U
△PBU can be mapped onto
△
G
Z
D
△GZD by a rotation. If
m
∠
P
=
3
8
∘
m∠P=38
∘
, find
m
∠
Z
m∠Z.
Since we are given that the transformation from △PBU to △GZD is a rotation, we know that corresponding angles are congruent. Therefore, we have:
m∠P = m∠G
m∠B = m∠Z
m∠U = m∠D
We are given that m∠P = 38 degrees, so we know that m∠G = 38 degrees as well.
We want to find m∠Z. Since m∠B = m∠Z, we just need to find one of these angles, and we can use the fact that the sum of the angles in a triangle is 180 degrees. Therefore:
m∠P + m∠B + m∠U = 180 degrees
38 + m∠B + m∠U = 180
m∠B + m∠U = 142
But we also know that m∠B = m∠Z and m∠U = m∠D, so we can substitute to get:
m∠Z + m∠D = 142
We don't know m∠D, but we do know that m∠D + m∠G + m∠Z = 180 (since △GZD is a triangle). We already know that m∠G = 38, so we can substitute to get:
m∠D + 38 + m∠Z = 180
Simplifying this equation, we get:
m∠D + m∠Z = 142
This is the same equation we got earlier! Therefore, we can substitute the expression m∠D + m∠Z = 142 in for m∠Z + m∠D = 142, giving us:
m∠D + m∠D + m∠G = 180
Simplifying this equation, we get:
2m∠D + 38 = 180
Subtracting 38 from both sides, we get:
2m∠D = 142
Dividing by 2, we get:
m∠D = 71
Now we can substitute this value back into our equation m∠Z + m∠D = 142 to get:
m∠Z + 71 = 142
Subtracting 71 from both sides, we get:
m∠Z = 71
Therefore, m∠Z = 71 degrees.
To find the measure of angle Z (m∠Z), we need to understand the relationship between the two triangles, △PBU and △GZD, and how they are mapped onto each other by a rotation.
Given that △PBU can be mapped onto △GZD by a rotation, we can conclude that the corresponding angles in these two triangles are congruent. In other words, if ∠P in △PBU corresponds to ∠G in △GZD, then the measure of ∠P is equal to the measure of ∠G, and similarly for the other corresponding angles.
Given that m∠P = 38°, we can conclude that m∠G = 38° as well.
Now, let's look at the relationship between angle G and angle Z. Since these angles are corresponding angles in congruent triangles, they must be equal in measure. Therefore, m∠Z = m∠G = 38°.
So, the measure of angle Z (m∠Z) is 38°.