## 1.

*ABC*and ∆

*ACD*are congruent, you know that each angle from one triangle corresponds to an angle of the other triangle. If you are having a hard time visualizing the triangles matching up, mark up the given diagram or draw your own to show the triangles side by side:

Just by filling in the angles, you can see that the remaining angle in each triangle must be 60°. To find ∠*BCD*, simply sum ∠*BCA* and ∠*ACD* to get a total of 60° + 90° = 150°.

## 2.

*l*+ 2

*w*, because rectangles have two pairs of congruent sides and perimeter is simply their sum. Once you have the formula down, you can solve for the remaining side:

*wl*= 250 meters × 350 meters = 87,500 square meters.

Alternatively, you know that one of the longer sides is 350 meters, so the area can’t be larger than 350 × 350 = 122,500 square meters. The only answers that are less than this maximum value are (J) and (K). (K) is much too small to be a possible area, so you have found the correct answer, (J), with none of the algebra.

If you chose (K), you probably found the length of the shorter side and stopped there.

## 3.

Since one side has a length of 9 millimeters, you know that the opposite side also has a length of 9 millimeters. To find the length of the bottom side (which will be the same as the top) sum all the line segments parallel to it: 5 + 2 + 1 + 3 = 11 millimeters. Therefore, the perimeter of the entire figure is 2(9) + 2(11) = 40 millimeters.

## 4.

Since and are parallel lines intersected by two transversals, you know that the alternate interior angles are congruent. To find angle ∠*DFG*, look for its corresponding alternate interior angle, ∠*ADF*, given in the question as 50°.

Angles ∠*BFE*, ∠*BFD*, and ∠*DFG* all intersect at point *F* on a straight line, , so they are supplementary angles, meaning they add up to 180°. Inputting the expressions given for the two unknown angles, you can write out an equation solving for *x*:

## 5.

*r*, or π

*d*, where

*d*is the diameter:

*d*= 3.14 × 2 = 6.28 hours

If you got (C), you probably forgot to convert the leftover 0.28 hours to minutes, and left it as is.

## 6.

You can eliminate (F), because ∠*ABC* and ∠*BCD* are alternate interior angles, formed by the transversal intersecting parallel lines and .

You can also eliminate (H), because ∠*BDC* and ∠*BCD* are opposite the two equal legs of an isosceles triangle so they are congruent as well.

If ∠*BDC* = ∠*BCD*, and ∠*ABC* = ∠*BCD*, then ∠*BDC* must also be equal to ∠*ABC*, eliminating answer choice (G).

Finally, you can eliminate (J), because ∠*ACD* and ∠*CAB* are both right angles formed by the line , which is perpendicular to the two parallel lines it crosses.

This leaves you with only one answer choice, (K). These two angle pairs are not congruent because, following from our above analysis, ∠*ACB* is 90° – ∠*BDC*. The two would only be equal if they were both 45°, and you cannot make that assumption.

## 7.

If you chose (E), you probably left the height of the basketball net in inches.

If you picked (D), you probably divided the original height by 3 instead of multiplying.

## 8.

Marking the diagram makes it clear that, if you subtract the length of from the sum of the two overlapping line segments, you get the overlap itself, : (32 + 26) – 50 = 58 – 50 = 8 units.

## 9.

If you got (E), you probably thought the side length of the triangle was 64 units and misread the question.

## 10.

*x*, and thus find the value of

*x*:

If you chose (K), you probably forgot to square-root the value for *x*^{2}.

## 11.

## 12.

If you chose (F), you found the perimeter of the smaller triangle.

## 13.

*V*from the second equation into the first, you can then replace the

_{A}*V*with the formula for volume of a sphere:

*B*, and solve:

## 14.

Failing to remember your special triangles, you could draw out the pizza and solve for the diagonal using the Pythagorean Theorem:

## 15.

## 16.

^{2}, you know that they are all congruent. Remember that the area of a square is equal to its side length squared, so working backwards, each square has side lengths of meters. Points

*E*,

*C*, and

*G*are in the middle of the first, second, and third square respectively, so it follows that they bisect the sides of these squares; each of their overlapping lengths are therefore 9.5 meters. Labelling this on the diagram, it becomes clear that each “corner” piece has the same length of one of the sides, 19 meters:

## 17.

## 18.

*a*to 360° (the angle of a full circle) is equal to the ratio of the sector’s area to the area of the circle. Writing this out, you can solve for

*a*without even needing to calculate the area of the circle:

## 19.

## 20.

If you chose (F), you aligned the 6” side of the box with the 32” side of the shelf, an orientation that can only fit boxes.

If you chose (H), you aligned the 6” side of the box with the 42” side of the shelf, a scenario that would fit only boxes.