Triangles Lesson Test

Answer: C

Correct answer: 57 degrees. The interior angles of a triangle add to 180 degrees. The known angles total

so the third angle is

• A: This subtracts the given angles from 360, which is the angle total for a quadrilateral, not a triangle.
• B: This is just the sum of the two given angles; it still has to be subtracted from 180.
• C (correct): The three angles total 180 degrees, and 180 - 52 - 71 = 57.
• D: This comes from a subtraction slip; 180 - 123 is 57, not 67.

Answer: D

Correct answer: Isosceles. Classify by sides by counting equal lengths. Exactly two sides are 9 inches, so at least two sides match and the triangle is isosceles.

• A: This is an angle classification, and the sides fail the test anyway: 81 + 81 = 162 is not 196.
• B: Equilateral requires all three sides to be equal; the 14-inch side is different.
• C: A scalene triangle has no equal sides, but this one has two equal 9-inch sides.
• D (correct): Two of the sides are equal (9 and 9), which is exactly the definition of isosceles.

Answer: A

Correct answer: 5 cm, 7 cm, and 10 cm. Check whether the two shorter sides add to more than the longest side. For 5, 7, and 10:

which is greater than 10, so a triangle can be formed. Each other set has a sum that is less than or equal to its longest side.

• A (correct): The two shorter sides sum to 12, which is greater than the longest side, 10.
• B: The two shorter sides sum to only 6, which is less than 7, so they cannot meet.
• C: The shorter sides sum to 9, which is less than the longest side of 10.
• D: The shorter sides sum to exactly 9; equal is not enough, so the pieces would lie flat.

Answer: C

Correct answer: 25. Use the Pythagorean theorem:

This is the 7-24-25 triple.

• A: This is the sum of the squares; the square root was never taken.
• B: This comes from subtracting the squares (576 - 49 = 527) and rounding its root; subtraction is for a missing leg.
• C (correct): Squaring and adding the legs gives 49 + 576 = 625, and the square root of 625 is 25.
• D: This adds the legs themselves instead of adding their squares.

Answer: C

Correct answer: 40 square cm. Use

Here

square centimeters.

• A: This multiplies base by height but forgets the one-half in the area formula.
• B: This adds the base and height instead of multiplying them.
• C (correct): One-half of 16 times 5 is 40.
• D: This takes half of the sum of base and height rather than half of their product.

Answer: B

Correct answer: 84 degrees. Add the angles and set the total to 180:

so 6x = 168 and x = 28. The angles are 28, 84, and 68 degrees (sum 180), so the largest is 84 degrees.

• A: This is the angle 2x + 12, which is smaller than 3x = 84.
• B (correct): Solving 6x + 12 = 180 gives x = 28, so the angles are 28, 84, and 68; the largest is 84.
• C: This is the value of x, which is also the smallest angle, not the largest.
• D: This ignores the + 12 term, giving 6x = 180 and x = 30, then 3x = 90.

Answer: D

Correct answer: 44 degrees. Let the vertex angle be v, so each base angle is v + 24. Then

3v = 132, and v = 44 degrees. Check:

• A: This splits 132 between two angles instead of dividing by 3 for the three v terms.
• B: This is the size of each base angle, not the vertex angle.
• C: This adds the 24 only once, solving 3v + 24 = 180, though both base angles include it.
• D (correct): With vertex v, v + 2(v + 24) = 180 gives 3v = 132 and v = 44.

Answer: B

Correct answer: 79 degrees. The exterior angle at L equals angle J plus angle K. So

and K = 79 degrees. Check: interior L is

• A: This is interior angle L, the angle that forms a straight line with the exterior angle.
• B (correct): An exterior angle equals the sum of the two remote interior angles, so K = 142 - 63 = 79.
• C: This adds the exterior angle and angle J; no interior angle of a triangle can exceed 180.
• D: This subtracts angle J from 180 and ignores the exterior angle completely.

Answer: B

Correct answer: PQ, PR, QR. Each side is named by the two vertices it joins and lies opposite the third vertex. PQ is opposite R (50), PR is opposite Q (58), and QR is opposite P (72). Smaller angles face shorter sides, so the order is PQ, PR, QR.

• A: This puts QR in the middle, but QR is opposite the largest angle, so it is the longest.
• B (correct): Shortest PQ faces the 50-degree angle R, PR faces the 58-degree angle Q, and longest QR faces the 72-degree angle P.
• C: This lists the sides from longest to shortest, the reverse of what was asked.
• D: This swaps PQ and PR; PQ is opposite the smallest angle, so it is the shortest side.

