Question 480768: In each of the following answer true or false. Justify your reasons.
(a) Congruent triangles are also similar.
(b) Two similar triangles are also congruent triangles.
(c) Any two equilateral triangles are similar.
(d) Two isosceles triangles are similar.
(e) A parallelogram has four acute angles.
(f) No square is a rectangle.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! (a) Congruent triangles are also similar.
yes - similar triangles have all corresponding angles equal which is a requirement for congruent triangles as well. similar triangle to not have to have corresponding sides equal. that's the difference.
(b) Two similar triangles are also congruent triangles.
no. congruent triangles are special cases of similar triangles where the proportion between the sides of the triangle is 1. in that case, the similar triangle are also congruent, but only in that case. any constant of proportionality other than 1 results in similar triangles that are not congruent.
(c) Any two equilateral triangles are similar.
yes. they have all corresponding angles that are equal which is a requirement for similar triangle.
(d) Two isosceles triangles are similar.
not necessarily. if the base angles of the isosceles triangles are different then the triangles are not similar.
(e) A parallelogram has four acute angles.
not necessarily. in fact never. the opposite angles of a parallelogram are supplementary. this means they have to add up to 180 degrees. if one of the angles is acute, the other will be obtuse, and vice versa.
if one of the angles is 90 degrees, then the other one is 90 degrees and you have a rectangle.
(f) No square is a rectangle.
not true. all squares are special forms of rectangles where the length and the width are the same.
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the requirements for triangles to be similar are that all corresponding angles are equal and all corresponding sides are proportional by the same ratio.
the requirements for triangle to be congruent are that all correspond angles are equal and all corresponding sides are congruent as well.
an equilateral triangle has all angles equal and all sides equal.
if one equilateral triangle is congruent to another equilateral triangle, then all the corresponding sides of the first triangle have to be the same measure of the corresponding sides of the other triangle.
ABC = all angles = 60 degrees and all sides = 3
DEF = all angles = 60 degrees and all sides = 3
GHI = all angles = 60 degrees and all sides = 6
ABC is congruent to DEF
ABC is similar to GHI
the ratio of the sides between ABC and GHI is 3:6 = 1:2
if multiply each side of ABC by 2, then you will get the sides of GHI.
note that ABC is also similar to GHI because all corresponding angles are equal and all corresponding sides are proportional in the ratio of 1:1.
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