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MATH / In the diagram, PRT and QRS are straight lines. What is the value of x? [asy] import olympiad; import math; size(6cm); // ADJUST pair r = (0, 0); pair t = dir(-55); pair s = dir(-5…

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problem

In the diagram, PRTPRT and QRSQRS are straight lines. What is the value of xx? [asy] import olympiad; import math; size(6cm); // ADJUST pair r = (0, 0); pair t = dir(-55); pair s = dir(-55 - 70); pair p = dir(-55 + 180); real qr = Sin(70) / Sin(40); pair q = qr * dir(-55 - 70 + 180); draw(r--p--q--s--t--r); label("RR", r, 2 * E); label("TT", t, SE); label("SS", s, SW); label("PP", p, W); label("QQ", q, E); label("x∘x^\circ", t, W + NW + NW); label("40∘40^\circ", q, 4 * SW); add(pathticks(p--q, spacing=1, s=4)); add(pathticks(q--r, spacing=1, s=4)); add(pathticks(r--s, 2, spacing=1, s=4)); add(pathticks(r--t, 2, spacing=1, s=4)); [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, PRT and QRS are straight lines. What is the value of x? [asy]
import olympiad;
import math;

size(6cm); // ADJUST

pair r = (0, 0);
pair t = dir(-55);
pair s = dir(-55 - 70);
pair p = dir(-55 + 180);
real qr = Sin(70) / Sin(40);

pair q = qr * dir(-55 - 70 + 180);

draw(r--p--q--s--t--r);

label("R", r, 2 * E);
label("T", t, SE);
label("S", s, SW);
label("P", p, W);
label("Q", q, E);

label("x^(∘)", t, W + NW + NW);
label("40^(∘)", q, 4 * SW);

add(pathticks(p--q, spacing=1, s=4));
add(pathticks(q--r, spacing=1, s=4));
add(pathticks(r--s, 2, spacing=1, s=4));
add(pathticks(r--t, 2, spacing=1, s=4));
[/asy]
Original LaTeX notation
In the diagram, $PRT$ and $QRS$ are straight lines. What is the value of $x$? [asy]
import olympiad;
import math;

size(6cm); // ADJUST

pair r = (0, 0);
pair t = dir(-55);
pair s = dir(-55 - 70);
pair p = dir(-55 + 180);
real qr = Sin(70) / Sin(40);

pair q = qr * dir(-55 - 70 + 180);

draw(r--p--q--s--t--r);

label("$R$", r, 2 * E);
label("$T$", t, SE);
label("$S$", s, SW);
label("$P$", p, W);
label("$Q$", q, E);

label("$x^\circ$", t, W + NW + NW);
label("$40^\circ$", q, 4 * SW);

add(pathticks(p--q, spacing=1, s=4));
add(pathticks(q--r, spacing=1, s=4));
add(pathticks(r--s, 2, spacing=1, s=4));
add(pathticks(r--t, 2, spacing=1, s=4));
[/asy]

level

Level 1

type

Geometry

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Source and history

task_id
d47248cb-8fdf-5016-a49b-9a0635798f00
task_key
geometry--test--d47248cb-8fdf-5016-a49b-9a0635798f00
task_revision_id
1
config / split
geometry / test

Official source

initial import