Angles 2

rand(2) rand(3) RAND_SWITCH2 + 10*RAND_SWITCH3

rand(15) + 40 rand(10) + 100 180 - Tri_Y - Tri_Z

Given the following:

\color{green}{\angle{ABC}} = Tri_Y°
\color{purple}{\angle{BAC}} = Tri_Z°
\overline{DE} \parallel \overline{BC}

What is \color{blue}{\angle{RAND_SWITCH2 === 0 ? "DAF" : "CAE"}} {?}

init({ range: [[-5, 5], [-3, 5]], scale: [40, 40] }); style({ stroke: "#888", strokeWidth: 2 }); // Draw a horizontal line and a crossing line // to form 2 opposing angles. path([ [-5, 0], [5, 0] ]); path([ [-5, -3], [5, 5] ]); path([ [-5, 3], [5, 3] ]); path([ [-1.2, 0], [-4, 3] ]); style({ fill: "grey" }, function() { // Label A, B, C label([-1.3, 0], "A", "below"); circle([-1.25, 0], 0.05); label([-4, 3], "B", "above"); circle([-4, 3], 0.05); label([2.5, 3], "C", "above"); circle([2.5, 3], 0.05); label([-4, 0], "D", "above"); circle([-4,0], 0.05); label([3, 0], "E", "above"); circle([3,0], 0.05); label([-3.75, -2], "F", "above"); circle([-3.75,-2], 0.05); }); // label angle ABC arc([-4,3], .75, 312, 360, { stroke: "green" }); label([-3.2, 3], "\\color{green}{Tri_Y°}", "below right", {color: "green"}); // label angle BAC arc([-1.3,0], .75, 38, 125, { stroke: "purple" }); label([-1.3, .7], "\\color{purple}{Tri_Z°}", "above", {color: "purple"}); // Label X according to problem variation if(RAND_SWITCH2 == 0 ) { //problem variation 1 arc([-1, 0], 1, 180, 210, { stroke: "blue"}); LABEL = label([-3.3, 0], "\\color{blue}{∠DAF} = {?}", "below", { color: "blue"}); } else { //problem variation 2 arc([-1, 0], 1, 0, 45, { stroke: "blue"}); LABEL = label([1, 0], "\\color{blue}{∠CAE} = {?}", "above", { color: "blue"}); }

NOTE: Angles not drawn to scale.

Tri_X

Remember that the measure of the angles in a triangle sum to 180°. Solve for \color{orange}{\angle{BCA}} by subtracting the measures of angles \color{purple}{\angle{BAC}} and \color{green}{\angle{ABC}} from 180°. We find that \color{orange}{\angle{BCA}} = Tri_X°. // label angle BAC arc([2.5, 3], .75, 180, 220, { stroke: "orange" }); label([1.8, 3], "\\color{orange}{Tri_X°}", "below left", {color: "orange"});

Solve for \color{blue}{\angle{DAF}} by using the fact that it is a corresponding angle to LABEL.remove(); LABEL = label([-3.3, 0], "\\color{blue}{\\angle{DAF}}=Tri_X°", "below"); \color{blue}{\angle{CAE}} by using the fact that it is an alternate interior angle to LABEL.remove(); LABEL = label([1, 0], "\\color{blue}{∠CAE} = Tri_X°", "above", { color: "blue"}); \color{orange}{\angle{BCA}}. That means those angles are equal because they are both created by the same set of parallel lines \overline{BC} and \overline{DE}, and transversal line \overline{CF}.

rand(20) + 100 180 - Y

Given the following:

\overline{AB} \parallel \overline{CD} (line AB is parallel to line CD)
\color{purple}{\angle{EGB}} = X°.
\color{purple}{\angle{AGH}} = X°.
\color{purple}{\angle{BGH}} = 180 - X°

