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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If triangles PQR and LMN are equilateral triangles, what is
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03 May 2020, 05:35
03 May 2020, 05:35
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If triangles PQR and LMN are equilateral triangles, what is the value of k in terms of x and y?
A) 60 + x – y
B) 120 – 2x + y
C) 120 – x + y
D) 180 – x – y
E) 180 – 2x + y
Re: If triangles PQR and LMN are equilateral triangles, what is
[#permalink]
03 May 2020, 05:56
03 May 2020, 05:56
1
to solve this question
Let us name the intersecting points between two triangles A,B
we must know two angles
: Angle Q(AQB) Is 60
: We have to find out Angle ABQ
to find out that we have to find out RBN
since they are vertically opposite angles RBN=ABQ
we can find out that BRN = 120-x
and similarly BNR = 120-y
that makes RBN = x+y-60
so now to find out k
K+ABQ+60=180
from this k=180-x-y
Let us name the intersecting points between two triangles A,B
we must know two angles
: Angle Q(AQB) Is 60
: We have to find out Angle ABQ
to find out that we have to find out RBN
since they are vertically opposite angles RBN=ABQ
we can find out that BRN = 120-x
and similarly BNR = 120-y
that makes RBN = x+y-60
so now to find out k
K+ABQ+60=180
from this k=180-x-y
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If triangles PQR and LMN are equilateral triangles, what is
[#permalink]
03 May 2020, 05:58
03 May 2020, 05:58
2
GreenlightTestPrep wrote:
If triangles PQR and LMN are equilateral triangles, what is the value of k in terms of x and y?
A) 60 + x – y
B) 120 – 2x + y
C) 120 – x + y
D) 180 – x – y
E) 180 – 2x + y
First of all, since the two triangles are equilateral triangles, we know that all of the their angles are 60°
So we'll add this to our diagram (in a few key places)
Next, we'll focus on two angles, which I have labelled a and b
Since angles on a line add to 180°, we can write: x + 60 + a = 180
Subtract 60 from both sides: x + a = 120
Subtract x from both sides to get: a = 120 - x
When we apply the same logic to the other angles, we get: b = 120 - y
Add this to our diagram:
Now let's focus on the red triangle below.
Since angles in a triangle always add to 180°, we can write: w + (120 - x) + (120 - y) = 180
Simplify to get: w - x - y + 240 = 180
Subtract 240 from both sides: w - x - y = -60
Add x and add y to both sides of the equation to get: w = x + y - 60
Add this to our diagram to get:
Since opposite angles are always equal, we know that the opposite angle must also be x + y - 60
Finally, we can focus on the red triangle below.
Since angles in a triangle always add to 180°, we can write: k + 60 + (x + y - 60) = 180
Simplify: k + x + y = 180
Subtract x and subtract y from both sides to get: k = 180 - x - y
Answer: D
Cheers,
Brent
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