Plane geometry (polygons, triangles, angles) — ISEE Upper Quantitative Reasoning practice questions

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Practice 181 Plane geometry (polygons, triangles, angles) questions in the app

What this topic is

Perimeter, area, angle sums and triangle properties make up Plane geometry (polygons, triangles, angles).

Percent changes to dimensions are a favorite: if a length grows by one-half and a width shrinks by one-half, the area does not stay the same but becomes three-quarters of the original, since the two factors multiply. The interior angles of a triangle add to one hundred eighty degrees, and an exterior angle equals the sum of the two remote interior angles.

Comparisons may set a square beside a rectangle with an equal perimeter. Figures are not always drawn to scale, so measurements should come from the stated facts.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

A rectangle’s length is decreased by $20 %$ , and its width is increased by $50 %$ . The area of the new rectangle is what percent of the area of the original rectangle?

70 %

✗ Not this one

100 %

✗ Not this one

120 %

✓ Correct

130 %

✗ Not this one

Show answer

Answer: C — $120 %$

The new length is 0.8 of the original and the new width is 1.5 of the original. The area scale factor is $0.8 \times 1.5 = 1.2$ , so the new area is $120 %$ of the original.

Question 2Mid

A rectangle has length twice its width. A square has the same perimeter as the rectangle.

Column A	Column B
The area of the rectangle	The area of the square

The quantity in Column A is greater✗ Not this oneThe quantity in Column B is greater✓ CorrectThe two quantities are equal✗ Not this oneThe relationship cannot be determined from the information given✗ Not this one

Show answer

Answer: B — The quantity in Column B is greater

Let the rectangle’s width be $w$ , so its length is $2 w$ and its perimeter is $6 w$ . The square’s side length is $\frac{6 w}{4} = \frac{3 w}{2}$ . The rectangle’s area is $2 w^{2}$ , while the square’s area is $\frac{9 w^{2}}{4}$ , so Quantity B is greater.

Question 3Mid

In triangle $A B C$ , $m ∠ A = 5 0^{\circ}$ and $m ∠ B > m ∠ C$ .

Column A	Column B
$m ∠ B$	$6 5^{\circ}$

The quantity in Column A is greater✓ CorrectThe quantity in Column B is greater✗ Not this oneThe two quantities are equal✗ Not this oneThe relationship cannot be determined from the information given✗ Not this one

Show answer

Answer: A — The quantity in Column A is greater

Angles $B$ and $C$ sum to $18 0^{\circ} - 5 0^{\circ} = 13 0^{\circ}$ . If they were equal, each would be $6 5^{\circ}$ . Since $m ∠ B > m ∠ C$ , angle $B$ must be greater than $6 5^{\circ}$ , so Quantity A is greater.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 181 Plane geometry (polygons, triangles, angles) questions in the app

Plane geometry (polygons, triangles, angles) — ISEE Upper Quantitative Reasoning practice questions

What this topic is

Sample questions

More ISEE Upper topics

Quantitative Reasoning