rand(2) rand(3) RAND_SWITCH2 + 10 * RAND_SWITCH3
rand(15) + 40 rand(10) + 100 180 - Tri_Y - Tri_Z

Given the following:

- `\green{\angle{ABC}} = Tri_Y^\circ`
- `\purple{\angle{BAC}} = Tri_Z^\circ`
- `\overline{BC} \parallel \overline{DE}` (line `BC` is parallel to line `DE`)
- `C` and `F` lie on a straight line.

What is `\blue{\angle{DAF}}{?}`

What is `\blue{\angle{CAE}}{?}`

init({ range: [[-5, 5], [-3, 5]], scale: [40, 40] }); // label angle ABC arc([-4, 3], 0.75, 312, 360, { stroke: GREEN }); label([-3.2, 3], "\\green{" + Tri_Y + "^\\circ}", "below right"); // label angle BAC arc([-1.3, 0], 0.75, 38, 125, { stroke: PURPLE }); label([-1.3, .7], "\\purple{" + Tri_Z + "^\\circ}", "above"); // Label X according to problem variation if (RAND_SWITCH2 == 0 ) { arc([-1, 0], 1, 180, 208, { stroke: BLUE }); LABEL = label([-3.3, 0], "\\blue{\\angle{DAF}} = {?}", "below"); } else { arc([-1.3, 0], 0.75, 0, 35, { stroke: BLUE }); LABEL = label([0.75, 0], "\\blue{\\angle{CAE}} = {?}", "above"); } // 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]]); ParallelLineMarkers(4.5, 0); ParallelLineMarkers(4.5, 3); label([-1.3, 0], "A", "below"); label([-4, 3], "B", "above"); label([2.5, 3], "C", "above"); label([-4, 0], "D", "above"); label([3, 0], "E", "above"); label([-3.75, -2], "F", "above"); var dotStyle = { r: 0.1, fill: BLUE, stroke: "none" }; circle([-1.25, 0], dotStyle); circle([-4, 3], dotStyle); circle([2.5, 3], dotStyle); circle([-4,0], dotStyle); circle([3,0], dotStyle); circle([-3.75,-2], dotStyle);

NOTE: Angles not drawn to scale.

Tri_X`\Large{{}^\circ}`

The measure of the angles in a triangle sum to `180^\circ`.

`\red{\angle{BCA}} = 180^\circ - \green{\angle{ABC}} - \purple{\angle{BAC}}`

arc([2.5, 3], 0.75, 180, 220, { stroke: RED }); graph.TriX = label([1.8, 3], "\\red{\\angle{BCA}}", "below left");

`\red{\angle{BCA}} = 180^\circ - \green{Tri_Y^\circ} - \purple{Tri_Z^\circ} = \red{Tri_X^\circ}`

graph.TriX.remove(); label([1.8, 3], "\\red{" + Tri_X + "^\\circ}", "below left");

`\blue{\angle{DAF}}` is a corresponding angle to `\red{\angle{BCA}}`.

`\blue{\angle{DAF}}` = `\red{\angle{BCA}} = Tri_X^\circ`

LABEL.remove(); label([-3.3, 0], "\\blue{\\angle{DAF}}=" + Tri_X + "^\\circ", "below");

`\blue{\angle{CAE}}` is an alternate interior angle to `\red{\angle{BCA}}`.

`\blue{\angle{CAE}}` = `\red{\angle{BCA}} = Tri_X^\circ`

LABEL.remove(); label([0.9, 0], "\\blue{\\angle{CAE}} = " + Tri_X + "^\\circ", "above");
rand(20) + 100 180 - Y

Given the following:

- `\overline{AB} \parallel \overline{CD}` (line `AB` is parallel to line `CD`)
- `\green{\angle{EGB}} = X^\circ`
- `\purple{\angle{AGH}} = X^\circ`
- `\green{\angle{BGH}} = 180 - X^\circ`
- `E` and `F` lie on a straight line.