Answer: C

Correct answer: 13. The third side must be strictly between

The whole numbers from 6 to 18 make

possible lengths.

• A: This includes one endpoint; both bounds are strict, so neither 5 nor 19 is allowed.
• B: This includes the endpoints 5 and 19, but a third side equal to the difference or the sum would lie flat.
• C (correct): The third side must be greater than 12 - 7 = 5 and less than 12 + 7 = 19, so it can be 6 through 18, which is 13 values.
• D: This counts every length from 1 to 18 and forgets that the third side must exceed the difference, 5.

Answer: C

Correct answer: 15 feet. The ladder is the hypotenuse and the wall height is a missing leg:

The ladder reaches 15 feet up the wall (an 8-15-17 triple).

• A: This subtracts the lengths themselves instead of subtracting their squares.
• B: This adds the lengths; the height must be shorter than the 17-foot ladder.
• C (correct): The height is a leg: the square root of 289 - 64 = 225, which is 15.
• D: This adds the squares, which is how to find a hypotenuse, not a missing leg; it is longer than the ladder.

Answer: D

Correct answer: Obtuse. Compare the squares with c = 14 as the longest side. Here

Because 185 < 196, the triangle is obtuse.

• A: Equal angles would require equal sides, and these three sides are all different.
• B: A right triangle would need 64 + 121 to equal exactly 196, and it does not.
• C: An acute triangle would need the sum of the smaller squares to exceed 196, but 185 falls short.
• D (correct): Since 64 + 121 = 185 is less than 14 squared, 196, the angle across from the longest side is greater than 90 degrees.

Answer: D

Correct answer: 26.4. Matching sides share one ratio. The scale factor is

The longest side 11 becomes 11 times 2.4 = 26.4.

• A: This adds 7 to each side (because 12 - 5 = 7); similar triangles multiply by a scale factor rather than adding.
• B: This scales the 8, the middle side, instead of the longest side 11.
• C: This rounds the scale factor down to 2 instead of using the exact value 2.4.
• D (correct): The scale factor is 12 / 5 = 2.4, so the longest side is 11 times 2.4 = 26.4.

Answer: A

Correct answer: 6√2. An isosceles right triangle is a 45-45-90 triangle with sides in the ratio $1:1:\sqrt{2}$. Each leg is

about 8.49.

• A (correct): In a 45-45-90 triangle, a leg equals the hypotenuse divided by the square root of 2, and 12 / $\sqrt{2}$ = $6\sqrt{2}$.
• B: This multiplies by the square root of 2, which would give the hypotenuse from a leg, not a leg from the hypotenuse.
• C: This uses the 30-60-90 long-leg ratio, but an isosceles right triangle is 45-45-90.
• D: This halves the hypotenuse, which is the rule for the short leg of a 30-60-90 triangle.

Answer: A

Correct answer: 32√3 square units. A right triangle with a 60-degree angle is a 30-60-90 triangle. The short leg is half the hypotenuse, 8, and the long leg is $8\sqrt{3}$. The legs are perpendicular, so the area is

about 55.4 square units.

• A (correct): The legs are 8 and $8\sqrt{3}$, and one-half of their product is $32\sqrt{3}$.
• B: This treats both legs as 8, leaving out the $\sqrt{3}$ factor that belongs to the long leg.
• C: This multiplies the two legs but forgets the one-half in the area formula.
• D: This uses the short leg 8 with the hypotenuse 16 as base and height, but the hypotenuse is not perpendicular to that leg.

Answer: B

Correct answer: 30 inches. Areas scale by $k^2$, so

Perimeters scale by k, so the larger perimeter is 20 times 1.5 = 30 inches.

• A: This uses the scale factor upside down (6/9), which shrinks the perimeter of what should be the larger triangle.
• B (correct): The area ratio 81 to 36 means a side ratio of 9 to 6, or 1.5, and 20 times 1.5 is 30.
• C: This adds the difference of the square roots (9 - 6 = 3) instead of multiplying by their ratio.
• D: This scales the perimeter by the area ratio 81/36 = 2.25; perimeters scale by k, not by k squared.

Answer: D

Correct answer: 115 degrees. The exterior angle equals the sum of the remote interior angles:

so 2x = 30 and x = 15. The exterior angle is 65 degrees, so the adjacent interior angle is

Check: the remote angles are 34 and 31, and

• A: This is the value of x, not an angle measure.
• B: This is the larger remote interior angle, 2x + 4, which does not touch the exterior angle.
• C: This is the exterior angle itself; the adjacent interior angle forms a straight line with it.
• D (correct): Solving 5x - 10 = 3x + 20 gives x = 15, so the exterior angle is 65 and the adjacent interior angle is 180 - 65 = 115.

