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CA Geometry: Similar triangles 1

Video transcript

actually just right after stopping that video I realized a very simple way of showing you that RP is congruent to TA a little bit more of a rigorous definition if we can show that this triangle if this triangle right there that one I drew in purple and this triangle I drew right here are congruent this triangle right they're congruent I think we can make a fairly reasonable argument that RP is going to be congruent to TA because they're essentially the corresponding sides of the two congruent triangles these congruent triangles would be kind of flipping each other right so how can we make that argument well on the purple triangle this angle is going to be equal to is going to be equal to this angle on the yellow triangle actually we got that from the fact that this is an isosceles trapezoid so the base angles are going to be the same they told us this an isosceles trapezoid so we know that this side right there is going to be congruent to this side right and then finally they both share this side right here they both share this side so we could use the argument once again side-angle-side that the side angle inside are congruent to this side angle inside so you could you know if we were doing it in dripped we could say by SAS triangle TRP triangle T are P is congruent to triangle what T AP t AP and if they're congruent then all of the corresponding sides are equal so then ta is cor is congruent to RP once again you didn't have to do all of that some multiple choice tests but I wanted to give you I felt bad that I wasn't giving you more rigorous definition but anyway more rigorous proof so anyway problem number 11 okay problem number eleven a conditional statement is shown below if a quadrilateral has perpendicular diagonals then it is a rhombus fair enough which of the following is a counter example to the statement above so they're saying if it's if it's perpendicular diagonals and it's a rhombus so if we could find something that has perpendicular diagonals that is not a rhombus then we have a counter example right then this would not be true so let's find with something with perpendicular diagonals that is not a rhombus well this one has perpendicular diagonals right the diagonals are perpendicular to each other all 90-degree angles and this is clearly not a rhombus it's like a kite and I mean you know this is not parallel to this and that is not probably so this is not a rhombus so this is definitely a counter example this one does have perpendicular diagonals but it's also a rhombus so it's not a counter example it's just an example of what they're trying to say this has perpendicular diagonals it's a square but a squares is a subset of rhombuses so this is this is another example and of course this one does not have perpendicular diagonals right this is not a right angle anyway so a is the counter example next question problem 12 problem 12 which triangles must be similar which triangles must be similar two obtuse triangles well that's I mean obtuse just means that they have you know two angles both of them have angles that are you know one obtuse angle might look like that where that is greater than 90 degrees there and then the other obtuse might be super obtuse it might be like that it might be super obtuse like that and clearly these aren't similar well this angle is obviously larger than that one okay so this is not similar similar means all the angles are the same so it's slight congruent but you can scale them in size that's all I think of them right like this triangle let me you know if could be congruent to I'm trying to draw it so it looks exactly the same that triangle well no but you can imagine that no matter if I cut cut and pasted that right it would only be similar to this triangle if I do everything to scale because the sides are different sizes but all the angles are the same that's what similar means so let's see two scalene triangles with congruent basis with congruent bases well no that's not true because I could have that's not similar because let's say this is their base it let's say they share the same base all right one scalene triangle might look like this might come out a little bit and then go down like that right and then the other scalene triangle let's say has the same base right there it might be something a little the sides might be a little bit closer to each other all right so clearly these two things aren't similar this angle is different than that angle all the angles are different so they're not similar triangles so that's not right two right triangles do those have to be similar well no you can have a right triangle that looks like this where maybe the two sides are equal right which would that's a 45-45-90 triangle or you could have something like this where you have a 30-60-90 triangle these clearly are not similar all the angles are not the same they just they both have a 90-degree angle so I'm already guessing the DS are right answer but let's see how it works out two isosceles triangle with congruent vertex angles two isosceles triangles with congruent vertex angles so I'm assuming when they say congruent vertex angles I'm assuming they meaning all of the angles are congruent maybe I'm miss miss so two isosceles triangles let me think about