What is `\blue{\angle{GHD}}{?}`

What is `\blue{\angle{CHF}}{?}`

init({ range: [[-6.5, 5.8], [-5, 4.2]], scale: [40, 40] }); // label angle X if(RAND_SWITCH2 === 0) { arc([-2.9, -2], 1, 0, 50, { stroke: BLUE }); LABEL = label([-2, -2], "\\blue{\\angle{GHD}}= {?}", "above right"); } else { arc([-2.9, -2], 1, 180, 220, { stroke: BLUE }); LABEL = label([-4, -2.5], "\\blue{\\angle{CHF}}= {?}", "below left"); } // label angle Y if(RAND_SWITCH3 === 0) { arc([1.2, 2], 1, 0, 54, { stroke: GREEN }); label([2.2, 2], "\\green{" + X + "^\\circ}", "above right"); } else if (RAND_SWITCH3 === 1) { arc([1.4, 2], 1, 180, 210, { stroke: PURPLE }); label([0, 2], "\\purple{" + X + "^\\circ}", "below left"); } else { arc([1.2, 2], 1, 216, 0, { stroke: GREEN }); label([1.5, 1.2], "\\green{" + (180 - X) + "^\\circ}", "below right"); } path([[-5, 2], [5, 2]]); path([[-5, -2], [5, -2]]); path([[-5, -4], [3, 4]]); ParallelLineMarkers(4.2, 2); ParallelLineMarkers(4.2, -2); label([-5, 2], "A", "below"); label([5, 2], "B", "below"); label([-5, -2], "C", "below"); label([5, -2], "D", "below"); label([3, 4], "E", "below"); label([-5, -4], "F", "right"); label([1, 2], "G", "below right"); label([-3, -2], "H", "below right"); var dotStyle = { r: 0.1, fill: BLUE, stroke: "none" }; circle([-5, 2], dotStyle); circle([5, 2], dotStyle); circle([-5, -2], dotStyle); circle([5, -2], dotStyle); circle([3, 4], dotStyle); circle([-5, -4], dotStyle); circle([1,2], dotStyle); circle([-3, -2], dotStyle);

NOTE: Angles not drawn to scale.

X`\Large{{}^\circ}`

`\purple{\angle{AGH}}` forms a linear pair with `\green{\angle{BGH}}`.

arc([1,2], 0.88, 180, 225, { stroke: PURPLE }); graph.AGH = label([0, 2], "\\purple{\\angle{AGH}}", "below left");

Therefore `\purple{\angle{AGH}} = 180^\circ - \green{Y^\circ} = X^\circ`.

graph.AGH.remove(); label([0,2], "\\purple{" + X + "^\\circ}", "below left");

`\blue{\angle{GHD}}` and `\green{\angle{EGB}}` are complementary angles.

Therefore `\blue{\angle{GHD}} = \green{\angle{EGB}}`.

`\purple{\angle{AGH}}` and `\green{\angle{EGB}}` are opposite angles.

arc([1,2], 0.88, 180, 225, { stroke: PURPLE }); graph.AGH = label([0, 2], "\\purple{\\angle{AGH}}", "below left");

Therefore `\purple{\angle{AGH}} = \green{\angle{EGB}} = X^\circ`.

graph.AGH.remove(); label([0,2], "\\purple{" + X + "^\\circ}", "below left");

`\blue{\angle{GHD}}` and `\purple{\angle{AGH}}` are alternate interior angles.

Therefore `\blue{\angle{GHD}} = \purple{\angle{AGH}}`.

`\blue{\angle{GHD}}` and `\purple{\angle{AGH}}` are alternate interior angles.

Therefore `\blue{\angle{GHD}} = \green{\purple{AGH}}`.

`\angle{GHD} = X^\circ`

LABEL.remove(); label([-2, -2], "\\blue{\\angle{GHD}}=" + X + "^\\circ", "above right");

`\blue{\angle{CHF}}` and `\purple{\angle{AGH}}` are corresponding angles.

Therefore `\blue{\angle{CHF}} = \purple{\angle{AGH}}`.