What is \color{blue}{\angle{RAND_SWITCH2 === 0 ? "GHD" : "CHF"}} {?}

init({ range: [[-7, 6], [-5, 5.2]], scale: [40, 40] }); style({ stroke: "#888", strokeWidth: 2 }); style({ fill: "grey" }, function() { // Draw 2 parallel horizontal lines path([ [-5, 2], [5, 2] ]); label([-5, 2], "A", "below"); circle([-5,2], 0.05); label([5, 2], "B", "below"); circle([5,2], 0.05); path([ [-5, -2], [5, -2] ]); label([-5, -2], "C", "below"); circle([-5,-2], 0.05); label([5, -2], "D", "below"); circle([5, -2], 0.05); // Draw a transversal line. path([ [-5, -4], [4, 5] ]); label([4, 5], "E", "below"); circle([4, 5], 0.05); label([-5, -4], "F", "right"); circle([-5, -4], 0.05); label([1, 2], "G", "below right"); circle([1,2], 0.05); label([-3, -2], "H", "below right"); circle([-3, -2], 0.05); }); // label angle X if(RAND_SWITCH2 == 0) { arc([-2.9, -2], 1, 0, 50, { stroke: "blue"}); LABEL = label([-2, -2], "\\color{blue}{\\angle{GHD}}= {?}", "above right"); } else { arc([-2.9, -2], 1, 180, 220, { stroke: "blue"}); LABEL = label([-4, -2.5], "\\color{blue}{\\angle{CHF}}= {?}", "below left"); } // label angle Y if(RAND_SWITCH3 == 0) { arc([1.2, 2], 1.5, 0, 50, { stroke: "purple" }); label([3.1, 2], "\\color{purple}{\\angle{EGB}}=X°", "above right"); } else if (RAND_SWITCH3 == 1) { arc([1.2, 2], 1.5, 180, 220, { stroke: "purple" }); label([-1, 2], "\\color{purple}{\\angle{AGH}}=X°", "below left"); } else { arc([1.2, 2], 1, 220, 0, { stroke: "purple" }); label([1.5, 1.2], "\\color{purple}{\\angle{BGH}}=180 - X°", "below right"); }

NOTE: Angles not drawn to scale.

X

\color{blue}{\angle{GHD}} = \color{purple}{\angle{EGB}}. We know this because they are 2 complementary angles of a set of parallel lines bisected by a single line.

First solve for \color{orange}{\angle{AGH}}. We know that \color{orange}{\angle{AGH}} = \color{purple}{\angle{EGB}} because opposite angles are equal. arc([1,2], .88, 180, 225, {stroke:"orange"}); label([0,2], "\\color{orange}{X°}", "below left");

\color{blue}{\angle{GHD}} = \color{purple}{\angle{AGH}} We know this because they are 2 alternate interior angles of a set of parallel lines bisected by a single line.

\color{blue}{\angle{CHF}} = \color{purple}{\angle{AGH}}. We know this because they are 2 corresponding angles of a set of parallel lines bisected by a single line.

First solve for \color{orange}{\angle{AGH}}. We know that \color{orange}{\angle{AGH}} = 180° - \color{purple}{\angle{BGH}}, because angles along a line plane add up to 180°. arc([1,2], .88, 180, 225, {stroke:"orange"}); label([0,2], "\\color{orange}{X°}", "below left");

\color{blue}{\angle{GHD}} = \color{orange}{\angle{AGH}}. We know those 2 angles are equal because they are alternate interior angles of 2 parallel lines.

\color{blue}{\angle{CHF}} = \color{orange}{\angle{AGH}}. We know those 2 angles are equal because they are corresponding angles formed by parallel lines, and a single bisecting lines.

