randFromArray([ "SSS", "SAS", "SAA", "ASA", "SSA", "SSA", "AAA" ]) { "SSS": $._("Side-Side-Side"), "SAS": $._("Side-Angle-Side"), "SAA": $._("Side-Angle-Angle"), "ASA": $._("Angle-Side-Angle"), "SSA": $._("Side-Side-Angle"), "AAA": $._("Angle-Angle-Angle") }[ TYPE ] randomTriangleAngles.triangle() 5 + random() * 2 randRange( 0, 1 ) === 1 ? true : false { "SSS": "Yes", "SAS": "Yes", "SAA": "Yes", "ASA": "Yes", "SSA": "No", "AAA": "No" }[ TYPE ] new Triangle( [ 0, 0 ], ANGLES, SCALE, {} ) randRange( 0, 360 )

Is NAME a triangle congruency postulate?

Answer the question by clicking and dragging the points below to see how many different triangles you can construct. If you can only construct congruent triangles, then the NAME postulate is true. The NAME postulate is not true if you can construct two or more different triangles using the given information.

Your triangle can be anywhere. There is no need to line up the two triangles.
init({ range: [ [ -6.2, 6.2 ], [ -5.9, 6.5 ] ] }); addMouseLayer(); initCongruence({ triangle: TRIANGLE, type: TYPE, reflected: REFLECTED }); TRIANGLE.rotate( ROTATION ); style({ stroke: "#b1c9f5", "stroke-width": 5 }); TRIANGLE.translate([ -5 - Math.min(TRIANGLE.points[0][0], TRIANGLE.points[1][0], TRIANGLE.points[2][0]), 6 - 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 );

- Yes — And I've constructed a congruent triangle, and this is the ONLY triangle I can construct with these conditions.
- No — And I've proven it by constructing an incongruent triangle (a counterexample).

[ $( "#solutionarea" ).find( "input:checked" ).val(), interactiveTriangle.points[0].coord, interactiveTriangle.points[1].coord, interactiveTriangle.points[2].coord, interactiveTriangle.points[3].coord ]
var saved = $.map( new Array( 4 ), function( el, n ) { return [ interactiveTriangle.points[ n ].coord ]; }); $.map( guess.slice( 1 ), function( el, n ) { interactiveTriangle.points[ n ].setCoord( el ); }); interactiveTriangle.update(); var isTriangle = interactiveTriangle.isTriangle; var isCongruent = abs( getDistance( guess[1], guess[2] ) - TRIANGLE.sideLengths[ 0 ] ) < 0.001 && abs( getDistance( guess[2], guess[3] ) - TRIANGLE.sideLengths[ 1 ] ) < 0.001 && abs( getDistance( guess[3], guess[4] ) - TRIANGLE.sideLengths[ 2 ] ) < 0.001; $.map( saved, function( el, n ) { interactiveTriangle.points[ n ].setCoord( el ); }); interactiveTriangle.update(); if ( guess[0] == null ) { return ""; } else if ( guess[0] !== ANSWER ) { return false; } else if ( !isTriangle ) { return $._("Your answer is almost correct, but you havent constructed a triangle."); } else if ( ANSWER === "No" && isCongruent ) { return $._("Your answer is almost correct, but the two triangles are congruent. Prove your answer by trying to construct an incongruent triangle."); } else { return true; }
$.map( guess.slice( 1 ), function( el, n ) { interactiveTriangle.points[ n ].setCoord( el ); }); interactiveTriangle.update();
$( "#solutionarea" ).find( "input:checked" ).prop( 'checked', false ); if ( guess[0] != null ) { $( "#solutionarea" ).find( "input[value=" + guess[0] + "]" ).prop( 'checked', true ); }

To be a congruency postulate, there must be one, and only one, way to make a triangle that's the same as the original triangle—except for being moved, rotated, or reflected.

With the constraints of NAME, there is exactly one way to make a triangle.

NAME is a congruency postulate. Be sure to construct the congruent triangle and think about why you're only able to construct the triangle in one way.

With the constraints of NAME, there is more than one way to construct a triangle. See if you can find both ways.

Both of these triangles have the same adjacent Side, Side, and Angle, but they are not congruent:

var triangle = new Triangle( [ 0, 0 ], ANGLES, SCALE, {} ); triangle.rotate( -ANGLES[1] ); init({ range: triangle.boundingRange( 0.4 ) }); addMouseLayer(); style({ stroke: "#b1c9f5", "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 points = [ [ TRIANGLE.sideLengths[ 1 ] - cos( ( 180 - ( 180 - ANGLES[ 0 ] ) - ANGLES[ 2 ] ) * PI / 180 ) * TRIANGLE.sideLengths[ 0 ], -sin( ( 180 - ( 180 - ANGLES[ 0 ] ) - ANGLES[ 2 ] ) * PI / 180 ) * TRIANGLE.sideLengths[ 0 ] ], [ TRIANGLE.sideLengths[ 1 ], 0 ], [ 0, 0 ] ]; var triangle = new Triangle( [], [], 0, {}, points ); init({ range: triangle.boundingRange( 0.4 ) }); addMouseLayer(); style({ stroke: "#b1c9f5", "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

Because we can create two triangles that are not congruent, we can show by counterexample that Side-Side-Angle is not a congruency postulate. Be sure to construct the incongruent triangle above to prove it.

With the constraints of NAME, 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 are not congruent (just similar):

style({ stroke: "#b1c9f5", "stroke-width": 5 }); init({ range: TRIANGLE.boundingRange( 0.4 ) }); addMouseLayer(); style({ stroke: "#b1c9f5", "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: "#b1c9f5", "stroke-width": 5 }); init({ range: triangle.boundingRange( 0.4 ) }); addMouseLayer(); style({ stroke: "#b1c9f5", "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

Because we can create triangles that are not congruent, we can show by counterexample that Angle-Angle-Angle is not a congruency postulate. Be sure to construct an incongruent triangle above to prove it.