`\angle{CHF} = X^\circ`

LABEL.remove(); label([-4, -2.5], "\\blue{\\angle{CHF}}=" + X + "^\\circ", "below left");
rand(10) + 30 rand(10) + 100 180 - ANGLE_2 - ANGLE_3

Given the following:

- `\green{\angle{BDC}} = ANGLE_2^\circ`
- `\red{\angle{DBE}} = ANGLE_1^\circ`

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

init({ range: [[-10, 10], [-7, 10]], scale: [25, 25] }); // 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], "\\green{" + ANGLE_2 + "^\\circ}", "above right"); arc([-6, -6], 1.3, 40, 70, { stroke: GREEN }); label([0, 7.4], "\\red{" + ANGLE_1 + "^\\circ}", "below"); arc([0, 9], 1.3, 247, 290, { stroke: RED }); // Label X according to variation on the problem if(RAND_SWITCH3 == 0) { LABEL = label([4.7, 1.0], "\\blue{\\angle{CHE}}= {?}", "right"); arc([3.2, 1.3], 1.5, 287, 35, { stroke: BLUE }); } else if (RAND_SWITCH3 == 1) { LABEL = label([4, 2.5], "\\blue{\\angle{GHC}}= {?}", "above"); arc([3.2, 1.3], 1, 35, 118, { stroke: BLUE }); } else { LABEL = label([2.5, -0.5], "\\blue{\\angle{DHE}}= {?}", "below"); arc([3.2, 1.3], 1, 219, 286, { stroke: BLUE }); } // Draw A Star path([[-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5]]);

NOTE: Angles not drawn to scale.

ANGLE_3 180 - ANGLE_3 `\Large{{}^\circ}`

The measure of the angles in a triangle sum to `180^\circ`.

`\purple{\angle{BHD}} = 180^\circ - \green{\angle{BDC}} - \red{\angle{DBE}}`

// label angle BHD arc([3.2, 1.3], 1, 118, 220, { stroke: PURPLE }); graph.angle3 = label([2.4, 2], "\\purple{\\angle{BHD}}", "below left");

`\purple{\angle{BHD}} = 180^\circ - \green{ANGLE_2^\circ} - \red{ANGLE_1^\circ} = \purple{ANGLE_3^\circ}`

graph.angle3.remove(); label([2.4, 2], "\\purple{" + ANGLE_3 + "^\\circ}", "below left");

`\blue{\angle{CHE}}` and `\purple{\angle{BHD}}` are opposite angles are equal.

Therefore `\blue{\angle{CHE}} = \purple{\angle{BHD}} = ANGLE_3^\circ`.

LABEL.remove(); label([4.8, 1.0], "\\blue{\\angle{CHE}}=" + ANGLE_3 + "^\\circ", "right");

`\blue{\angle{CHG}}` `\blue{\angle{DHE}}` and `\purple{\angle{BHD}}` form a linear pair.

Therefore `\blue{\angle{CHG}} \blue{\angle{DHE}} = 180 ^\circ - \purple{ANGLE_3^\circ}`.

`\blue{\angle{GHC}}` LABEL.remove(); label([4, 2.5], "\\blue{\\angle{GHC}}=" + (180 - ANGLE_3) + "^\\circ", "above"); `\blue{\angle{DHE}}` LABEL.remove(); label([2.5, -0.5], "\\blue{\\angle{DHE}}=" + (180 - ANGLE_3) + "^\\circ", "below"); `= 180 - ANGLE_3^\circ`

rand(10) + 30 rand(10) + 70 180 - ANGLE_1 - ANGLE_2

Given the following:

- `\green{\angle{BDC}} = ANGLE_1^\circ`
- `\red{\angle{AIC}} = 180 - ANGLE_2^\circ`

- `\green{\angle{GCH}} = ANGLE_1^\circ`
- `\red{\angle{FGH}} = 180 - ANGLE_2^\circ`

