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## Integrated math 1

### Course: Integrated math 1>Unit 17

Lesson 2: Triangle congruence from transformations

# Proving the SAS triangle congruence criterion using transformations

We can prove the side-angle-side (SAS) triangle congruence criterion using the rigid transformation definition of congruence. Created by Sal Khan.

## Want to join the conversation?

• So high school geometry is grade 10 math right?
• It would be 10th grade if you don't have pre-ap or ap classes. So if you do have gate geometry you will do it in 9th grade and do Algebra 2 next year. If you don't have a gate then you do it in 10th grade and have Algebra 2 next year.
• I wish I had Sal as a teacher in 8th grade for algebra I failed algebra that year and I thought I wasn't doing good enough but when I retook algebra in 9th I got all A's on the test. Maybe I was just my 8th-grade algebra teacher I don't know
• So there's this little apostrophe-looking thing that keeps appearing on certain letters (around the triangles) and I'm a little confused as to what that is. If anyone can tell me what that stands for, I would really appreciate it! :)

(It's throughout the video when those symbols show up.)
• that is the prime mark (think of transformers and optimus prime) which is an indication that a transformation has been performed on a point or shape. So the pre-image (starting point) would have points such as ABC, but after it is transformed (through translation, rotation, or reflection), the new figure would be A'B'C' which would be congruent, but not necessarily the same orientation. The video talks about A going to A', B going to B', and C going to C'.
• How do you prove triangles congruent with attitude?
Do it with SAS.
• @naowomi1497

Great Question! If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.

If you didn't know, there are five ways to prove triangles are congruent.

1.Side-Side-Side (SSS) theorem.
2.Side-Angle-Side (SAS) theorem.
3.Angle-Angle-Side (AAS) theorem.
4.Angle-Side-Angle (ASA) theorem.
5.Hypotenuse-Leg (HL) theorem.

If two pairs of corresponding angles and the pair of included sides are congruent, then the triangles are congruent. If two pairs of corresponding angles and a pair of non-included sides are congruent, then the triangles are congruent.

To prove that altitudes of a triangle are concurrent, we have to prove that the line segment joining the orthocentre and a vertex considering the altitudes drawn from the other two vertices of triangle meet at the orthocentre.

Hope this helps.
(1 vote)
• Wouldn't c' be mapped to F anyway without this extra stuff?
• The way that Sal showed it, yes it would be. However it would not have been if a reflection was already used. Say a was translated to d then the triangle was rotated about point a however many degrees it was needed to make b on the opposite side of a' from e, then reflected on a vertical line through point a', c' would be on the opposite side of f on line a', b'. If this was the case then it would need to be reflected.
• Video won't play. Help?
• Try going on youtube.com. All of the videos are found there. Check your internet connection also.
• Hi, Sal
Could you tell me which compass you think is the best (most durable) as mine keeps moving (out of the circle) and breaking?
• a compass with stiff arms and a adjustable nut
(1 vote)
• Around Sal says that the angle "looks something like this" and then he draws it using a straightedge. Is there a more precise way to construct the angle for the proof using a compass and then the straightedge?

For example, can something like this be done? We know F=C' and E=B'. We can draw a line between them. So, line FE = line C'B'. Can we use line FE as a radius around point E to create a circle that will intersect with a circle around point D with line DF as a radius and then connect the points and reflect over line CE?