![]() ![]() The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (base and perpendicular).We also know that the angles BAD and CAD are equal.(AD bisects BC, which makes BD equal to CD). We know that angles B and C are equal (Isosceles Triangle Property). Therefore, a hypotenuse and a leg pair in two right triangles, are satisfying the definition of the HL theorem. AD = AD because they are common in both the triangles. AB and AC are the respective hypotenuses of these triangles, and we know they are equal to each other. Proof:ĪD, being an altitude is perpendicular to BC and forms ADB and ADC as right-angled triangles. Given: Here, ABC is an isosceles triangle, AB = AC, and AD is perpendicular to BC. Observe the following isosceles triangle ABC in which side AB = AC and AD is perpendicular to BC. And we're done.The proof of the hypotenuse leg theorem shows how a given set of right triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal. Saying these are my statements, statement, and this is my The two-column proofs, I can make this look a little bit more like a two column-proof by In previous videos, and just to be clear, sometimes people like So we now know that triangleĭCA is indeed congruent to triangle BAC because of angle-angle-side congruency, which we've talked about And so now, we have two angles and a side, two angles and a side, that are congruent, so we can now deduce byĪngle-angle-side postulate that the triangles are indeed congruent. So, just to be clear, this angle, which is CAB, is congruent to this angle, which is ACD. Part of a transversal, so we can deduce that angle CAB, lemme write this down, I shouldīe doing different color, we can deduce that angle CAB, CAB, is congruent to angle ACD, angle ACD, because they are alternate,Īlternate interior, interior, angles, where a transversal Parallel to DC just like before, and AC can be viewed as ![]() Saying that something is going to be congruent to itself. ![]() We know that segment AC is congruent to segment AC, it sits in both triangles,Īnd this is by reflexivity, which is a fancy way of Well we know that AC is in both triangles, so it's going to be congruent to itself, and let me write that down. Triangle DCA is congruent to triangle BAC? So let's see what we can deduce now. Over here is 31 degrees, and the measure of this angle Let's say we told you that the measure of this angle right The information given, we actually can't prove congruency. Looks congruent that they are, and so based on just Information that we have, we can't just assume thatīecause something looks parallel, that, or because something Make some other assumptions about some other angles hereĪnd maybe prove congruency. If you did know that, then you would be able to 'cause it looks parallel, but you can't make thatĪssumption just based on how it looks. Side that are congruent, but can we figure out anything else? Well you might be tempted to make a similar argument thinking that this is parallel to that To be congruent to itself, so in both triangles, we have an angle and a ![]() We also know that both of these triangles, both triangle DCA and triangleīAC, they share this side, which by reflexivity is going Parallel to segment AB, that's what these little arrows tell us, and so you can view this segment AC as something of a transversalĪcross those parallel lines, and we know that alternate interior angles would be congruent, so we know for example that the measure of this angle is the same as the measure of this angle, or that those angles are congruent. Pause this video and see if you can figure Like to do in this video is to see if we can prove that triangle DCA is congruent to triangle BAC. ![]()
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