Lines, Angles, and Triangles MCQ Quiz - Objective Question with Answer for Lines, Angles, and Triangles - Download Free PDF

Similar triangles have congruent corresponding angles. Therefore, if angle $X$ in $\mathrm{△}XYZ$ is $45\text{degree}$, angle $P$ in $\mathrm{△}PQR$ must also be $45\text{degree}$. Option 3 is the correct choice. The other options represent incorrect angles that don't satisfy the property of similar triangles.

Triangles $RST$ and $UVW$ have sides $RS=UV=8$, $ST=VW=6$, and $TR=WU=10$. Since all corresponding sides are equal, the triangles are congruent by the SSS (Side-Side-Side) congruence criterion. Therefore, option 4 is correct. Options 1 and 2 are incorrect because the triangles not only have equal sides but also equal angles, making them congruent, not just similar. Option 3 is incorrect because the triangles are indeed congruent.

The ladder, wall, and ground form a right triangle. The side opposite the ${30}^{\circ }$ angle is the height the ladder reaches on the wall, which is $8$ feet. The hypotenuse is the ladder itself. Using the sine function, $\sin ({30}^{\circ })=\frac{1}{2}$, we have: $\sin ({30}^{\circ })=\frac{8}{\text{hypotenuse}}$. This implies $\frac{1}{2}=\frac{8}{\text{hypotenuse}}$, leading to $\text{hypotenuse}=16$. Thus, the length of the ladder is $16$ feet. Option 2 is correct.

The sum of the interior angles of any triangle is ${180}^{\circ }$. Given two angles measuring ${50}^{\circ }$ and ${70}^{\circ }$, the third angle $x$ can be found by solving $50+70+x=180$. This simplifies to $x=180-120=60$. Thus, the measure of the third angle is ${60}^{\circ }$ (Option 3). Options 1, 2, and 4 do not meet the requirement of the total angle sum of ${180}^{\circ }$.

Since triangles $GHI$ and $JKL$ are similar, their corresponding angles are equal. Thus, angle $H$ corresponds to angle $K$, and both have a measure of $70\text{degree}$. Option 2 is correct because it aligns with the properties of similar triangles. The other options suggest incorrect angle measures that do not match the given similarity condition.

The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio $2:1:\sqrt{3}$. If x is the measure of the shortest leg, then the measure of the other leg is $\sqrt{3}x$ and the measure of the hypotenuse is 2x. The perimeter of the triangle is given to be $18+6\sqrt{3}$, and so the equation for the perimeter can be written as $2x+x+\sqrt{3}x=18+6\sqrt{3}$. Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives $(3+\sqrt{3})x=18+6\sqrt{3}$. Rewriting the right-hand side of the equation by factoring out 6 gives $(3+\sqrt{3})x=6(3+\sqrt{3})$. Dividing both sides of the equation by the common factor $(3+\sqrt{3})$ gives x = 6. The longest side of the right triangle, the hypotenuse, has a length of 2x , or 2(6), which is 12.

The correct answer is 0. Note that no matter where point W is on $\overline{RT}$, the sum of the measures of ∠RSW and ∠WST is equal to the measure of ∠RST, which is 90° Thus, ∠RSW and ∠WST are complementary angles. Since the cosine of an angle is equal to the sine of its complementary angle, cos (∠RSW) = sin (∠WST). Therefore, cos (∠RSW) - sin (∠WST) = 0.

The correct answer is 6 . Since $\tan B=\frac{3}{4}$, ΔABC and ΔDBE are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since BC = 15, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of ΔABC to the 3-4-5 triangle is 3 : 1.

Therefore, the length of $\overline{AC}$ (the side opposite to ∠B) is 3 × 3 = 9, and the length of $\overline{AB}$ (the side adjacent to ∠B) is 4 × 3 = 12. It is also given that DA = 4. Since AB = DA + DB and AB = 12, it follows that DB = 8, which means that the similarity ratio of ΔDBE to the 3-4-5 triangle is 2 : 1 ($\overline{DB}$ is the side adjacent to ∠B). Therefore, the length of $\overline{DE}$, which is the side opposite to ∠B, is 3 × 2 = 6.

Choice C is correct. It's given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is 20 centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of 5 of the sides of the triangles, the length of each side of an equilateral triangle is $\frac{20}{5}=4\mathrm{cm}$. Dividing up one equilateral triangle into two right triangles yields a pair of congruent 30° - 60° - 90° triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or 2 cm . Using the Pythagorean Theorem, a² + b² = c², the height of the equilateral triangle can be found. Substituting a = 2 and c = 4 and solving for b yields $\frac{1}{2}(12)(2\sqrt{3})$. The area of one equilateral triangle is $\frac{1}{2}bh$, where b = 2 and $h=2\sqrt{3}$. Therefore, the area of one equilateral triangle is $\frac{1}{2}(4)(2\sqrt{3})=4\sqrt{3}{\mathrm{cm}}^2$. The shaded area consists of two such triangles, so its area is $(2)(4)\sqrt{3}=8\sqrt{3}{\mathrm{cm}}^2$.

Alternate approach: The area of a trapezoid can be found by evaluating the expression $\frac{1}{2}(b_1+b_2)h$, where b₁ is the length of one base, b₂ is the length of the other base, and h is the height of the trapezoid. Substituting b₁ = 8, b₂ = 4, and $h=2\sqrt{3}$ yields the expression $\frac{1}{2}(8+4)(2\sqrt{3})$, or $\frac{1}{2}(12)(2\sqrt{3})$, which gives an area of $12\sqrt{3}{\mathrm{cm}}^2$ for the trapezoid. Since two-thirds of the trapezoid is shaded, the area of the shaded region is $\frac{2}{3}\times 12\sqrt{3}=8\sqrt{3}$.

