hey there it's Kalei in this lesson we will be talking about congruent and
similar figures we've been using the word congruent a lot
remember congruent means exactly the same and we've been using words like
equilateral to talk about line segments that are the same length an
equiangular to talk about angles that are the same measure of degree but we can
apply the word congruent to entire shapes when two figures are exactly the
same size and shape they are congruent looking at these two triangles they're
the same in every way these two triangles are congruent what about these
two triangles are they congruent even though the triangles are oriented
differently we can still find that each line segment on the green triangle
has a matching line segment on the blue triangle so yes these are congruent and
these two Pentagon's are congruent we have just taken the orange Pentagon and
rotated it a little to the right shapes are congruent if they have the same
shape in size even if they've been rotated flipped or turned upside down
now what about these two triangles they are the same shape right yes but the
pink triangle is obviously smaller than the green triangle so these triangles
are not congruent when two figures are the same shape but different sizes we
say the shapes are similar so these two triangles are similar how would you
identify these blue parallelograms are they the same shape and size yes so we
can say that these two shapes are congruent
now what about the orange trapezoids are they of the same shape and size well
they aren't the same size but they are the same shape so these aren't congruent
they are similar but what about these two triangles are they the same size no
are they the same shape well they're both triangles but the big one is
equilateral triangle and the small one is a right triangle so these aren't even
the same shape these two figures are not congruent and they are not similar now
that we know the difference between congruent and similar figures we can
talk about the corresponding parts corresponding parts refer to the sides
and angles that match on two congruent figures we use this symbol an equal sign
with the swigille top to represent corresponding and congruent sides and
angles now let's name the corresponding parts on these two congruent triangles
well I see that line segment a B is the same as line segment d so I can write
that as line segment a B remember I put my line second on top is corresponding
and congruent to line segment D E now what other sides of these two triangles
do you see are the same they have the same hypotenuse here this long side so I
can say a line segment AC is congruent and corresponding to line segment D and
lastly I see that I have the same base here so I can say line segment BC is
congruent and corresponding to line segment E so now I've got all my line
segments what do you think about these angles yeah so on congruent shapes the
angles are corresponding and also congruent
so we would have this small angle here on both will be the same so we can say
angle a c b is corresponding and congruent to angle DFE remember I
put my vertex in the center there and I could say the same for all the rest of
these angles these angles will be corresponding and these angles will be
corresponding nice job now if we go back to similar figures we know that they are
not the same size but they are the same shape so there has to be something
corresponding about them even though the side lengths aren't congruent they are
different lengths we can see which sides on one triangle correlate to the other
and it might be hard to see it first but similar figures have matching angles as
well here I have a red triangle inside of a blue triangle and this one to a red
triangle inside of a blue triangle now what do you notice about the angles of
the red triangle compared to the angles of the blue triangle if I change the
size do you notice that the angles of the red triangle and the blue triangle
are always matching and notice that the side lengths of my blue triangle and my
red triangle are always corresponding even as the size of the red triangle
changes this means that they are similar figures we use this symbol to represent
corresponding sides on similar figures so here are my two similar triangles
let's write the corresponding sides so I see that a be the base of my triangle
here will be corresponding to G H so I can write line segment a B is
corresponding but not congruent to line segment G H what else
see the basis those are gonna be corresponding so line segment BC is
corresponding to line segment H I see I'm using my similar symbol here not my
congruent because they're not the same length and then I also see the line
segment AC is corresponding to G I so I can say AC corresponding and similar to
G I now what about my angles here so let's start with this small angle here
so I could say angle a/c B is corresponding to this angle over here
and we know on similar figures they are congruent so corresponding and congruent
I can use my other symbol to say that they're congruent to angle G I H and I
can say this for all angles so this one here I'm gonna say angle a B C is
congruent and corresponding to angle G H I and lastly my angles up here are
corresponding and congruent so I'm going to say angle BAC is corresponding and
congruent to angle H G I so once I can identify which sides of my similar
triangles are corresponding I can write these corresponding sides as ratios
remember a ratio is a fraction and this fraction will help me compare the side
lengths so I could write a B is similar to G H as a B over line segment gh and I
can write this one here as BC over H I and I'll write my other one over here
line segment AC over line segment GI so now that I have my ratios I can go
ahead and plug in my side lengths here that I have so first line segment a-b
which is this one here tells me that it's a length of four and that
corresponding side gh here has a length of two and here I have BC my base is a
length of six and corresponding base is three and over here my hypotenuse AC has
a length of eight and my corresponding has a length of four now what do you
notice about all of these ratios here I have 4 over 2 6 over 3 and 8 over 4 all
of these can be simplified right so 2 goes into 4 how many times 2 times 3
goes into 6 how many times 2 times and 8 goes into 4 how many times 2 times
so notice that all of these ratios from our similar line segments are equal to 2
and that tells us that every side length on my ABC triangle is 2 times the side
length of my GHI triangle nice work we found that each of these ratios were
equal in fact that is always true for similar figures these ratios or the
ratio of the side lengths are proportional proportional means
corresponding in size or measure and in math proportional means a pair of equal
ratios so we can say the ratio of a B over gh is equal to line segment BC over
H I is equal to line segment AC over G I and we know that this is true because
all of these were equal - - now I can use this knowledge about
proportional sides being equal for similar figures to find missing side
lengths like this X here so if I know that XY is corresponding to JK I can say
10 over 5 is going to be equal to Y Z which corresponds to KL will be 20 over
10 which is then equal to my hypotenuse which are corresponding x over 15 now
I'm solving for my unknown value X so if I simplify both of these proportions
here I'm going to get 10 over 5 and 20 over 10 are both equal to 2 and since I
know that all these proportions are equal I can say 2 is equal to x over 15
and now I'm just solving for my unknown value X so I'm going to multiply both
sides by 15 and that will give me 30 is equal to I can cross out these 15s X so
now I know that my missing value here is equal to 30 good job great job learning
the difference between congruent and similar figures practice what you've
learned by doing the online games and quizzes have fun and remember to always
be clever