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Congruent triangle postulates and right triangle congruence

Congruent triangles (two or more triangles) have 3 sets of congruent (of equal length) sides and 3 sets of congruent (of equal measure) angles.

Congruent triangle postulates

SSS (side-side-side) – Two triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other.

This statement basically says that you cannot draw two different triangles that have all three sides of the same length. That means that every triangle that has matching sides the same length will be congruent. It doesn’t matter in which position are those triangles, they can be moved, rotated, mirrored, anything, but if they have the same length matching sides, they are congruent. If they have three matching sides, which means that they also have three matching angles.

SSS postulate

SAS (side-angle-side) – Two triangles are congruent if two sides in one triangle are congruent to the corresponding sides in the other, and if the angles they enclose are of equal measure.

For example let’s say you’ve got c = 5 cm, b = 7 cm, α = 60^{\circ}.

First, draw the sketch.

SAS sketch

Draw side c. Now, let’s get to the construction of an angle.
This angle is enclosed by sides c and b, so you’ll construct your angle in point A. When you constructed your angle, take you compass, and draw another circle, with center in A radius 7 (length of b), you got point C, and all that there is left to do, is connect those points.

SSA statement

As you can see, there is only one triangle that can be drawn if two sides and angle between them are default, which means that triangles you draw that have those three properties equal will be congruent.

Any two triangles that have all these conditions met will have all three sides of equal length and all three angles of equal measure.

For example: following triangles are congruent.

SAS postulate

Example 1: Are triangles ABC and PQR congruent if \mid AB \mid = 13, \mid BC \mid = 14, \mid CA \mid = 15,
\mid PQ \mid = 13, \mid QR \mid = 14 \mid RP \mid = 15.2

So, we have two triangles, one with length of sides 13, 14 and 15, and another with length of sides 13, 14, 15.2. By SSS theorem, two triangles are congruent if and only if length of all sides of the first triangle matches length of sides of other triangles. Since one side has no match, those two triangles are not congruent.

What needs to be changed for them to be congruent? You compare sides, so \mid AB \mid = \mid PQ \mid,\mid BC \mid = \mid QR \mid, so, \mid CA \mid must be equal to \mid RP \mid.

Example 2: Are triangles ABC and PQR congruent if \mid AB \mid = 8, \mid BC \mid = 12, \angle abc = 40^{\circ}.
As you can notice, all given side lengths from first triangle match ones from the other, and the angle matches also. But this is not enough- you have to check if the angle is in the right place, which means to check if he is enclosed by the sides with the same length as the same angle in the first triangle.

The easiest way is to draw a sketch; it doesn’t have to be precise, just sufficient for you to make conclusions out of it. So, angles with the same measure are enclosed by the sides with matching equal length. By SAS these triangles are congruent.

sketch congruence

How would you draw another triangle congruent to the one that you have? You would simply construct another triangle with same measurements.

 

ASA (angle-side-angle) – Two triangles are congruent if two angles and one side in one triangle are congruent to two angles and one side of second triangle.

Again, this statement says, that you cannot draw two different triangles that have two congruent angles and one side that they ‘share’.

For example, let’s say you have \ c = 5 cm, \alpha = 90^{\circ}, \beta = 60^{\circ}.

First, you draw side c. And then you construct angle α in point A and \beta in point B, and the point where their arms intersect will be your point C.

ASA postulate

From this construction you can conclude that there is only one triangle with these properties. And that means that every two triangles that have two angles of same size and matching side of equal length will be congruent- all three pairs of angles will be of equal measure, and all three sides equal length.

SSA (side-side-angle) – Two triangles are congruent if two angles and angle that is opposite to the larger side are congruent.

This statement says that you can only draw one triangle if you know two sides and angle that is opposite to the larger side.

Example: let’s say that \ a < b, a = 3, b = 4 and \beta = 60^{\circ}. Draw the triangle ABC.

The sketch:
First we’ll draw a, then the angle \beta = 60^{\circ}, and then b.

SSA sketchSSA statement

 

Example: Are two triangles ABC and PQR congruent if \mid AB \mid = 6, \mid PQ \mid = 6, \angle bac = 40^{\circ}, \angle acb = 110^{\circ}, \angle qpr = 40^{\circ}, \angle pqr = 30^{\circ}
You have to remember not to rush into conclusions. It doesn’t matter if you can’t see congruent sides or angles at first, solution can’t always be obvious.

Draw a sketch.

SSA example sketch

From the sketch you realize that you don’t know matching angles, but you can calculate them knowing that the sum of measurements of angles in every triangle equals to 180^{\circ}.

Example: Are two triangles ABC and PQR congruent if \mid AB \mid = 7, \mid BC \mid = 4, \angle acb = 30^{\circ}

SSA example sketch 2Theorems for defining congruence in right triangles

Right triangles also have special congruent postulates that apply to them.

LL (leg, leg) – Two right triangles are congruent if their matching legs are of equal length.

This theorem is equivalent to SAS, because you know two sides(legs) and angle between them.

When you know both legs, then there is only one line that connects two points at their ends.

LL postulate

HL (hypotenuse-leg) – Two right triangles are congruent if their hypotenuses are of equal length, and matching leg of equal length.

This theorem is equivalent to SSA, because you know one side(leg), larger side(hypotenuse) and angle that is opposite to the larger side (the right angle).

First you would draw leg, and then an angle, and bisect the arm of an angle with a circle that has radius equal to the length of a hypotenuse. There is only one way to do that.

HL postulate

HA (hypotenuse-angle) – Two right triangles are congruent if their hypotenuses are of equal length, and one angle of equal measure.

This theorem is equivalent to AAS, because you know two angles (the right angle and the given angle) and one side-hypotenuse.

To construct the right triangle whose hypotenuse and angle are given, we would firstly draw a hypotenuse, then angle, and construct perpendicular line on drawn leg from the other end of hypotenuse. Of course, because in your right angle you have to have one right angle, and the sum of angles In a triangle equals to 180^{\circ}, the given angle must be less than 90^{\circ}.

We’ll construct one with 60^{\circ}. The steps of constructions are marked with numbers.

HA postulate

 

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