randomTriangleAngles.triangle() 5 + random() * 2 randRange( 0, 1 ) === 1 ? true : false new Triangle([0, 0], ANGLES, SCALE, { points: ["A", "B", "C"] }) randRange( 0, 360 ) "#b1c9f5"

By clicking and dragging the points below, is it possible to make a triangle with lengths/angles different from the lengths/angles in `\triangle{ABC}?`.
Your triangle can be anywhere. There is no need to line up the two triangles.

init({ range: [[-6.2, 6.2], [-2, 10]] }); addMouseLayer(); initCongruence({ triangle: TRIANGLE, type: TYPE, reflected: REFLECTED }); TRIANGLE.rotate( ROTATION ); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); TRIANGLE.translate([ -5 - Math.min(TRIANGLE.points[0][0], TRIANGLE.points[1][0], TRIANGLE.points[2][0]), 9 - Math.max(TRIANGLE.points[0][1], TRIANGLE.points[1][1], TRIANGLE.points[2][1]) ]); path([ kline.midpoint(TRIANGLE.sides[2]), TRIANGLE.points[2], TRIANGLE.points[1], TRIANGLE.points[0], kline.midpoint(TRIANGLE.sides[2]) ]); addTriangleDecorations( TRIANGLE, TYPE ); TRIANGLE.color = FAINT_BLUE; TRIANGLE.drawLabels();

ANSWER

- Yes
- No

"SSS" "No"
"SAS" "No"
"SAA" "No"
"ASA" "No"
"SSA" "Yes"

With these constraints, there are two ways to construct a triangle. See if you can find both ways.

Both of these triangles have sides with lengths `\overline{AB}` and `\overline{BC}`, and an angle with the same measure as `\angle C`. However, angles `\angle A` and `\angle B` are not the same.

var triangle = new Triangle([0, 0], ANGLES, SCALE, { points: ["A", "B", "C"] }); triangle.rotate( -ANGLES[1] ); init({ range: triangle.boundingRange(1) }); addMouseLayer(); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); path([ kline.midpoint(triangle.sides[2]), triangle.points[2], triangle.points[1], triangle.points[0], kline.midpoint(triangle.sides[2]) ]); addTriangleDecorations(triangle, TYPE); triangle.color = FAINT_BLUE; triangle.drawLabels(); KhanUtil.currentGraph = $("div#congruent-triangles").data().graphie
var angle = (180 - (180 - ANGLES[0]) - ANGLES[2]) * PI / 180; var points = [ [TRIANGLE.sideLengths[1] - cos(angle) * TRIANGLE.sideLengths[0], -sin(angle) * TRIANGLE.sideLengths[0]], [TRIANGLE.sideLengths[ 1 ], 0], [0, 0] ]; var triangle = new Triangle( [], [], 0, { points: ["A", "B", "C"] }, points ); init({ range: triangle.boundingRange(1) }); addMouseLayer(); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); path([ kline.midpoint(triangle.sides[2]), triangle.points[2], triangle.points[1], triangle.points[0], kline.midpoint(triangle.sides[2]) ]); addTriangleDecorations(triangle, TYPE); triangle.color = FAINT_BLUE; triangle.drawLabels(); KhanUtil.currentGraph = $("div#congruent-triangles").data().graphie

So, yes, we can construct a triangle with the same angles as `\triangle ABC` but different side lengths.

"AAA" "Yes"

With these constraints, there is more than one way to construct a triangle. See if you can find some different ways.

Both of these triangles have the same three angles, but they have different side lengths:

style({ stroke: FAINT_BLUE, "stroke-width": 5 }); init({ range: TRIANGLE.boundingRange(1) }); addMouseLayer(); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); path([ kline.midpoint(TRIANGLE.sides[2]), TRIANGLE.points[2], TRIANGLE.points[1], TRIANGLE.points[0], kline.midpoint(TRIANGLE.sides[2]) ]); addTriangleDecorations( TRIANGLE, TYPE ); KhanUtil.currentGraph = $("div#congruent-triangles").data().graphie
var triangle = new Triangle([0, 0], ANGLES, SCALE - 4, {}); triangle.rotate( ROTATION ); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); init({ range: triangle.boundingRange(1) }); addMouseLayer(); style({ stroke: FAINT_BLUE, "stroke-width": 5 }); path([ kline.midpoint(triangle.sides[2]), triangle.points[2], triangle.points[1], triangle.points[0], kline.midpoint(triangle.sides[2]) ]); addTriangleDecorations(triangle, TYPE); KhanUtil.currentGraph = $("div#congruent-triangles").data().graphie

So, yes, we can construct a triangle different from `\triangle ABC` with these constraints.

No, with these constraints, it is not possible to make a triangle with different lengths or angles from `\triangle ABC`.