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MATH / In the diagram, PW is parallel to QX, S and T lie on QX, and U and V are the points of intersection of PW with SR and TR, respectively. If ∠SUV=120^(∘) and ∠VTX=112^(∘), what is th…

Problem

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problem

In the diagram, PWPW is parallel to QXQX, SS and TT lie on QXQX, and UU and VV are the points of intersection of PWPW with SRSR and TRTR, respectively. If ∠SUV=120∘\angle SUV = 120^\circ and ∠VTX=112∘\angle VTX = 112^\circ, what is the measure of ∠URV\angle URV? [asy] size(6cm); defaultpen(linewidth(0.75) + fontsize(11)); // Useful function path extend(pair a, pair b, int n=10) { return a--(a + n * (b - a));} // Points real ust = 60; real vtx = 112; pair q = (0, 0); pair p = (0, 0.5); pair w = (2, p.y); pair x = (2, 0); pair s = (0.3, 0); pair t = s + (1, 0); pair u = shift(s) * (p.y / Tan(ust), p.y); pair v = shift(t) * (p.y / Tan(vtx), p.y); pair r = IP(extend(s, u), extend(t, v)); // Draw lines draw(s--r--t); draw(p--w, Arrow(position=0.8, size=5)); draw(q--x, Arrow(position=0.8, size=5)); // Labels label("PP", p, W); label("QQ", q, W); label("WW", w, E); label("XX", x, E); label("RR", r, N); label("UU", u, NW); label("VV", v, NE); label("SS", s, NW); label("TT", t, NE); [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, PW is parallel to QX, S and T lie on QX, and U and V are the points of intersection of PW with SR and TR, respectively. If ∠SUV=120^(∘) and ∠VTX=112^(∘), what is the measure of ∠URV? [asy]
size(6cm); defaultpen(linewidth(0.75) + fontsize(11));
// Useful function
path extend(pair a, pair b, int n=10) { return a--(a + n * (b - a));}

// Points
real ust = 60; real vtx = 112;
pair q = (0, 0); pair p = (0, 0.5); pair w = (2, p.y); pair x = (2, 0);
pair s = (0.3, 0); pair t = s + (1, 0);
pair u = shift(s) * (p.y / Tan(ust), p.y); pair v = shift(t) * (p.y / Tan(vtx), p.y);
pair r = IP(extend(s, u), extend(t, v));

// Draw lines
draw(s--r--t);
draw(p--w, Arrow(position=0.8, size=5));
draw(q--x, Arrow(position=0.8, size=5));

// Labels
label("P", p, W); label("Q", q, W); label("W", w, E); label("X", x, E);
label("R", r, N); label("U", u, NW); label("V", v, NE); label("S", s, NW);

label("T", t, NE);

[/asy]
Original LaTeX notation
In the diagram, $PW$ is parallel to $QX$, $S$ and $T$ lie on $QX$, and $U$ and $V$ are the points of intersection of $PW$ with $SR$ and $TR$, respectively. If $\angle SUV = 120^\circ$ and $\angle VTX = 112^\circ$, what is the measure of $\angle URV$? [asy]
size(6cm); defaultpen(linewidth(0.75) + fontsize(11));
// Useful function
path extend(pair a, pair b, int n=10) { return a--(a + n * (b - a));}

// Points
real ust = 60; real vtx = 112;
pair q = (0, 0); pair p = (0, 0.5); pair w = (2, p.y); pair x = (2, 0);
pair s = (0.3, 0); pair t = s + (1, 0);
pair u = shift(s) * (p.y / Tan(ust), p.y); pair v = shift(t) * (p.y / Tan(vtx), p.y);
pair r = IP(extend(s, u), extend(t, v));

// Draw lines
draw(s--r--t);
draw(p--w, Arrow(position=0.8, size=5));
draw(q--x, Arrow(position=0.8, size=5));

// Labels
label("$P$", p, W); label("$Q$", q, W); label("$W$", w, E); label("$X$", x, E);
label("$R$", r, N); label("$U$", u, NW); label("$V$", v, NE); label("$S$", s, NW);

label("$T$", t, NE);

[/asy]

level

Level 4

type

Prealgebra

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Source and history

task_id
a1e742e9-a9bb-5a79-bf87-3e2e2dda1fc6
task_key
prealgebra--test--a1e742e9-a9bb-5a79-bf87-3e2e2dda1fc6
task_revision_id
1
config / split
prealgebra / test

Official source

initial import