In isosceles triangle FGH, FG is congruent to GH. If Elijah draws a line segment from the vertex G to the midpoint of FH, labeli
ng the midpoint K, what can he prove about triangles FGK and HGK?
A= The triangle are neither congruent nor similar
B= The triangles are congruent, but not similar
C= The triangles are similar, but not congruent
D= The triangles are both congruent and similar
A= The triangle are neither congruent nor similar
B= The triangles are congruent, but not similar
C= The triangles are similar, but not congruent
D= The triangles are both congruent and similar
2 answers:
Answer:
<h2>B) The triangles are congruent, but not similar.</h2>Step-by-step explanation:
Givens
is isosceles.
.
In the image attached you can observe what Elijah did. The vertical line goes from vertex G to the midpoint K. When this happens, the resulting right triangles are congruents, that is, because we already know that FG is congruent to GH, GK is common for both triangles and the pair of angles in the base of the isosceles are congruent, by definiton.
Therefore, the triangles are congruent, but no similar. Choice B is correct.
Remember, when we talk about similarity, it refers to proportional corresponding sides of the triangles. When we talk about congruence, it refers to the exact equivalence between corresponding sides of both triangles. That's the difference. So, as you can deduct, a triangle cannot be similar and congruent at the same time.
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When , we have
and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)
Suppose this is true for , that
Now for , we have
so we know the left side is at least divisible by by our assumption.
It remains to show that
which is easily done with Fermat's little theorem. It says
where is prime and
is any integer. Then for any positive integer
,
Furthermore,
which goes all the way down to
So, we find that
QED
Answer:
The correct option is;
Corresponding angles theorem
Step-by-step explanation:
Type of lines of lines a and b = Horizontal and parallel lines
The transversal to a and b = Line c
The angles between a and c labelled clockwise from the upper left quarter segment = 1, 2, 4 and 3
The angles between b and c labelled clockwise from the upper left segment = 5, 6, 8 and 7
Therefore, we have;
Statement Reason
1. a║b, c is a transversal Given
2. ∠6 ≅ ∠2 Corresponding angles theorem
3. m∠6 = m∠2 Definition of congruent
4. ∠6 is supp. to ∠8 Definition of linear pair
5. ∠2 is supp. to ∠8 Congruent supplement theorem
Corresponding angles are the angles located in spatially similar or matching corners of two lines that have been crossed by the same transversal. When the two lines having a common transversal are parallel, the corresponding angles will be congruent.