Apply Shape Properties - Praxis Core Math

In parallelogram EFGH, the measure of angle E is $(3x + 10)°$ and the measure of angle G is $(5x - 30)°$ . What is the measure of angle F?
Answer: 110° In a parallelogram, opposite angles are equal in measure. Therefore, $m\angle E = m\angle G$ . We can set up the equation: $3x + 10 = 5x - 30$ . To solve for $x$ , subtract $3x$ from both sides: $10 = 2x - 30$ . Add 30 to both sides: $40 = 2x$ . Thus, $x = 20$ . Now, find the measure of angle E: $m\angle E = 3(20) + 10 = 60 + 10 = 70°$ . In a parallelogram, consecutive angles are supplementary (add up to 180°). Therefore, $m\angle E + m\angle F = 180°$ . Substituting the value for angle E: $70° + m\angle F = 180°$ . Solving for angle F gives $m\angle F = 110°$ .
The lengths of the sides of a triangle are 9, 40, and 41. What is the sum of the measures of the two smallest angles of the triangle?
Answer: 90° First, determine the type of triangle by checking the Pythagorean theorem. The two shorter sides are 9 and 40, and the longest side is 41. Check if $9^2 + 40^2 = 41^2$ . $81 + 1600 = 1681$ , and $41^2 = 1681$ . Since the condition is met, the triangle is a right triangle, and the angle opposite the hypotenuse (the side of length 41) is 90°. The sum of all three angles in a triangle is 180°. The two smallest angles are the two acute angles. Their sum is $180° - 90° = 90°$ .
In the figure shown, equilateral triangle $ABC$ has side length $12$ . Point $D$ is on $\overline{BC}$ such that $BD=4$ . What is the length of $\overline{AD}$ ?
Answer: $4\sqrt{7}$ Drop a perpendicular from $A$ to $\overline{BC}$ meeting at midpoint $M$ (since triangle is equilateral), so $BM=6$ and $AM=6\sqrt{3}$ . Then $DM=BM-BD=6-4=2$ , and $AD=\sqrt{AM^2+DM^2}=\sqrt{108+4}=\sqrt{112}=4\sqrt{7}$ . Alternatively, use the Law of Cosines in triangle $ABD$ : $AD^2=12^2+4^2-2(12)(4)\cos 60°=144+16-48=112$ . (B) $8\sqrt{2}=\sqrt{128}$ uses cos 60° = ½ sign wrong. (C) $4\sqrt{3}$ is the altitude length mistaken. (D) $2\sqrt{37}=\sqrt{148}$ uses $+48$ instead of $-48$ .
Refer to the figure. In the figure, $\triangle ABC\sim\triangle ADE$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . If $AD=4$ , $DB=6$ , and the area of trapezoid $DBCE$ is $84$ , what is the area of $\triangle ADE$ ?
Answer: $16$ The ratio of similarity is $AD:AB=4:10=2:5$ , so the ratio of areas is $4:25$ . Let the area of $\triangle ADE=4k$ ; then area of $\triangle ABC=25k$ and area of trapezoid $DBCE=25k-4k=21k=84$ , giving $k=4$ . Area of $\triangle ADE=4k=16$ . (B) 24 uses linear ratio $2/5$ of 84−... error. (C) 33.6 uses ratio 2:5 on 84 directly. (D) 56 uses $21k$ wrong subtraction.
A triangle has two sides of length 7 cm and 12 cm. If the length of the third side, $x$ , is an integer, what is the difference between the maximum and minimum possible values of $x$ ?
Answer: 12 According to the Triangle Inequality Theorem, the length of the third side $x$ must be greater than the difference of the other two sides and less than their sum. So, $12 - 7 < x < 12 + 7$ , which simplifies to $5 < x < 19$ . Since $x$ must be an integer, the minimum possible value for $x$ is 6 and the maximum possible value is 18. The difference between the maximum and minimum values is $18 - 6 = 12$ .
Which of the following conditions is sufficient to prove that a parallelogram is a rhombus, but NOT sufficient to prove that it is a square?
Answer: The diagonals are perpendicular. If the diagonals of a parallelogram are perpendicular, it is a rhombus. This condition alone does not guarantee the angles are 90°, so it is not sufficient to prove it is a square. A is incorrect because congruent diagonals in a parallelogram prove it is a rectangle. C is incorrect because if one angle in a parallelogram is a right angle, all angles must be right angles, proving it is a rectangle. D is incorrect because if the area is $s^2$ , it implies the height is equal to the side, which means it must be a rectangle. If it's a parallelogram and a rectangle, it might not be a rhombus. Perpendicular diagonals is the specific property that elevates a parallelogram to a rhombus.
Each interior angle of a regular polygon measures 150°. How many diagonals does this polygon have?
