SAT Congruence and Similarity
To successfully solve problems about congruent and similar figures on the SAT, you need to know how to:
- identify congruent triangles
- identify and compare similar triangles
- identify similar polygons
- find area and perimeter relations in similar polygons
Examples
Which list of corresponding parts in two triangles does NOT guarantee that the two triangles are congruent?
(A)\(\ \ \) Two sides and the included angle
(B)\(\ \ \) Two angles and the included side
(C)\(\ \ \) Two angles and a non-included side
(D)\(\ \ \) Three corresponding angles
(E)\(\ \ \) Three corresponding sides
Correct Answer: D
Solution:
Tip: Know how to prove triangles are congruent.
If three angles in one triangle are congruent to three angles in another triangle, then the two triangles will be similar, but not necessarily congruent.
Incorrect Choices:
(A), (B), (C), and (E)
SAS, ASA, AAS, and SSS are all ways to prove two triangles are congruent.
Which of the following figures is similar to quadrilateral \(ABCD?\)
Correct Answer: E
Solution:
Two polygons are similar if
1) the corresponding sides of each are in the same proportion and
2) the corresponding interior angles are congruent.Looking through the choices, only choice (E) offers a figure which satisfies both statements above.
Both figures are rectangles -- in a rectangle, each internal angle is \(90^\circ\) -- and therefore corresponding angles in the two figures are congruent.
The ratio between the given corresponding sides in the two figures is the same: \(\frac{2}{1} = \frac{3}{1.5} = 2.\) And since the figures are rectangles, this is true for all corresponding sides.
Incorrect Choices:
(A)
Here corresponding sides aren't proportional. The short sides have a ratio \(\frac{2}{2} = 1\) but the long sides have a ratio \(\frac{3}{4} \neq 1.\)(B)
Here corresponding sides aren't proportional. The short sides have a ratio \(\frac{2}{2} = 1\) but the long sides have a ratio \(\frac{3}{2} \neq 1.\)(C)
Here, corresponding angles are not congruent. \(60\neq 90\) and \(120\neq 90.\)(D)
Here, corresponding angles are not congruent. \(91\neq 90.\)
In the figure above, the three squares have the same center. What is the ratio of the perimeter of the outermost square to the perimeter of the innermost square?
(A) \(\ \ 1\)
(B) \(\ \ 1.5\)
(C) \(\ \ 2\)
(D) \(\ \ 3\)
(E) \(\ \ 9\)
Correct Answer: D
Solution 1:
Tip: If two figures are similar, and their scale factor is \(a:b,\) then the ratio of their perimeters is \(a:b\) and the ratio of their areas is \(a^2:b^2\).
The side of the smallest circle is \(2\cdot 2= 4\) and the side of the largest circle is \((2+2+2) \cdot 2 = 12.\)Since squares are similar figures, \(\frac{P_{\text{outermost}}}{P_{\text{innermost}}}=\frac{12}{4}= 3.\)
Solution 2:
Tip: The perimeter of a square with side length \(s\): \(P_{\square} = 4s.\)
The side of the innermost circle is \(2\cdot 2= 4\) and the side of the outermost circle is \((2+2+2) \cdot 2 = 12.\)The perimeter of the innermost square is \(P_{\text{innermost}} = 4\cdot 4 = 16.\)
The perimeter of the outermost square is \(P_{\text{outermost}}=4\cdot 12 =48.\)
Then,
\(\frac{P_{\text{outermost}}}{P_{\text{innermost}}}=\frac{48}{16}= 3.\)
Incorrect Choices:
(A)
This is just \(\frac{2}{2}=1.\) But as the diagram indicates, the innermost square is not congruent to the outermost square.(B)
This is the ratio of the perimeter of the outermost square to the perimeter of the middle square.(C)
This is the ratio of the perimeter of the middle square to the perimeter of the innermost square.(E)
This is the ratio of the area of the outermost square to the area of the innermost square.
Review
If you thought these examples difficult and you need to review the material, these links will help:
SAT Tips for Congruence and Similarity
- If two figures are similar, and their scale factor is \(a:b,\) then the ratio of their perimeters is \(a:b\) and the ratio of their areas is \(a^2:b^2\).
- Know how to prove triangles are similar.
- Know how to prove triangles are congruent.
- Know the Properties of Proportions.
- The circumference of a circle with radius \(r\) and diameter \(d: C = 2\pi r = \pi d.\)
- Area of a circle with radius \(r: A_{\bigodot} = \pi r^2.\)
- SAT General Tips