QuestionDetermine if the following polygon pairs are always, sometimes, or never similar: two obtuse triangles a trapezoid and a parallelogram two right triangles two isosceles triangles a scalene triangle and an isosceles triangle two equilateral triangles
Studdy Solution
STEP 1
Assumptions
1. Two polygons are similar if their corresponding angles are equal and their corresponding sides are in proportion.
2. An obtuse triangle is a triangle with one angle greater than 90 degrees.
3. A trapezoid is a quadrilateral with at least one pair of parallel sides.
4. A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
5. A right triangle is a triangle with one 90-degree angle.
6. An isosceles triangle is a triangle with at least two sides of equal length.
7. A scalene triangle is a triangle with all sides of different lengths.
8. An equilateral triangle is a triangle with all three sides of equal length.
STEP 2
For two obtuse triangles to be similar, they must have their corresponding angles equal and their corresponding sides in proportion. While all obtuse triangles share the property of having one angle greater than 90 degrees, their other angles and side lengths can vary. Therefore, two obtuse triangles are not always similar, but they can be similar if their angles and sides are in the correct proportion.
STEP 3
A trapezoid and a parallelogram have different properties regarding their sides and angles. A trapezoid has only one pair of parallel sides, while a parallelogram has both pairs of opposite sides parallel. Because their fundamental properties are different, a trapezoid and a parallelogram cannot be similar.
STEP 4
Two right triangles are similar if their corresponding angles are equal. Since all right triangles have one 90-degree angle, they only need to have one other corresponding angle equal for them to be similar (as the third angle will automatically be equal due to the angle sum property of triangles). Therefore, two right triangles are sometimes similar when their non-right angles are in correspondence.
STEP 5
Two isosceles triangles are similar if their corresponding angles are equal. Since isosceles triangles have at least two equal angles (the angles opposite the equal sides), they can be similar if the angles and the sides opposite to them are in the correct proportion. Therefore, two isosceles triangles are sometimes similar.
STEP 6
A scalene triangle and an isosceles triangle generally have different side lengths and angles. A scalene triangle has no equal sides or angles, while an isosceles triangle has at least two equal sides and two equal angles. It is possible for a scalene triangle to have angles that match those of an isosceles triangle, but this is not generally the case. Therefore, a scalene triangle and an isosceles triangle are sometimes similar, but this is not typical.
STEP 7
Two equilateral triangles are always similar because all their angles are equal (each angle measures 60 degrees), and their sides are in proportion (all sides are equal in length). Therefore, any two equilateral triangles will always be similar.
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