Answer: B

Correct answer: 72 cm. Find the missing leg first. 18 and 30 are 6 times 3 and 5, a scaled 3-4-5 triple, so the other leg is 6 times 4 = 24 cm. The perimeter is

• A: This takes the missing leg as 30 - 18 = 12, subtracting lengths instead of their squares.
• B (correct): The missing leg is the square root of 900 - 324 = 576, which is 24, so the perimeter is 18 + 24 + 30 = 72.
• C: This adds the squares to get a third side of about 35 cm, but that would be longer than the hypotenuse.
• D: This adds only the two given sides and leaves out the missing leg.

Answer: A

Correct answer: AB = DE, BC = EF, and angle A = angle D. Check where the angle sits relative to the sides. Two sides with the angle between them (SAS), two angles with the side between them (ASA), and two angles with a non-included side (AAS) all prove congruence. Two sides with an angle that is not between them (SSA) does not, so AB = DE, BC = EF, and angle A = angle D is not enough.

• A (correct): Angle A is not between sides AB and BC, so this is the SSA pattern, which can fit two different triangles.
• B: Side BC is not between angles A and B, so this is AAS, which is still a valid congruence test.
• C: Angle B lies between sides AB and BC, so this is SAS, a valid congruence test.
• D: Side AB lies between angles A and B, so this is ASA, a valid congruence test.

Answer: A

Correct answer: 24 cm. An isosceles triangle has two equal sides, so the third side is either 4 or 10. Sides 4, 4, 10 fail because

is not greater than 10. Sides 10, 10, 4 work because

The perimeter is

• A (correct): The third side must repeat one of the given lengths. Lengths 4, 4, 10 fail the triangle inequality, so the sides are 10, 10, 4 and the perimeter is 24.
• B: Only one of these is possible; the 4-4-10 case breaks the triangle inequality.
• C: This doubles the sum of the two given sides, as for a rectangle, but a triangle has only three sides.
• D: This uses sides 4, 4, and 10, but 4 + 4 = 8 is less than 10, so those lengths cannot form a triangle.

Answer: D

Correct answer: 68°. An exterior angle equals the sum of the two remote interior angles, so

°.

• A: 63° is the interior angle adjacent to the exterior angle (180 - 117), not a remote interior angle.
• B: This adds the two given angles; the exterior angle is already the sum of the remote angles.
• C: This subtracts both given angles from 180°, treating 117° as if it were an interior angle.
• D (correct): The exterior angle equals the sum of the remote interior angles: 117 - 49 = 68°.

Answer: A

Correct answer: 49√3 square cm. A height drawn from one vertex splits the triangle into two 30-60-90 triangles with hypotenuse 14 and short leg 7, so the height (the long leg) is $7\sqrt{3}$.

The area is $49\sqrt{3}$ square centimeters, about 84.9.

• A (correct): Base 14 times height $7\sqrt{3}$, halved, gives $49\sqrt{3}$.
• B: This forgets the one-half in the area formula and doubles the answer.
• C: $7\sqrt{3}$ cm is the height of the triangle, not its area.
• D: This uses the 45-45-90 ratio; half an equilateral triangle is a 30-60-90 triangle, so the factor is $\sqrt{3}$.

Answer: B

Correct answer: 175 square inches. The scale factor is

Areas of similar triangles scale by

so the larger area is

• A: This multiplies by k = 2.5, but area depends on two lengths and scales by $k^2$.
• B (correct): Area scales by $2.5^2=6.25$, and 28 × 6.25 = 175.
• C: This multiplies by 6/15, shrinking the area instead of enlarging it.
• D: This multiplies by $(6/15{)}^2$, using the scale factor upside down.

Answer: D

Correct answer: Obtuse. A triangle is obtuse if one angle is greater than 90°. Since 93° is more than 90°, the triangle is obtuse, even though its other two angles are acute.

• A: An acute triangle needs all three angles under 90°; 93° is not.
• B: A right triangle needs an angle of exactly 90°; 93° is close but larger.
• C: Equiangular means all three angles are equal (60° each), which is not the case here.
• D (correct): One angle, 93°, is greater than 90°, which makes the triangle obtuse.

Answer: A

Correct answer: 51 feet. The pole, the ground and the wire form a right triangle with legs 24 and 45, which are 3 times the 8-15-17 triple. So the wire is

Check:

• A (correct): The legs 24 and 45 are 3 times 8 and 15, so the hypotenuse is 3 × 17 = 51 feet.
• B: This adds the legs; the Pythagorean theorem adds their squares.
• C: This subtracts the squares, which finds a leg, but the wire is the hypotenuse.
• D: 2,601 is $c^2$; the square root still has to be taken.