Choice A is incorrect. This is the height of the trapezoid. Choice B is incorrect. This is the area of one of the equilateral triangles, not two. Choice D is incorrect and may result from using a height of 4 for each triangle rather than the height of $2\sqrt{3}$.

The correct answer is $\frac{101}{2}$. In the figure, lines q, r, and 8 form a triangle. One interior angle of this triangle is vertical to the angle marked a°; therefore, the interior angle also has measure a°. It's given that a = 43. Therefore, the interior angle of the triangle has measure 43°. A second interior angle of the triangle forms a straight line, q, with the angle marked b°. Therefore, the sum of the measures of these two angles is 180°. It's given that b = 122. Therefore, the angle marked b° has measure 122° and the second interior angle of the triangle has measure (180 - 122)°, or 58°. The sum of the interior angles of a triangle is 180°. Therefore, the measure of the third interior angle of the triangle is (180 - 43 - 58)°, or 79°. It's given that parallel lines q and t are intersected by line r. It follows that the triangle's interior angle with measure 79° is congruent to the same side interior angle between lines q and t formed by lines t and r. Since this angle is supplementary to the two angles marked w°, the sum of 79°, w°, and w° is 180°. It follows that 79 + w + w = 180, or 79 + 2w = 180. Subtracting 79 from both sides of this equation yields 2w 101. Dividing both sides of this equation by 2 yields $w=\frac{101}{2}$. Note that 101/2 and 50.5 are examples of ways to enter a correct answer.

Choice C is correct. Since RT = TU, it follows that ΔRTU is an isosceles triangle with base RU. Therefore, ∠TRU and ∠TUR are the base angles of an isosceles triangle and are congruent. Let the measures of both ∠TRU and ∠TUR be t°. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is 180°. Therefore, 114° +2t° = 180°, so t = 33.

Note that ∠TUR is the same angle as ∠SUV. Thus, the measure of ∠SUV is 33°. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, x° is equal to the sum of the measures of ∠VSU and ∠SUV; that is, 31° + 33° = 64°. Thus, the value of x is 64.

Choice B is incorrect. This is the measure of ∠STR, but ∠STR is not congruent to ∠SVR. Choices A and D are incorrect and may result from a calculation error. 

The correct answer is 30 . It is given that the measure of ∠QPR is 60°. Angle MPR and ∠QPR are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is 180°, and so the measure of ∠MPR is 120°. The sum of the angles in a triangle is 180°. Subtracting the measure of ∠MPR from 180° yields the sum of the other angles in the triangle MPR. Since 180 - 120 = 60, the sum of the measures of ∠QMR and ∠NRM is 60°. It is given that MP = PR, so it follows that triangle MPR is isosceles. Therefore ∠QMR and ∠NRM must be congruent. Since the sum of the measure of these two angles is 60°, it follows that the measure of each angle is 30°.

An alternate approach would be to use the exterior angle theorem, noting that the measure of ∠QPR is equal to the sum of the measures of ∠QMR and ∠NRM. Since both angles are equal, each of them has a measure of 30°. 

The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of X° is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of 23° and the other, which is supplementary to the angle with measure 106°, has measure of 180° - 106° = 74°. Therefore, the value of x is 23 + 74 = 97.

The correct answer is 123. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180 degrees. It's given that the measure of ∠SQX is 48° and the measure of ∠SXQ is 86°. Since points S, Q, and X form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of ∠QSX is 180 - 48 - 86, or 46 . It's also given that the measure of ∠SWU is 85°. Since ∠SWU and ∠SWR are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of ∠SWR is 180 - 85, or 95 . Since points R, S, and W form a triangle, and ∠RSW is the same angle as ∠QSX, it follows from the triangle angle sum theorem that the measure, in degrees, of ∠WRS is 180 - 46 - 95, or 39. It's given that the measure of ∠VTU is 162°. Since ∠VTU and ∠STU are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of ∠STU is 180 - 162, or 18 . Since points R, T, and U form a triangle, and ∠URT is the same angle as ∠WRS, it follows from the triangle angle sum theorem that the measure, in degrees, of ∠TUR is 180 - 39 - 18, or 123. 

The correct answer is 83. It's given that in the figure, AC = CD. Thus, triangle ACD is an isosceles triangle and the measure of angle CDA is equal to the measure of angle CAD. The sum of the measures of the interior angles of a triangle is 180°. Thus, the sum of the measures of the interior angles of triangle ACD is 180°. It's given that the measure of angle ACD is 104°. It follows that the sum of the measures of angles CDA and CAD is (180 - 104)°, or 76°. Since the measure of angle CDA is equal to the measure of angle CAD, the measure of angle CDA is half of 76°, or 38°. The sum of the measures of the interior angles of triangle BDE is 180°. It's given that the measure of angle EBC is 45°. Since the measure of angle BDE, which is the same angle as angle CDA, is 38°, it follows that the measure of angle DEB is (180 - 45 - 38)°, or 97°. Since angle DEB and angle AEB form a straight line, the sum of the measures of these angles is 180°. It's given in the figure that the measure of angle AEB is x°. It follows that 97 + x = 180. Subtracting 97 from both sides of this equation yields x = 83.