Therefore, \angle{GHD} = X° LABEL.remove(); label([-2, -2], "\\color{blue}{\\angle{GHD}}=X°", "above right"); \angle{CHF} = X° LABEL.remove(); label([-4, -2.5], "\\color{blue}{\\angle{CHF}}=X°", "below left"); .

rand(10) + 30 rand(10) + 100 180 - Tri1_Y - Tri1_Z

Given the following:

\color{green}{\angle{BDC}°} = Tri1_Y
\color{orange}{\angle{DBE}°} = Tri1_X

What is \color{blue}{\angle{RAND_SWITCH3 === 0 ? "CHE" : ( RAND_SWITCH3 === 1 ? "GHC" : "DHE" )}} {?}

init({ range: [[-10, 10], [-7, 10]], scale: [30, 30] }); style({ stroke: "#888", strokeWidth: 2 }); // Draw A Star path([ [-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5] ]); // Label pts on the star. label([-8, 5], "A", "left"); label([0, 9], "B", "above"); label([8, 5], "C", "right"); label([-6, -6], "D", "below"); label([6, -6], "E", "below"); label([-1.8, 5], "F", "above left"); label([1.8, 5], "G", "above right"); label([3.2, 1.3], "H", "below right"); label([0, -1.3], "I", "below"); label([-3.2, 1.3], "J", "below left"); // Label the given angles label([-5.5, -5.2], "\\color{green}{Tri1_Y°}", "above right"); arc([-6, -6], 1.3, 40, 71, { stroke: "green" }); label([0, 7.4], "\\color{orange}{Tri1_X°}", "below"); arc([0, 9], 1.3, 245, 290, { stroke: "orange" }); // Label X according to variation on the problem if(RAND_SWITCH3 == 0) { LABEL = label([4.8, 1.0], "\\color{blue}{\\angle{CHE}}= {?}", "right"); arc([3.2, 1.3], 1.7, 287, 35, { stroke: "blue" }); } else if (RAND_SWITCH3 == 1) { LABEL = label([4, 2.5], "\\color{blue}{\\angle{GHC}}= {?}", "above"); arc([3.2, 1.3], 1, 35, 118, { stroke: "blue" }); } else { LABEL = label([2.5, -0.5], "\\color{blue}{\\angle{DHE}}= {?}", "below"); arc([3.2, 1.3], 1.1, 219, 286, { stroke: "blue" }); }

NOTE: Angles not drawn to scale.

Tri1_Z 180-Tri1_Z

\color{purple}{\angle{BHD}} = 180° - \color{green}{\angle{BDC}} - \color{orange}{\angle{DBE}} = Tri1_Z°. This is because the interior angles of a triangle add up to 180°. // label angle BHD arc([3.2, 1.3], .75, 118, 220, { stroke: "purple" }); label([2.6, 2], "\\color{purple}{Tri1_Z^\\circ}", "below left");

\color{blue}{\angle{CHE}} = \color{purple}{\angle{BHD}}. This is because they are opposite each other, and opposite angles are equal.

\color{blue}{\angle{CHE}} = Tri1_Z°

LABEL.remove(); label([4.8, 1.0], "\\color{blue}{\\angle{CHE}}=Tri1_Z^\\circ", "right");

\color{blue}{\angle{CHG}} \color{blue}{\angle{DHE}} = 180° - \color{purple}{\angle{BHD}}. This is because angles along a line add up to 180°.

\color{blue}{\angle{GHC}} LABEL.remove(); label([4, 2.5], "\\color{blue}{\\angle{GHC}}=180 - Tri1_Z^\\circ", "above"); \color{blue}{\angle{DHE}} LABEL.remove(); label([2.5, -0.5], "\\color{blue}{\\angle{DHE}}=180 - Tri1_Z^\\circ", "below"); = 180 - Tri1_Z°

rand(10) + 30 rand(10) + 70 180 - Tri2_Y - Tri2_Z

Given the following:

\color{green}{\angle{BDC}°} = Tri2_Y°
\color{orange}{\angle{AIC}°} = 180 - Tri2_Z°

\color{green}{\angle{GCH}°} = Tri2_Y°
\color{orange}{\angle{FGH}°} = 180 - Tri2_Z°

