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BBC - KS2 Bitesize Maths - Shapes : Fullscreen

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Maths shape and space exercise - Shapes activity

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Triangles are congruent when they have
exactly the same three sides and exactly the same three angles.

What is "Congruent" ... ?

It means that one shape can become another using Turns, Flips and/or Slides:

(see Congruent for more info)

Congruent Triangles

When two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there.

Same Sides

When the sides are the same then the triangles are congruent.

For example:

is congruent to:

and also to:

because they all have exactly the same sides.

But:

is NOT congruent to:

because the two triangles do not have exactly the same sides.

Same Angles

Does this also work with angles? Not always!

Two triangles with the same angles might be congruent:

is congruent to:

only because they are the same size

But they might NOT be congruent because of different sizes:

is NOT congruent to:

because, even though all angles match, one is larger than the other.

So just having the same angles is no guarantee they are congruent.

Other Combinations

There are other combinations of sides and angles that can work ...
... read more at How To Find if Triangles are Congruent

Marking

When two triangles are congruent we often mark corresponding sides and angles like this:

is congruent to:

The sides marked with one line are equal in length. Similarly for the sides marked with two lines. Also for the sides marked with three lines.

The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs. Also for the angles marked with three arcs.

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Triangles are congruent when they have exactly the same three sides and exactly the same three angles. It means that one shape can become another using Turns, Flips and/or Slides: (see Congruent for more info) When two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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The square on the hypotenuse is equal to the sum of the squares on the other two sides.

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If the hypotenuse and one side of a right-angled triangle are congruent to the corresponding parts of another right triangle, the right-angled triangles are congruent.[br][br]Remember during SSS we also experience that the triangle became a right-angle triangle when the sides were Pythagorean Triples. Read more about Pythagorean Triples by clicking here:[br][br][list][*][url=http://www.mathsisfun.com/pythagorean_triples.html]Pythagorean Triples: Basics[/url]﻿[br][/*][/list][list][*][url=http://www.mathsisfun.com/numbers/pythagorean-triples.html]Pythagorean Triples: Advanced[/url]﻿[br][/*][/list][br][color=#0000ff][b]We can always use the Pythagorean Theorem to work out the third side and the use the SSS-Approach to construct this. [/b][/color][color=#ff0000][b]But, t[/b][/color][b][color=#ff0000]his is a special case of ASS that will be discussed in more detail under the section about problematic methods. (See the "Donkey Theorem" for more details)[/color][/b][br][br]Below is an example how to construct this. If you change anything in the construction, just click on the arrows on the top right to restore the construction.

[b][center][color=#ff0000]Now you try to draw a triangle congruent to the previous one﻿[/color][br][br][color=#38761d]You need to draw a triangle with side AB=8cm, right angle CAB and hypotenuse of 10cm. [br]Try to do this in the "Applet" below[/color][/center][/b][br] [br][list=1] [*]Use [icon]/images/ggb/toolbar/mode_segmentfixed.png[/icon] to draw segment AB and if you are requested to give the length type in 8[/*] [*]Use [icon]/images/ggb/toolbar/mode_circlepointradius.png[/icon] to draw a circle at point B and if requested to enter a radius type in 10[br][/*][*]Use [icon]/images/ggb/toolbar/mode_orthogonal.png[/icon] to draw a line perpendicular to line AB through point A. (Click on line AB and then point A.)[/*] [*]Use [icon]/images/ggb/toolbar/mode_intersect.png[/icon] to place point C at the intersection of the perpendicular line and the circle[/*] [*]Use [icon]/images/ggb/toolbar/mode_polygon.png[/icon] to draw triangle ABC[/*][/list]

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If the hypotenuse and one side of a right-angled triangle are congruent to the corresponding parts of another right triangle, the right-angled triangles are congruent. Remember during SSS we also experience that the triangle became a right-angle triangle when the sides were Pythagorean Triples.

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Visual proof of SAS triangle congruence using rigid transformations.

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Discover the properties of all Isosceles triangles.

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4) An exterior angleof a triangle= sum of its two interior opp. angles.

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4) An exterior angleof a triangle= sum of its two interior opp. angles.

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ii) An exterior angle of a triangle is greater than either of the two interior opp. angles

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Use this applet to discover the properties of the exterior angles of a triangle.

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Investigate properties of various kinds of triangles.