What do congruent

And to figure that out, I'm just over here going to write our triangle congruency postulate. So we know that two triangles are congruent if all of their sides are the same-- so side, side, side. We also know they are congruent if we have a side and then an angle between the sides and then another side that is congruent-- so side, angle, side. If we reverse the angles and the sides, we know that's also a congruence postulate.

So if we have an angle and then another angle and then the side in between them is congruent, then we also have two congruent triangles. And then finally, if we have an angle and then another angle and then a side, then that is also-- any of these imply congruency.

So let's see our congruent triangles. So let's see what we can figure out right over here for these triangles. So right in this triangle ABC over here, we're given this length 7, then 60 degrees, and then 40 degrees.

Or another way to think about it, we're given an angle, an angle and a side-- 40 degrees, then 60 degrees, then 7. And in order for something to be congruent here, they would have to have an angle, angle, side given-- at least, unless maybe we have to figure it out some other way.

But I'm guessing for this problem, they'll just already give us the angle. So they'll have to have an angle, an angle, and side. And it can't just be any angle, angle, and side. It has to be 40, 60, and 7, and it has to be in the same order. It can't be 60 and then 40 and then 7. If the degree side has-- if one of its sides has the length 7, then that is not the same thing here.

Here, the degree side has length 7. So let's see if any of these other triangles have this kind of 40, 60 degrees, and then the 7 right over here. So this has the 40 degrees and the 60 degrees, but the 7 is in between them. So this looks like it might be congruent to some other triangle, maybe closer to something like angle, side, angle because they have an angle, side, angle. So it wouldn't be that one.

This one looks interesting. This is also angle, side, angle. So maybe these are congruent, but we'll check back on that. We're still focused on this one right over here.

And this one, we have a 60 degrees, then a 40 degrees, and a 7. This is tempting. We have an angle, an angle, and a side, but the angles are in a different order. Here it's 40, 60, 7. Here it's 60, 40, 7. So it's an angle, an angle, and side, but the side is not on the degree angle.

It's on the degree angle over here. So this doesn't look right either. Here we have 40 degrees, 60 degrees, and then 7. So this is looking pretty good. We have this side right over here is congruent to this side right over here. Then you have your degree angle right over here. Save Word. Essential Meaning of congruent. Full Definition of congruent. Other Words from congruent congruently adverb. Examples of congruent in a Sentence Their goals are not congruent with the goals of the team.

Seek ongoing feedback. Dettmar, The Atlantic , 2 Sep. First Known Use of congruent 15th century, in the meaning defined at sense 1. History and Etymology for congruent Middle English, from Latin congruent-, congruens , present participle of congruere — see congruous. Learn More About congruent. Time Traveler for congruent The first known use of congruent was in the 15th century See more words from the same century. Style: MLA.

Kids Definition of congruent. Three such triangles are shown below, and they are clearly not congruent. This shows that just knowing that two pairs of sides are equal is not enough information to establish congruence.

Constructing a triangle with three given sides. When all three sides of a triangle are given, however, there is no longer any freedom of movement, and only one such triangle can be constructed up to congruence. To demonstrate this, suppose that we are asked to construct a triangle ABC with three given sides lengths:.

In the module, Constructions we have seen that we can carry out this construction using ruler and compasses, as in the diagram below. This construction has yielded two triangles with the given measurements. However, these two triangles, are congruent. This establishes that it is reasonable to take the SSS congruence test as an axiom of geometry. Notice that this congruence test tells us that the three angles of a triangle are completely determined by its three sides.

After trigonometry has been introduced, the cosine rule can be used to find the sizes of these angles. The second dotpoint in the box above does not imply that given any three lengths, a triangle can be constructed with those lengths as side lengths.

What restriction must be placed on the three side lengths in order for a triangle with those side lengths to exist? The example below shows that a quadrilateral with opposite sides equal is a parallelogram. Many results of this type will be discussed in the module, Parallelograms and Rectangles. In the diagram to the right, the opposite sides of the quadrilateral ABCD are equal, and the diagonal BD has been drawn.

Using congruence to prove the validity of constructions. We claimed in the module, Constructions that the validity of the ruler-and-compasses constructions described there could be established once congruence had been introduced. The following exercises proves that two of the constructions work. This exercise proves the validity of the standard construction to bisect a given angle.

The diagram to the right shows an angle AOB. This exercise proves the validity of the standard construction to copy a given angle.

A circle with centre O is drawn cutting the arms at A and B , and a second circle with the same radius is drawn with centre P cutting PZ at F. A circle with centre F and radius AB is drawn, cutting the second circle at G. The other way to stop the two sides flapping is to specify the angle between them. This angle between the sides is called the included angle. Constructing a triangle given two sides and the included angle. To illustrate this, let us construct a triangle ABC in which.

This establishes that it is reasonable to take the SAS congruence test as an axiom of geometry. The SAS congruence test. This congruence test tells us that the sides and angles of a triangle are completely determined by any two of its sides and the angle included between them. The cosine rule can be used to find the length of the third side and the sizes of the other two angles.

This example demonstrates a method of constructing a parallelogram from the diameters of two concentric circles. Conclude that APBQ is a parallelogram. Proving the validity of the construction of the perpendicular bisector of an interval. The following exercise uses the SSS and SAS congruence tests to prove the validity of the standard ruler-and-compasses construction of the perpendicular bisector of a given interval.

The circles in the diagram below have centres A and B and the same radius. The circumcentre of a triangle. The following exercise proves that the three perpendicular bisectors of the sides of a triangle are concurrent.

It also shows that this point is equidistant from all three vertices, so it is the centre of the circle passing through all three vertices of the triangle.

The circle is called the cirumcircle and its centre is called its circumcentre. Part b proves that O is the circumcentre of the triangle. Part c proves that the perpendicular bisectors are concurrent. Demonstrating that the angle in the SAS test must be the included angle. The SAS congruence test requires that the angle be included. The following exercises demonstrate that the test would fail if we allowed non-included angles.

Use ruler and compasses to construct two non-congruent triangles ABC with. The triangle ABC to the right is isosceles, with. The point X is any point on the side BC. Assuming wrongly that the SAS test can be applied when the angles are non-included, prove that. Now let us turn attention to the angles of a triangle. But knowing all three angles of a triangle does not determine the triangle up to congruence. To demonstrate this, suppose that we were asked to construct a triangle ABC in which.

Clearly nothing controls the size of the resulting triangle ABC. Thus knowing that two triangles have the same angle sizes is not enough information to establish congruence. In the module, Scales Drawings and Similarity we will see that the two triangles are similar. Constructing a triangle with two angles and a given side. When the angles of a triangle and one side are known, however, there is no longer any freedom for the size to change, so that only one such triangle can be constructed up to congruence.

To demonstrate this, suppose that we are asked to construct a triangle ABC with these angles and sides length:. The most straightforward way is to draw the interval BC and then construct the angles at the endpoints B and C.

A further two congruent triangles can be formed by reflecting in a line through C perpendicular to BC. This establishes that it is reasonable to take the AAS congruence test as an axiom of geometry. Notice that this congruence test tells us that the other two sides of a triangle are completely determined by one side and two angles.

The sine rule can be used to find the other two side lengths. This exercise proves that if one diagonal of a quadrilateral bisects both vertex angles, then the quadrilateral is a kite.

We have seen that two sides and a non-included angle are, in general, not enough to determine a triangle up to congruence. When the non-included angle is a right angle, however, we do obtain a valid test.

In this situation, one of the two specified sides lies opposite the right angle, and so is the hypotenuse.