What is \color{blue}{\angle{RAND_SWITCH2 === 0 ? "AJF" : "IHE"}} {?}

init({ range: [[-10, 10], [-7, 10]], scale: [30, 30] }); style({ stroke: "#888", strokeWidth: 2 }); // Draw A Star path([ [-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5] ]); // Label pts on the star. label([-8, 5], "A", "left"); label([0, 9], "B", "above"); label([8, 5], "C", "right"); label([-6, -6], "D", "below"); label([6, -6], "E", "below"); label([-1.8, 5], "F", "above left"); label([1.8, 5], "G", "above right"); label([3.2, 1.3], "H", "below right"); label([0, -1.3], "I", "below"); label([-3.2, 1.3], "J", "below left"); // Label Angles and X according to variation if( RAND_SWITCH2 == 0) { // Label the given angles label([-5.5, -5.2], "\\color{green}{Tri2_Y°}", "above right"); arc([-6, -6], 1.3, 40, 71, { stroke: "green" }); label([0, -.2], "\\color{orange}{180 - Tri2_Z°}", "above"); arc([0, -1], 1, 28, 152, { stroke: "orange" }); // Label X LABEL = label([-3.7, 2.5], "\\color{blue}{\\angle{AJF}}= {?}", "above"); arc([-3.2, 1.3], 1, 65, 142, { stroke: "blue" }); } else { // Label the given angles label([6.5, 5], "\\color{green}{Tri2_Y°}", "below left"); arc([8, 5], 1.3, 180, 220, { stroke: "green" }); label([1.3, 4.5], "\\color{orange}{180 - Tri2_Z°}", "below left"); arc([1.6, 5], 0.75, 180, 289, { stroke: "orange" }); // Label X LABEL = label([4.0, -0.3], "\\color{blue}{\\angle{IHE}}= {?}", "below left"); arc([3.1, 1.2], .75, 220, 290, { stroke: "blue" }); }

NOTE: Angles not drawn to scale.

Tri2_X

\color{purple}{\angle{DIJ}} = 180° - \color{orange}{\angle{AIC}}. This is because angles along a line total 180°. // label angle JID arc([0, -1.2], .75, 143, 220, { stroke: "purple" }); label([-.75, -1.2], "\\color{purple}{Tri2_Z°}", "left");

\color{purple}{\angle{HGC}} = 180° - \color{orange}{\angle{FGH}}. This is because angles along a line or flat plane total 180°. // label angle HGC arc([1.8, 5], 1, 280, 0, { stroke: "purple" }); label([2.5, 4.3], "\\color{purple}{Tri2_Z°}", "below right");

\color{teal}{\angle{DJI}} = 180° - \color{green}{\angle{BDC}} - \color{purple}{\angle{DIJ}}. We know this because the sum of angles inside a triangle add up to 180°. // label angle JID arc([-3.2, 1.3], .75, 260, 320, { stroke: "teal" }); label([-3.2, 0.50], "\\color{teal}{Tri2_X°}", "below right");

\color{teal}{\angle{CHG}} = 180° - \color{green}{\angle{ACD}} - \color{purple}{\angle{HGC}}. We know this, because the sum of angles inside a triangle add up to 180°. // label angle CHG arc([3.2, 1.3], .75, 38, 120, { stroke: "teal" }); label([3.4, 1.78], "\\color{teal}{Tri2_X°}", "above");

\color{blue}{\angle{AJF}} = \color{teal}{\angle{DJI}}. We know they are equal because they are opposite angles.

\color{blue}{\angle{IHE}} = \color{teal}{\angle{CHG}}. We know they are equal because they are opposite angles.

Therefore, \angle{AJF} = Tri2_X° LABEL.remove(); label([-3.7, 2.5], "\\color{blue}{\\angle{AJF}}=Tri2_X°", "above"); \angle{IHE} = Tri2_X° LABEL.remove(); label([4.0, -0.3], "\\color{blue}{\\angle{IHE}}=Tri2_X°", "below left"); .