What is `\blue{\angle{AJF}} {?}`

What is `\blue{\angle{IHE}} {?}`

init({ range: [[-10, 10], [-7, 10]], scale: [25, 25] }); // 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, 1.4], "H", "below right"); label([0, -1.3], "I", "below"); label([-3.2, 1.4], "J", "below left"); // Label Angles and X according to variation if( RAND_SWITCH2 == 0) { // Label the given angles label([-5.5, -5.2], "\\green{" + ANGLE_1 + "^\\circ}", "above right"); arc([-6, -6], 1.3, 39, 69, { stroke: GREEN }); label([0, -0.4], "\\red{" + (180 - ANGLE_2) + "^\\circ}", "above"); arc([0, -1.2], 0.75, 32, 145, { stroke: RED }); // Label X LABEL = label([-3.7, 2.5], "\\blue{\\angle{AJF}}= {?}", "above"); arc([-3.2, 1.3], 1, 65, 142, { stroke: BLUE }); } else { // Label the given angles label([6.5, 5], "\\green{" + ANGLE_1 + "^\\circ}", "below left"); arc([8, 5], 1.3, 180, 220, { stroke: GREEN }); label([1.3, 4.5], "\\red{" + (180 - ANGLE_2) + "^\\circ}", "below left"); arc([1.7, 5], 1, 180, 289, { stroke: RED }); // Label X LABEL = label([4.0, -0.3], "\\blue{\\angle{IHE}}= {?}", "below left"); arc([3.1, 1.2], 1, 220, 290, { stroke: BLUE }); } // Draw A Star path([[-8, 5], [8, 5], [-6, -6], [0, 9], [0, 9], [6,-6], [-8, 5]]);

NOTE: Angles not drawn to scale.

ANGLE_3 `\Large{{}^\circ}`

`\purple{\angle{DIJ}}` forms a linear pair with `\red{\angle{AIC}}`.

Therefore, `\purple{\angle{DIJ}} = 180^\circ - \red{180 - ANGLE_2^\circ}`.

// label angle JID arc([0, -1.2], 0.75, 143, 220, { stroke: PURPLE }); label([-.75, -1.2], "\\purple{" + ANGLE_2 + "^\\circ}", "left");

The measure of the angles in a triangle sum to `180^\circ`.

Therefore, `\pink{\angle{DJI}} = 180^\circ - \green{\angle{BDC}} - \purple{\angle{DIJ}}`.

// label angle JID arc([-3.2, 1.3], 1, 256 , 318, { stroke: PINK }); graph.DJI = label([-3.2, 0.50], "\\pink{\\angle{DJI}}", "below right");

`\pink{\angle{DJI}} = 180^\circ - \green{ANGLE_1^\circ} - \purple{ANGLE_2^\circ} = \pink{ANGLE_3^\circ}`

graph.DJI.remove(); label([-3.2, 0.50], "\\pink{" + ANGLE_3 + "^\\circ}", "below right");

`\blue{\angle{AJF}}` and `\pink{\angle{DJI}}` are opposite angles.

Therefore, `\angle{AJF} = \angle{DJI} = ANGLE_3^\circ`.

LABEL.remove(); label([-3.7, 2.5], "\\blue{\\angle{AJF}}=" + ANGLE_3 + "^\\circ", "above");

`\purple{\angle{HGC}}` forms a linear pair with `\red{\angle{FGH}}`.

Therefore, `\purple{\angle{HGC}} = 180^\circ - \red{180 - ANGLE_2^\circ}`.

// label angle HGC arc([1.7, 5], 1, 285, 0, { stroke: PURPLE }); label([2.4, 4.6], "\\purple{" + ANGLE_2 + "^\\circ}", "below right");

The measure of the angles in a triangle sum to `180^\circ`.

Therefore, `\pink{\angle{CHG}} = 180^\circ - \green{\angle{ACD}} - \purple{\angle{HGC}}`.

// label angle CHG arc([3.2, 1.3], 0.75, 35, 118, {stroke: PINK }); graph.CHG = label([3.5, 1.8], "\\pink{\\angle{CHG}}", "above");

`\pink{\angle{CHG}} = 180^\circ - \green{ANGLE_1^\circ} - \purple{ANGLE_2^\circ} = \pink{ANGLE_3^\circ}`

graph.CHG.remove(); label([3.5, 1.8], "\\pink{" + ANGLE_3 + "^\\circ}", "above");

`\blue{\angle{IHE}}` and `\pink{\angle{CHG}}` are opposite angles.

Therefore, `\angle{IHE} = \pink{\angle{CHG}} = ANGLE_3^\circ`.

LABEL.remove(); label([4.0, -0.3], "\\blue{\\angle{IHE}}=" + ANGLE_3 + "^\\circ", "below left");