definition of congruent segment
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Answers:base angles of an isosceles triangle are congruent is the reason. D
Answers:Instead of numbering the angles, please refer to them using the letters (e.g. Angle ILJ). Otherwise, we don't know which angles are referred to by those numbers.
Answers:It has to be isosceles. Which means it can also be equilateral, though it does not have to be. Approach it this way: One can find an infinite number of lines (line segments) that intersect AC and make right angles. One of these line segments will also intersect point B. Now, just knowing this alone gets one no further. But there is further information. Assume one has found the line segment mentioned above. Then assume it also divides AC so that AD and CD are congruent. If we later show this matches the last condition given in the problem, then this single possibility is actually the case. If we go on to show this leads to something not matching the last condition given, well, then it's a bad approach... So, we have a line segment BD that we constructed to intersect AC where it makes right angles, and we are assuming it also divides AC into congruent portions. If so, what else do we know about the triangles ABD and CBD? We know BD is self-congruent (it is the same segment and so is congruent in each triangle). So using the Pythagorean Theorum, we find that AB and CB are congruent (the "a" and "b" for each triangle are congruent segments so a^2 + b^2 is the same for each triangle, which means c^2 is also the same and therefore "c" is the same: so the segments represented by "c" are congruent as well. If all three sides are congruent, then the triangles are congruent. If the triangles are congruent, then the corresponding angles are also congruent. Hence, angles ABD and CBD are congruent. How does that compare to the "last condition" the problem gives? That last condition was that segment BD bisects angle ABC. Bisecting angle ABC means producing two component angles, ABD and CBD, that are congruent. Which is just what we have. So we have successfully used assumptions to give us our answer. At this point, we know the triangle is exactly as the assumptions described: AB and CB are congruent so the triangle is definitively isosceles. Of course, it could still be equilateral since an equilateral triangle is also isosceles. One could argue "who cares?" but, of course, the teacher does, and it also often useful to fully characterize something if one needs to characterize it at all. So, we know it can be equilateral, but does it have to be? Is there any information, or an approach like above, that can show it is? No. AC is not defined any more deeply than as the third side of the original triangle. The second condition, that BD bisects AC does nothing to limit AC's length, except in the extreme (ABC is still a triangle, so AC's length must be lass than the sum of AB and BC). The last condition, that BD bisect angle ABC does not require AC to be any particular length. It may be congruent to AB or BC, but does not have to be. So we cannot assert it is always an equilateral triangle from the given information. There is an assumption we could make, that AC is congruent to AB and BC. But this does not lead to any condition unique to an equilateral triangle that we can test against the given information, so it is not a useful approach. Hence, the most one can say for certain is: a) it is in all circumstances an isosceles triangle and b) it may also be equilateral.
Answers:You never asked the question but 3 of the 4 statements do not make for congruent segments. congruent angles form congruent segments - think of a triangle and put the picture in a copy machine and enlarge it 200%. both triangles have the same exact angles but the sides are in the ratio of 1 :2. Obviously not congruent definition of perpendicular segments are segment that meet at right angles. But the length of the segments could be anything. No good here - reflexive property says something is the same as itself. So AD is congruent to AD but AD is not the same line as DB The answer must be "Definition of a midpoint" A MIDPOINT divides a line into 2 congruent parts. Draw a line AB and in the middle is D. If D is the midpoint of AB then the 2 segments AD and DB are congruent