Why not aas and ssa used for cougruence rule.
Why not aas and ssa used for cougruence rule.
Why SSA and AAA Don't Work as Congruence Shortcuts?
There are four shortcuts allow students to know two triangles must be congruent: SSS, SAS, ASA, and AAS.
However, knowing only Side-Side-Angle (SSA) does not work because the unknown side could be located in two different places.
Knowing only Angle-Angle-Angle (AAA) does not work because it can produce similar but not congruent triangles.
Triangle Congruence - why SSA doesn't always work An exploration of the concept of side-side-angle (SSA) congruence for triangles
AAS, unlike ASA, is not unoversal when you consider non-Euclidean geometries.
Imagine the Earth is a sphere. Draw the meridian from the North Pole NNthrough London LL to the South Pole SS. Construct the great circle perpendicular to NLSNLS through LL, drawing this route westwards until it meets the Equator at pount MMwest of South America. Then you construct median NMSNMS.
You now have two triangles LMNLMNand LMSLMS that satisfy AASAAS, as each triangle has two right angles and a common side not between the right angles. But the third angles are different because London is not on the Equator, so the triangles are not congruent.
The Greeks used geometry and trigonometry to model the Earth and sky as well as an idealized plane, so the failure of AASAAS on spheres was a big deal.
There are four shortcuts allow students to know two triangles must be congruent: SSS, SAS, ASA, and AAS.
However, knowing only Side-Side-Angle (SSA) does not work because the unknown side could be located in two different places.
Knowing only Angle-Angle-Angle (AAA) does not work because it can produce similar but not congruent triangles.
Triangle Congruence - why SSA doesn't always work An exploration of the concept of side-side-angle (SSA) congruence for triangles
AAS, unlike ASA, is not unoversal when you consider non-Euclidean geometries.
Imagine the Earth is a sphere. Draw the meridian from the North Pole NNthrough London LL to the South Pole SS. Construct the great circle perpendicular to NLSNLS through LL, drawing this route westwards until it meets the Equator at pount MMwest of South America. Then you construct median NMSNMS.
You now have two triangles LMNLMNand LMSLMS that satisfy AASAAS, as each triangle has two right angles and a common side not between the right angles. But the third angles are different because London is not on the Equator, so the triangles are not congruent.
The Greeks used geometry and trigonometry to model the Earth and sky as well as an idealized plane, so the failure of AASAAS on spheres was a big deal.
