Which one of the following pairs of triangle must be similar?
a) two isosceles triangles with congruent vertex angles
b) two right triangles
c) two scalene triangles with congruent bases
d) two obtuse triangles
a) if the two vertex angles are the same then the other two angles of both both triangles have to be the same and identical since all three have to add to 180 then since the two base angles are the same the sides opposite them are congruent and the two triangles are similar be SAS(side angle side)
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To determine which pair of triangles must be similar, we need to understand the conditions for triangle similarity.
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.
Let's analyze each pair of triangles:
a) Two isosceles triangles with congruent vertex angles:
If the vertex angles of the isosceles triangles are congruent, it means that the base angles are also congruent. Since all three angles of a triangle must add up to 180 degrees, the remaining angle in each triangle must be congruent. Therefore, both the corresponding angles and the corresponding sides are congruent, leading to similarity.
b) Two right triangles:
Right triangles can be similar if one angle is congruent (and the other two angles are not necessarily congruent) and the corresponding sides are proportional. However, right triangles can also have varying proportions of sides, making them not automatically similar. Therefore, two right triangles are not necessarily similar.
c) Two scalene triangles with congruent bases:
Congruent bases in scalene triangles do not ensure similarity. The congruency of bases only guarantees that one pair of corresponding sides is congruent, but the remaining sides may not be proportional. Therefore, two scalene triangles with congruent bases are not necessarily similar.
d) Two obtuse triangles:
Obtuse triangles can also be similar if their corresponding angles are congruent and their corresponding sides are proportional. The fact that the triangles are obtuse does not affect their similarity.
Based on the analysis, the pair of triangles that must be similar is:
a) Two isosceles triangles with congruent vertex angles.