it a little bit too i sighs oh actually I think what they mean they mean the angle in the middle and they take vertex angle so well let me if that's one of my isosceles triangles that's one of mine an isosceles triangle it means it that sides equal to that side and that angle is equal to that angle the vertex angle I'm guessing they mean this angle right there so if I had another isosceles triangle let's say maybe was a little bit smaller see it looks something like that and their vertex angles are the same if that angle is equal to this angle well if those angles if that angle is equal to this angle and we know it's isosceles so if we know size I says that's equal to that then that has to be equal to that we know that all the angles are the same how do we know that this angle is equal to this angle well think about it whatever angle this is let's call this X right let's call this angle Y and this angle Y we know that X plus 2y is equal to 180 right or that 2y is equal to 180 minus X or Y is equal to 90 minus x over 2 right now if this is X and then D let's call these Z and Z so we know that X plus 2z is equal to 180 all the angles in a triangle have to add up to 180 subtract X from both sides you get 2 Z is equal to 180 minus X divide by 2 you get Z is equal to 90 minus x over 2 so Z and Y are going to be the same angles so all the angles are the same so we're dealing with similar triangles so choice choice D was definitely correct 1313 13 okay which of the following facts would be sufficient to prove the triangles ABC so it's ABC that's the big triangle and triangle DBE so that's the small one are similar so we have to prove that all of their angles are similar and I can not even look at the choices and I can guess where this is we want to prove that those are similar so first of all they share the same angle right angle ABC this angle is the same as angle DBE so they share that same angle so we got one angle down right and let's think about it if we knew that this angle was equal to that angle and that angle is equal to that angle would be done and the best way we could come to that conclusion is if somehow if they told us that this and this are parallel I'm guessing that's where they're going now I might have gone on a completely wrong candidate because if those are two parallel then these two lines are transversals of the parallel so that and that would be corresponding that line that angle and that angle be corresponding angle so they'd be congruent and then that angle and that angle would be corresponding congruent corresponding angles so they'd also be congruent so if they told us it either prowler we're done these are definitely similar triangles and sure enough choice C they tell us that AC and de are parallel these are parallel that's a transversal this is a corresponding ail that so they're congruent this is a corresponding angle to this congruent so all of the angles are congruent so we have a similar triangle problem problem 1414 okay parallelogram ABCD is shown below fair enough parallelogram that tells us that the opposite sides are parallel that's parallel to that and then this is parallel to that okay and all the choices got clipped at the bottom but I'll copy them over maybe I'll copy them above the question well let me see what I can do if I wrote it if I paste it there I think that's good enough a little unconventional okay parallel is shown below and they say which pair of triangles can be established to be congruent to prove which pair of triangles can be established to be congruent to prove that angle D a/b is congruent to angle D C D so they want us to show that D a B D a B is so da B is this let me do another color D a B is that angle is congruent to b c d to be C D so they want us to show that those have the same angle measure okay and what do we have to show and they say what pair of triangles can establish to be congruent to prove that okay so we have to show two if these are both part of two different congruent triangles and they're the corresponding angles then we know that they're congruent and we'd be done so let's see what they say triangle ABC and BCD ad C ad C and B C D BCD has this angle in it right BCD it does help us because it has this angle but triangle a ADC does not have this angle in it right triangle ADC has this the smaller angle and it has this ADC doesn't involve this whole thing so that's not going to help us triangle AED AED once again does not involve this larger angle does not involve the angle D a B right it only involves a little smaller angle so that's not going to help us triangle da B that looks good that has this whole angle in it D a B and then BCD right if we show that this that that that triangle is congruent to this triangle right here I think we're done that would be enough to show that this is congruent this angle is congruent to that angle because it would be the corresponding angles of congruent of a congruent triangle so I think C is where we're going to go let's just look at choice D Dec Dec once again Dec triangle Dec let me make this point clear triangle Dec does not involve either of the angles we care about it clearly does not involve this angle and it only involves part of this angle only this part it doesn't involve this whole angle so that's not going to help us either so the answer is C anyway see in the next video