CA Geometry: More proofs

all right we're on problem number seven and when I copied and pasted it I made a little bit smaller so I'm going to read it for you just in case this is too small for you to read but says use the proof to answer the question below so they gave us that angle to is congruent to angle three so angle two or the measure of angle two is equal to the measure of angle three I'm starting to get the knack of the language that they use in geometry class which I will admit that language kind of sends to disappear as you leave your geometry class but since we're in geometry class we'll use that language so the angle angle 2 is congruent to angle 3 which means that they're measure is the same or that they kind of did the same to angle essentially fair enough so let's see statement 1 angle 2 is congruent to angle 3 that's given I drew that already up here statement 2 angle 1 is congruent to angle 2 angle 3 is congruent to angle 4 so they're saying that angle 2 is congruent to angle 1 or angle 1 is congruent angle 2 that's there right there and that angle 4 is congruent to angle 3 fair enough and they say what's the reason that you could give and I don't know what what I forgot the actual terminology but I in my head I was thinking oh opposite angles are equal or the measures are equal or they're congruent right and you could just imagine two sticks and change the angles of the intersection you'll see that opposite angles are always going to be congruent but let's see that is the reason I would give opposite angles are congruent let's see which which statement of the choices is most is most like what I just said complements of congruent angles are congruent complements supplement vertical angles are congruent I think that's what they call opposite angles vertical angles I think that's what they mean by opposite angles let me supplements of congruent angles are congruent that's not true corresponding angles are congruent these are corresponding I think this is what they mean by vertical angles complements of congruent angles are congruent you know what I'm going to look this up with you on Wikipedia let me see vertical angles as you can see at the age of 32 some of the terminology starts to escape what matters is that you understand the intuition and then you can do these Wikipedia search to just make sure that you remember the right terminology let's see what Wikipedia has to say about it vertical angles a pair of angles is said to be vertical or opposite oh I guess I use the British English opposite angles if the angle share the same vertex and are bounded by the same pair of lines but are opposite to each other right so somehow growing up in Louisiana I somehow picked up the British English version of it maybe because the word opposite made a lot more sense to me than the word vertical with that said they're the same thing Wikipedia has shown us the light and so my logic of opposite opposite angles is the same as their logic of vertical angles are congruent next problem next problem I'll start using the US terminology although I think there are a good number of people outside of the US who watt to these so maybe it's good that I somehow picked up the British English version of it okay once again might be hard for you to read I'll read it out for you two lines in a plane always intersect in exactly one point fair enough which of the following best describes a counter example to the assertion above so I like to think of the answer even before seeing the choices so can I think of two lines in a plane that always interact in exactly one point well what if they're parallel right what if you know I have let me see that line and that line they're never going to intersect with each other they're parallel that's the definition of parallel lines the other the other example I can think of is if they're the same point if they're the same line I mean I guess you might not want to call them two lines then but you know I would buy this line and then I had you know another well that's parallel but matter if they're right on top of each other they would intersect everywhere so either of those would be counter examples to the idea that two lines in a plane always exactly always intersect in exactly one point and if we look at their choices well okay they have the first thing I just write they're parallel lines see there are two lines in a plane but they don't intersect in one point problem eight I'm going to make it a little bigger from now on just you can read it okay this is problem nine problem nine which figure can serve as a counter example to the conjecture below if one pair of opposite sides of a quadrilateral is parallel then the quadrilateral is a parallelogram so once again a lot of terminology and I do remember these for my geometry days quadrilateral means four sides right a four-sided figure and a parallelogram means that all the opposite sides are parallel so for example this is a parallelogram and if you remember or well you know this let me see if I can how well I can do this then like well that's if you ignore this little part that's hanging on there that's a parallelogram and if all the sides were the same it's a rhombus and all of that but that's a parallelogram and that's parallel because that's a parallelogram because this side is parallel to that side and this side is parallel to that side all the sides are parallel now they say if one pair of opposite sides of a quadrilateral is parallel then the quadrille is a parallelogram so I want to give a counter example so let me make one draw a figure that has two sides