Answer: B

Correct answer: 47 m. The third side must be less than

and greater than

The largest whole number below 24 is 23, so the greatest perimeter is

• A: This lets the third side equal 24 m, but a side equal to the sum of the other two lies flat; the inequality is strict.
• B (correct): The third side can be at most 23 m, giving 9 + 15 + 23 = 47 m.
• C: 31 m is the smallest possible perimeter (third side 7 m), not the greatest.
• D: 23 m is the longest possible third side, not the perimeter.

Answer: A

Correct answer: 56°. Interior angle

°. Angles B and C are the base angles opposite the equal sides, so angle C = 62°. Then angle

°. (Check with the exterior angle theorem:

)

• A (correct): With both base angles 62°, angle A = 180 - 124 = 56°.
• B: 62° is each base angle; angle A is the vertex angle between the equal sides.
• C: 124° is the sum of the two base angles, not angle A.
• D: This splits the exterior angle in half, but the two remote angles A and C are not equal.

Answer: C

Correct answer: 45 feet. The sun's rays make similar triangles, so height and shadow are in the same ratio:

• A: This sets up the ratio upside down, multiplying 27 by 3/5.
• B: This adds the 2-foot difference between height and shadow instead of using a ratio.
• C (correct): Height is 5/3 of shadow length, and 27 × 5 / 3 = 45 feet.
• D: This multiplies 27 by 5 but forgets to divide by 3.

Answer: B

Correct answer: 53°. The interior angles of a triangle add to 180°, so the third angle is

°.

• A: 127° is the sum of the two given angles, not the third angle.
• B (correct): 180 - 39 - 88 = 53°.
• C: This subtracts from 360°, but a triangle's angles total 180°.
• D: This is an arithmetic slip; 39 + 88 + 63 = 190, not 180.

Answer: D

Correct answer: 72 inches. Convert to one unit first: 2 feet = 24 inches. The perimeter is

• A: This adds 2 + 18 + 30 without converting the 2 feet to inches.
• B: This counts 2 feet as 20 inches; a foot is 12 inches, not 10.
• C: The perimeter is 6 feet, not 6 inches.
• D (correct): 2 feet is 24 inches, and 24 + 18 + 30 = 72 inches.

Answer: A

Correct answer: 44 square meters.

The area is 44 square meters.

• A (correct): One-half of base 11 times perpendicular height 8 is 44.
• B: This leaves out the one-half in the formula.
• C: This adds the base and height instead of multiplying.
• D: This takes half of the sum 11 + 8 rather than half of the product.

Answer: D

Correct answer: Acute. Compare the squares with c as the longest side:

Since 225 is greater than 196, the triangle is acute.

• A: A right triangle needs the squares to be equal; 225 is not 196. (9-12-15 would be right.)
• B: Obtuse requires $a^2+b^2<c^2$, but here the sum is larger than $c^2$.
• C: 9 + 12 = 21 is greater than 14, so these lengths do form a triangle.
• D (correct): The squares of the two shorter sides total 225, more than $14^2=196$, so all angles are acute.

Answer: B

Correct answer: 72°. Let the vertex angle be v, so each base angle is 2v. The angles total 180°:

so 5v = 180 and v = 36. Each base angle is 2(36) = 72°.

• A: 36° is the vertex angle, not a base angle.
• B (correct): With v = 36°, each base angle is 72°, and 36 + 72 + 72 = 180.
• C: This counts only one base angle (v + 2v = 180), forgetting that there are two.
• D: 60° would make the triangle equilateral, where base angles equal the vertex angle.

Answer: B

Correct answer: 25 square inches. In a 45-45-90 triangle each leg is the hypotenuse divided by $\sqrt{2}$:

The legs are the base and height, so the area is

• A: This forgets the one-half in the area formula.
• B (correct): Each leg is $5\sqrt{2}$, and half of $5\sqrt{2}\times 5\sqrt{2}=50$ is 25.
• C: This takes each leg as 5 (half the hypotenuse); the legs are $5\sqrt{2}$.
• D: This leaves a stray $\sqrt{2}$; $5\sqrt{2}\times 5\sqrt{2}$ is 50, a whole number.

Answer: C

Correct answer: 30 inches. The width, height and diagonal form a right triangle with hypotenuse 50. Subtract squares to find the missing leg:

The screen is 30 inches tall (a 3-4-5 triangle scaled by 10).