Answer: 54 First, find the number of sides, $n$ . The formula for the measure of an interior angle of a regular n-gon is $\frac{(n-2) \times 180}{n}$ . Set this equal to 150: $\frac{(n-2) \times 180}{n} = 150$ . Multiply both sides by $n$ : $180(n-2) = 150n$ . Distribute: $180n - 360 = 150n$ . Subtract $150n$ from both sides: $30n - 360 = 0$ . Add 360 to both sides: $30n = 360$ . Divide by 30: $n = 12$ . The polygon is a dodecagon. The formula for the number of diagonals in an n-gon is $D = \frac{n(n-3)}{2}$ . Substitute $n=12$ : $D = \frac{12(12-3)}{2} = \frac{12(9)}{2} = \frac{108}{2} = 54$ .
A quadrilateral has diagonals that are congruent, but they do not bisect each other. What is the most specific name for this quadrilateral?
Answer: Isosceles Trapezoid Let's analyze the properties. In a parallelogram, diagonals bisect each other. Since the diagonals do not bisect each other, it cannot be a parallelogram, and therefore cannot be a rectangle, rhombus, or square. An isosceles trapezoid has congruent diagonals. In a general isosceles trapezoid, the diagonals do not bisect each other. Therefore, the most specific name for a quadrilateral with these properties is an isosceles trapezoid.
The measure of an exterior angle of a triangle is 110°. One of the non-adjacent interior angles measures 40°. Which of the following correctly classifies the triangle?
Answer: Isosceles and acute The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Let the interior angles be A, B, and C. Let the exterior angle at vertex C be 110°. Then $A + B = 110°$ . We are given one of these is 40°, let's say $A = 40°$ . Then $40° + B = 110°$ , which means $B = 70°$ . The interior angle at vertex C is supplementary to the exterior angle, so $C = 180° - 110° = 70°$ . The three interior angles are 40°, 70°, and 70°. Since two angles are equal, the triangle is isosceles. Since all angles are less than 90°, the triangle is acute.
In parallelogram ABCD, the diagonals intersect at point E. If AE = $2x + 3$ and EC = $4x - 9$ , what is the length of the diagonal AC?
Answer: 30 A property of parallelograms is that their diagonals bisect each other. This means that segment AE must be equal in length to segment EC. Set up the equation: $2x + 3 = 4x - 9$ . Subtract $2x$ from both sides to get $3 = 2x - 9$ . Add 9 to both sides to get $12 = 2x$ . Solve for $x$ to find $x = 6$ . The question asks for the length of the entire diagonal AC. First, find the length of AE by substituting $x=6$ : $AE = 2(6) + 3 = 12 + 3 = 15$ . Since the diagonals bisect each other, $AC = 2 \times AE = 2 \times 15 = 30$ .
Which of the following sets of side lengths could form an isosceles right triangle?
Answer: 6, 6, $6\sqrt{2}$ An isosceles right triangle must satisfy two conditions: two sides must be equal (isosceles), and the side lengths must satisfy the Pythagorean theorem $a^2 + b^2 = c^2$ (right triangle). A: It is isosceles, but $5^2 + 5^2 = 50 \neq 8^2 = 64$ . B: It is a right triangle ( $6^2+8^2=100=10^2$ ), but it is not isosceles. C: It is isosceles with two sides of length 6. Let's check the Pythagorean theorem: $6^2 + 6^2 = 36 + 36 = 72$ . The third side squared is $(6\sqrt{2})^2 = 36 \times 2 = 72$ . Since $6^2 + 6^2 = (6\sqrt{2})^2$ , it is a right triangle. D: This represents the side ratios of a 30-60-90 triangle, which is a right triangle but not isosceles.
Refer to the figure. In the figure, $ABCD$ is a square with side length $10$ , and semicircles are drawn with $\overline{AB}$ and $\overline{CD}$ as diameters, both opening into the interior of the square. What is the area of the region inside the square but outside both semicircles?
Answer: $100-25\pi$ Each semicircle has radius $5$ , so area $\frac{1}{2}\pi(5)^2=\frac{25\pi}{2}$ . The two semicircles meet at the center of the square (both diameters are 10, and the square has side 10, so each semicircle extends 5 units into the square — exactly reaching the center line). The two semicircles do not overlap; they are tangent at the center. Total area of semicircles $=2\cdot\frac{25\pi}{2}=25\pi$ . Region outside both $=100-25\pi$ . (B) forgets to halve the full circle areas. (C) halves the square area erroneously. (D) halves 25π incorrectly.
A square is inscribed in a circle. Which of the following represents the ratio of the length of the square's diagonal to the circle's radius?
Answer: $2:1$ When a square is inscribed in a circle, the vertices of the square lie on the circle. The diagonal of the square is a line segment that passes through the center of the circle and connects two opposite vertices. This segment is also a diameter of the circle. The diameter of a circle is twice its radius ( $d=2r$ ). Therefore, the length of the square's diagonal is equal to the circle's diameter. The ratio of the square's diagonal to the circle's radius is the ratio of $2r$ to $r$ , which simplifies to $2:1$ .
