then, the riangles ABC and EFG are congruent, ABC = EFG
Rule 2: The SAS rule: Side – Angle – Side rule
The side-angle-side rule states that if two sides and the angle between those two sides are equal to the two sides and the angle between them of the other triangle then those two triangles are congruent.
AB = EF, BC = FG
$ \displaystyle \widehat{B}=\widehat{F}$
Then, the riangles ABC and EFG are congruent, ABC = EFG
Rule 3: The AAS rule: Angle – Angle – Side rule
The angle-angle-side rule states that if two angles and one of the side in front of one of the angles of the triangle are equal to the two angles and the other side of the other triangle then those two triangles are congruent.
AB = EF
$\displaystyle \widehat{B}=\widehat{F}$ ; $\displaystyle \widehat{C}=\widehat{G}$
Then the triangles ABC and EFG are congruent, ABC = EFG.
Rule 4: The ASA rule: Angle – Side – Angle rule
The angle-side-angle rule states that if one side and the two angles sideways this side of the triangle are equal to the side and the two angles sideways this side of the other triangle then those triangles are congruent.
AB = EF,
$\displaystyle \widehat{A}=\widehat{E}$ ; $\displaystyle \widehat{B}=\widehat{F}$
Then the triangles ABC and EFG are congruent ABC = EFG.
Worked example 1: We are given the parallelogram ABCD. Prove that the diagonal AC divides the parallelogram in two congruent triangles. ABC = ADC
Solution: Based on the properties of the parallelogram we know that the opposite sides are parallel and congruent.
We also know that when two parallel lines are intersected by a third one we know that the alternate internal angles have equal measures, also the alternate external angles have equal measures.
In our case we have two corresponding internal angles that are equal with each other.
$ \displaystyle \widehat{BCA}=\widehat{CAD}$
$\displaystyle \widehat{BAC}=\widehat{ACD}$
When we look into this two triangles ABC and ADC we found that we have two corresponding angles that are equal.
We also see that the diagonal of the parallelogram is a common side to both of our triangles.
So, we have one equal side and the two angles sideways the side that are equal.
We recall that this is the angle – side – angle rule states that if one side and the two angles sideways this side of the triangle are equal to the side and the two angles sideways this side of the other triangle then those triangles are congruent.
So, $\displaystyle \Delta $ABC and $\displaystyle \Delta $ADC are congruent.
Worked Example 2: The segments $ \displaystyle \left[ AE \right]$ and $\displaystyle \left[ BC \right]$ intersect in the point D, which is the middle point of each of this segments. Find the AB, if CE = 10 cm.
Solution: If we see the figure we have that:
1. $\displaystyle \left[ AD \right]=\left[ DE \right]$
Because the point D is the middle point of the segment $ \displaystyle \left[ AE \right]$
2. $\displaystyle \left[ BD \right]=\left[ DC \right]$
Because the point D is the middle point of the segment
$\displaystyle \left[ BC \right]$
3. $\displaystyle \widehat{ADB}=\widehat{CDE}$ because they are opposite angles.
That’s why based on the the side – angle – side rule states that if two sides and the angle between those two sides are equal to the two sides and the angle between them of the other triangle, then those two triangles are congruent.
So, $\displaystyle \Delta $ABC and $\displaystyle \Delta $ CED are congruent.
In congruent triangles in front of congruent angles $\displaystyle \widehat{ADB}=\widehat{CDE}$
There are congruent side lengths $\displaystyle \left[ AB \right]=\left[ CE \right]$
From this we have that AB = CE, which means that AB = 10 cm.