that are parallel so let's say that side and that side are parallel and then I don't want the other two to be parallel then it wouldn't be a parallelogram now let's say the other sides are not parallel so they look like let's say they look like that and like that and once again that you know just digging in my head of definitions of shapes that looks like a trapezoid to me so let's see yeah good you have a trapezoid as a choice trapezoid all the rest are parallelograms right a rectangle is all the sides are parallel and we have all 90-degree angles right rectangles are actually a subset of parallelograms rhombus are we have a parallelogram where all of the sides are the same length all the angles aren't necessarily equal Square is all the sides are parallel equal and all the angles are 90 degrees so all of these are subsets of Prada gram this is not a parallelogram although it does have two sides that are parallel so this is the counter example to the conjecture I think you're already seeing the pattern a lot of geometry the the the terminology is often the hard part the ideas aren't as deep as a terminology might suggest okay given trap that all right that already makes me worried okay given trap is an isosceles trapezoid with diagonals RP and ta which of the following must be true and isosceles tribe ok let's see what we can do here so an isosceles trapezoid means that the two sides on the on the the two kind of sides that lead up from the base to the top side are equal kind of like an isosceles triangle so let me draw that actually I'm kind of guessing that I haven't seen the definition of an isosceles triangle anytime in the recent past but an isosceles trapezoid but it sounds right so I'll go with it and that's a good skill in life whoops to make your best see so I think what they say when they say Solly's trapezoid they're essentially saying that this side you know it's a trapezoid so that's going to be equal to that and they're saying that this side is equal to that side so isosceles trapezoid okay which of the following and they say RP and ta are diagonals of it so let me draw that so let me actually write the whole trap so this is T are a P is a trapezoid and let me draw the diagonals RP is that diagonal and ta is this diagonal right here okay all right let's see what we can do which of the following must be true RP is perpendicular to TA well I can already tell you that that's not going to be true and you don't even have to prove it because you can even visualize is R if you squeeze the top part down right imagine some device where this is kind of the cross-section if you were to squeeze the top down right they didn't tell us how high then these angles you can imagine that these at both of these angles let me see if I could draw it that angle and that angle which are opposite or vertical angles which we know you know that's the u.s. word for it those are going to get smaller and smaller if we squeeze it down right in order for both of these to be perpendicular those would have to be 90-degree angles and we already can see that that's definitely not the case all right our P is parallel to TA well that's clearly not the case they intersect all right they're the diagonals our P is congruent to TA well now that looks pretty good to me right because it's an isosceles trapezoid you get this hole if we do a line of symmetry here if we draw a line of symmetry this everything you see on this side is going to be kind of congruent to its mirror image on that side right so both of these lines you can this is going to be equal to this and I could make the argument but basically we know that our P since this is an isosceles trapezoid you can imagine kind of continuing a triangle and making an isosceles triangle here then we would know that that angle is equal to that angle let's see we could make a see if that angle is equal to that angle then well actually I'm not going to go down that path because I think that'll be but you can I mean you can almost look at it from inspection although maybe I should be it do a little more rigorous definition of it but RP is definitely going to be parallel that's definitely going to be congruent to TA right because both sides of these trapezoids are going to be symmetric and so there's no way you could have RP being a different length than TA right since this trapezoid is perfectly symmetric since it's isosceles and then DRP bisects TA RP bisects TA and I think there again you can you can show that if you were to let say if I were to draw this trapezoid slightly differently if it looks something like this let me draw it like this if this was the trapezoid and then this is also an isosceles trapezoid and then the diagonals would look like this whoops the diagonals would look like this so here it's pretty clear that they're not bisecting each other right it's pretty in order from the bisect each other this length would have to be equal to that length and that's clear just by looking at it that that's not the case right that is not equal to that so they're definitely not bisecting each other so you can really in this problem knock out choices a B and D and say oh well choice C looks pretty good but you can actually deduce that by just saying by using an argument of all of the angles that this angle is going to be equal to let me see if I can make the arc thats going to waste your time but that's a good exercise for you is to make the formal proof argument of why this is true although you can make a pretty good intuitive argument just based on the symmetry of the triangle itself anyway see in the next video