• A: This subtracts the lengths, 50 - 40, instead of their squares.
• B: This adds the squares, which finds a hypotenuse, but the diagonal is already the hypotenuse.
• C (correct): $2500-1600=900$, and the square root of 900 is 30.
• D: 900 is $h^2$; the square root still has to be taken.

Answer: A

Correct answer: Angle Z. The largest angle lies opposite the longest side. The longest side is XY, and the vertex not on XY is Z, so angle Z is the largest.

• A (correct): Angle Z is opposite XY, the longest side.
• B: Angle X is opposite YZ, the shortest side, so it is the smallest angle.
• C: Angle Y is opposite XZ, the middle side, so it is the middle-sized angle.
• D: Angles are equal only when their opposite sides are equal, and YZ and XZ are not.

Answer: C

Correct answer: Scalene. No two of the side lengths are equal, so the triangle is scalene.

• A: An isosceles triangle needs at least two equal sides, and no two of these lengths are equal.
• B: An equilateral triangle needs all three sides equal.
• C (correct): All three side lengths are different.
• D: Right describes angles, and these sides fail anyway: 144 + 225 = 369, not 400.

Answer: B

Correct answer: 12 + 12√3. The long leg (opposite 60°) is the short leg times $\sqrt{3}$, so the short leg is

The hypotenuse is twice the short leg, $8\sqrt{3}$.

Perimeter:

units.

• A: This treats 12 as the short leg (opposite 30°), but the given leg is opposite 60°.
• B (correct): The sides are 12, $4\sqrt{3}$ and $8\sqrt{3}$, which add to $12+12\sqrt{3}$.
• C: This treats 12 as the hypotenuse; the hypotenuse is opposite the 90° angle.
• D: This includes the hypotenuse but leaves out the short leg.

Answer: C

Correct answer: HL. For right triangles, a matching hypotenuse and a matching leg are enough to prove congruence; this is the HL (hypotenuse-leg) shortcut.

• A: SSA is not a valid shortcut in general; only the right-triangle case, HL, works.
• B: AAA is not a congruence test; matching angles fix the shape but not the size.
• C (correct): HL applies to right triangles with an equal hypotenuse and an equal leg, exactly the facts given.
• D: ASA needs two angles and the side between them; here only one angle (the right angle) is known to match.

Answer: A

Correct answer: 8, 15, 17. Test

with c the longest side. For 8, 15, 17:

= $17^2$, so it is right. It is one of the common Pythagorean triples.

• A (correct): 64 + 225 = 289, which equals $17^2$.
• B: 36 + 64 = 100, which is less than 144, so this triangle is obtuse.
• C: 25 + 144 = 169, which is less than 196; the right triangle would be 5-12-13.
• D: 49 + 576 = 625, which is less than 676; the right triangle would be 7-24-25.

Answer: D

Correct answer: 87°. Add the angles and set the sum to 180:

so 6x = 174 and x = 29. The angles are 41°, 52° and 87°, and the largest is 87°.

• A: 29 is x, not an angle measure.
• B: This sets the sum equal to 360 (x = 59), but a triangle's angles total 180°.
• C: 52° is the (2x - 6)° angle, which is the middle-sized one.
• D (correct): With x = 29 the angles are 41°, 52° and 87°, which total 180°.

Answer: D

Correct answer: 33 inches. The first triangle's perimeter is

Perimeters scale by the same factor as sides, so

and the longest side becomes

• A: 11 is the longest side of the original triangle; it must be scaled by 3.
• B: This multiplies by $k^2=9$, but lengths scale by k; only areas scale by $k^2$.
• C: This adds the 50-unit perimeter difference to one side instead of using a ratio.
• D (correct): The scale factor is 75 / 25 = 3, so the longest side is 3 × 11 = 33.

Answer: C

Correct answer: 22.5 square cm. The height must be perpendicular to the base, so use 5 cm, not the slanted 7 cm side.

The area is 22.5 square centimeters.

• A: This uses the slanted 7 cm side as the height.
• B: This leaves out the one-half in the formula.
• C (correct): One-half of base 9 times perpendicular height 5 is 22.5.
• D: This adds 9, 7 and 5 instead of using one-half base times height.

Answer: C

Correct answer: 11 inches. The third side must be greater than

and less than

Only 11 inches falls strictly between 8 and 18.

• A: 8 equals the difference, so 5 + 8 = 13 and the sides would lie flat; the inequality is strict.
• B: 18 equals the sum 5 + 13, so the sides would lie flat instead of forming a triangle.
• C (correct): 11 is greater than 8 and less than 18, so the three sides can close into a triangle.
• D: 20 is more than 5 + 13, so the two shorter sides could not reach across.