High school geometry
- Geometric constructions: congruent angles
- Geometric constructions: parallel line
- Geometric constructions: perpendicular bisector
- Geometric constructions: perpendicular line through a point on the line
- Geometric constructions: perpendicular line through a point not on the line
- Geometric constructions: angle bisector
- Justify constructions
- Congruence FAQ
Frequently asked questions about congruence
What is the relationship between transformations and congruence?
Two shapes are congruent if they have the same size and shape. When we use rigid transformations (translations, rotations, and reflections) to move a shape, it keeps the same size and shape.
So if there is a series of rigid transformations that show that one shape matches up with another shape, the figures are congruent. If they cannot match up, the figures are not congruent.
Try it yourself with our Congruence & transformations exercise.
How do transformations help us develop triangle congruence criteria?
Transformations can help us develop triangle congruence criteria that only need three congruent corresponding measures instead of needing to know the measures of all three sides and all three angles. By using transformations, we can essentially "move" one triangle on top of the other to see if they are congruent.
For example, if we know that two triangles have two congruent sides and a congruent angle between them, we can use a transformation to rotate and/or reflect one triangle so that it lines up with the other triangle. Since the two triangles match up, we can conclude that they are congruent, even if we don't know the measures of the other sides or angles.
We can apply this same logic to other triangle congruence criteria as well. For example, if we know the measures of two angles and the side between them for two triangles, we can use transformations to see that the triangles are congruent. So, by using transformations, we are able to develop triangle congruence criteria that rely on fewer measurements.
Try it yourself with our Justify triangle congruence exercise.
What are the triangle congruence criteria?
There are three main criteria for triangle congruence. If two triangles meet any one of these three criteria, they are considered congruent.
- Side-Side-Side (SSS) criterion: Two triangles are congruent if all three of their corresponding side lengths are equal.
- Side-Angle-Side (SAS) criterion: Two triangles are congruent if two of their corresponding side lengths and the angle between those sides are equal.
- Angle-Angle-Side (AAS) criterion: Two triangles are congruent if two of their corresponding angles and the side between those angles are equal.
In the case of right triangles, the Pythagorean theorem means that we can calculate the length of the third leg given the lengths of any other two. So right triangles are a special case of the Side-Side-Side criterion where we only need two pairs of congruent side lengths.
- Hypotenuse-Leg (HL) criterion: Two right triangles are congruent if their hypotenuse lengths are equal and the lengths of one of their legs are equal.
We don't need a Leg-Leg criterion for right triangles, because that would just be another case of Side-Angle-Side congruence.
Try it yourself with our Prove triangle congruence exercise.
Why can't I use Angle-Side-Side or Angle-Angle-Angle to show triangle congruence?
The simple answer is that these two methods don't always work. Angle-Side-Side seems like it could work, but it's possible to have two triangles with the same angle and two side measurements that aren't congruent. The same goes for Angle-Angle-Angle. Two triangles can have the same three angles but not be congruent.
This is why we use the four methods that do work: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Hypotenuse-Leg (HL) for right triangles. These methods have been proven to work every time, so we can rely on them to show that two triangles are congruent.
How do the triangle congruence criteria help us work with other figures?
The triangle congruence criteria help us compare and work with other figures in several ways.
First, they allow us to determine whether two triangles are congruent, which can be helpful when we want to determine whether two other figures are also congruent.
Second, the triangle congruence criteria can help us break down other figures into triangles, which can be easier to work with. For example, we might divide a quadrilateral into two triangles, determine whether the two triangles are congruent, and then use that information to infer something about the quadrilateral.
Lastly, the triangle congruence criteria can also help us solve problems involving other figures by allowing us to use properties of congruent triangles. For example, if we know that two triangles are congruent, we can use the corresponding parts of the congruent triangles to find missing measurements in one of the figures.
Try it yourself with our Prove parallelogram properties exercise.
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