A triangle has angles measuring $x°$ , $2x°$ , and $3x°$ . Which of the following is a true statement about the triangle?
Answer: It is a right triangle. The sum of the angles in a triangle is 180°. Setting up the equation: $x + 2x + 3x = 180$ . Combining like terms: $6x = 180$ . Dividing by 6: $x = 30$ . Substituting back: the angles are 30°, 60°, and 90°. Since one angle measures exactly 90°, this is a right triangle.
The side lengths of a triangle are given by $x$ , $x+4$ , and $2x-1$ . Which of the following is a possible value for $x$ ?
Answer: 3.0 For a valid triangle, all side lengths must be positive, and the sum of the lengths of any two sides must be greater than the length of the third side. First, ensure positive side lengths: $x > 0$ , $x+4 > 0$ (true for $x>0$ ), and $2x-1 > 0$ (so $x > 0.5$ ). The strictest condition is $x > 0.5$ . Now, apply the Triangle Inequality Theorem: 1) $x + (x+4) > 2x-1 \implies 2x+4 > 2x-1 \implies 4 > -1$ (This is always true). 2) $x + (2x-1) > x+4 \implies 3x-1 > x+4 \implies 2x > 5 \implies x > 2.5$ . 3) $(x+4) + (2x-1) > x \implies 3x+3 > x \implies 2x > -3 \implies x > -1.5$ (This is redundant). The combined conditions are $x > 0.5$ and $x > 2.5$ . The stricter condition is $x > 2.5$ . Of the given choices, only 3.0 is greater than 2.5.
In rectangle ABCD, the diagonals intersect at point E. If the measure of angle BDC is 40°, what is the measure of angle AEB?
Answer: 100° In rectangle ABCD, opposite sides are parallel, so AB || DC. When diagonal BD intersects these parallel sides, alternate interior angles are equal. Therefore, angle ABD = angle BDC = 40°. The diagonals of a rectangle bisect each other, so AE = BE, making triangle ABE isosceles. In an isosceles triangle, the base angles are equal, so angle BAE = angle ABE = 40°. Using the triangle angle sum: angle AEB = 180° - 40° - 40° = 100°.
A certain quadrilateral has exactly one pair of parallel sides and its diagonals are not congruent. Which of the following must be true?
Answer: The non-parallel sides are not congruent. The quadrilateral has exactly one pair of parallel sides, so it is a trapezoid. A property of isosceles trapezoids is that their diagonals are congruent. Since the diagonals of this trapezoid are not congruent, it cannot be an isosceles trapezoid. The definition of an isosceles trapezoid is that its non-parallel sides (legs) are congruent. Therefore, if it is not an isosceles trapezoid, its non-parallel sides must not be congruent. The other statements are not necessarily true for a non-isosceles trapezoid.
A triangle has angles measuring $x°$ , $2x°$ , and $3x°$ . Which of the following is a true statement about this triangle?
Answer: It is a right triangle. The sum of the angles in any triangle is 180°. Therefore, we can write the equation $x + 2x + 3x = 180$ . Combining the terms on the left gives $6x = 180$ . Dividing both sides by 6 gives $x = 30$ . Now we can find the measure of each angle: the first angle is $x = 30°$ , the second angle is $2x = 2(30) = 60°$ , and the third angle is $3x = 3(30) = 90°$ . Since one of the angles is exactly 90°, the triangle is a right triangle.
The sum of the measures of the interior angles of a polygon is 1,080°. If the polygon is regular, what is the measure of one of its exterior angles?
Answer: 45° The formula for the sum of the interior angles of a polygon with $n$ sides is $S = (n-2) \times 180°$ . We are given $S = 1080°$ . So, $1080 = (n-2) \times 180$ . Dividing both sides by 180 gives $6 = n-2$ , so $n = 8$ . The polygon is an octagon. For any convex polygon, the sum of the exterior angles is 360°. In a regular polygon, all exterior angles are equal. Therefore, the measure of one exterior angle is $360° / n = 360° / 8 = 45°$ .
According to the inclusive definition, a trapezoid is a quadrilateral with at least one pair of parallel sides. Using this definition, which of the following statements is true?
Answer: All rectangles are trapezoids. The inclusive definition of a trapezoid requires at least one pair of parallel sides. A rectangle, by definition, has two pairs of parallel sides. Since two is 'at least one', all rectangles fit the definition of a trapezoid. A: A rhombus has two pairs of parallel sides, so it is a trapezoid by this definition. C: A kite has no parallel sides unless it is also a rhombus. D: A parallelogram has two pairs of parallel sides, so it is always a trapezoid under the inclusive definition.

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