A. One triangle has angles measuring, in degrees, 63 and 80. The third angle measures: 180 - 63 - 80 = 37 Since one angle of the first triangle measures 37, and an angle of the other triangle measures 27, the triangles cannot be similar.
B.
Both triangles are isosceles.The angles opposite the congruent sides are congruent angles.
The upper left triangle has an angle measuring 36 deg.The sum of the measures of the other two angles is 180 - 36 = 144.The base angles are congruent, so each base angle measures 72.
The lower right triangle is also isosceles.One base angle measures 72, so the other base angle also measure 72 deg. There are two 72-deg angles in both triangles, so by AA Similarity, the triangles are similar.
C. If polygons are similar, then the ratios of the lengths of corresponding sides are equal.For the quadrilaterals to be congruent, the ratios 6/4 and 4/2 would have to be equal. 6/4 = 1.5 4/2 = 2
1.5 is not equal to 2, so the ratios are not equal, and the quadrilaterals are not similar.
D. All squares are similar. A square and a non-square rectangle are never similar since the ratios of the lengths of corresponding sides are not equal. The upper quadrilateral is a square, and the lower quadrilateral is a non-square rectangle. They are not similar.
Let S=larger square side and s=smaller square side. The area between the larger and smaller is simple the larger area minus the smaller area. The area of any square being s^2. So our remaining area is:
