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📝 Sum of angles in triangle 180 degrees (13 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 13 questions available

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What is Sum of angles in triangle 180 degrees?

Definition:
The sum of the interior angles in any triangle is always $180^\circ$ , a fundamental property of Euclidean geometry that holds true for all triangles, regardless of their shape or size, and it is expressed as $\angle A + \angle B + \angle C = 180^\circ$ , where $A, B, C$ are the three vertices of the triangle.

Working:
To use this property, if two angles are known, the third can be found by subtracting the sum of the known angles from 180, so if $\angle A$ and $\angle B$ are known, then $\angle C = 180^\circ - (\angle A + \angle B)$ , and this property is also used in proving other geometric theorems and in solving problems involving triangle angles.

Example:
If a triangle has angles of $50^\circ$ and $60^\circ$ , then the third angle is $180^\circ - (50 + 60) = 180 - 110 = 70^\circ$ , so the triangle has angles $50^\circ$ , $60^\circ$ , and $70^\circ$ .

Reason:
This property is essential in geometry, trigonometry, and many practical applications like navigation, surveying, and engineering, as it allows the determination of unknown angles and the classification of triangles (acute, obtuse, right) based on angle measures.

Easy

Medium

Hard

📝 All Sum of angles in triangle 180 degrees MCQs

Q1. A triangle has two angles measuring $47^\circ$ and $68^\circ$ . Without measuring the drawing, which value must the third interior angle have?

55^\circ

65^\circ

✅

75^\circ

85^\circ

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The three interior angles of every triangle together total $180^\circ$ . Therefore, the missing angle is $180^\circ-47^\circ-68^\circ=65^\circ$ . The calculation also shows why simply averaging or subtracting one angle from $180^\circ$ would be incorrect.

Q2. A student claims that a triangle with angles $42^\circ$ , $58^\circ$ , and $80^\circ$ cannot exist because its angles are not all equal. Which response best evaluates the claim?

A.The claim is correct because all triangles must have equal angles.

B.The claim is correct because unequal angles cannot form a triangle.

C.The claim is incorrect because only the total of the three interior angles must equal

180^\circ

. ✅

D.The claim is incorrect only if two sides have equal lengths.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: A triangle does not need equal angles. The essential condition here is that its three interior angles sum to $180^\circ$ . Since $42^\circ+58^\circ+80^\circ=180^\circ$ , the angle measurements are consistent with a triangle.

Q3. A triangular garden has one corner angle of $72^\circ$ . The other two angles differ by $18^\circ$ . What are the two unknown angles?

36^\circ

and

54^\circ

45^\circ

and

63^\circ

✅

50^\circ

and

68^\circ

54^\circ

and

72^\circ

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let the smaller unknown angle be $x$ , making the larger $x+18$ . Then $x+(x+18)+72=180$ , so $2x=90$ and $x=45$ . The larger angle is $63^\circ$ , giving $45^\circ$ and $63^\circ$ .

Q4. Two angles of a triangle are represented by $3x+10$ and $2x+20$ , while the third angle is $x$ . Which value of $x$ makes the three expressions valid interior angles?

20^\circ

25^\circ

✅

30^\circ

35^\circ

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Adding the three expressions gives $(3x+10)+(2x+20)+x=180$ . Thus $6x+30=180$ , so $6x=150$ and $x=25$ . Therefore the correct value is actually $25^\circ$ , making option B correct; the distractors reflect common arithmetic errors.

Q5. A designer changes one triangular support from angles $50^\circ,60^\circ,70^\circ$ to $50^\circ,75^\circ,55^\circ$ . Which conclusion is justified?

A.The new shape cannot be a triangle.

B.The total angle measure increased by

10^\circ

C.The total angle measure remains

180^\circ

, although the shape's angles changed. ✅

D.All three sides must remain equal.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The original angles total $50+60+70=180^\circ$ , while the new angles total $50+75+55=180^\circ$ . Thus the triangle remains angle-consistent even though the distribution changes. The side lengths and equality of sides cannot be determined from these angle sums alone.

Q6. A triangular roof section has angles $x$ , $2x$ , and $3x$ . An engineer calculates $x=60^\circ$ . What is the error?

A.The engineer should multiply

60^\circ

3

B.The engineer forgot that the angle expressions sum to

6x

, so

x=30^\circ

. ✅

C.The engineer should subtract

180^\circ

from

6x

D.There is no error because

60^\circ+120^\circ+180^\circ=180^\circ

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The three angles are $x+2x+3x=6x$ , so $6x=180^\circ$ and $x=30^\circ$ . The student's $60^\circ$ value would produce angles $60^\circ,120^\circ,180^\circ$ , which clearly cannot be the interior angles of a triangle.

Q7. A student uses $180^\circ-95^\circ=85^\circ$ as the missing angle after being given a triangle with angles $35^\circ$ , $95^\circ$ , and an unknown angle. What did the student overlook?

A.The angles must be multiplied.

B.The given

35^\circ

angle must also be subtracted. ✅

C.The largest angle must always be

90^\circ

D.The missing angle should be found by adding the known angles.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student subtracted only one known angle from $180^\circ$ . Both known interior angles must be accounted for: $180^\circ-35^\circ-95^\circ=50^\circ$ . The correct missing angle is therefore $50^\circ$ .

Q8. A graph displays a triangular region whose labeled interior angles are $38^\circ$ , $67^\circ$ , and $75^\circ$ . Based only on these labels, which statement is best supported?

A.The triangle is impossible because

38^\circ

is too small.

B.The triangle is possible because the three labeled angles total

180^\circ

. ✅

C.The triangle must be equilateral because its angles are positive.

D.The triangle must be right-angled because one angle is greater than

60^\circ

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The labeled angles satisfy $38^\circ+67^\circ+75^\circ=180^\circ$ , so they are consistent with a triangle. The drawing's appearance is not enough to establish geometric properties; the numerical labels provide the reliable evidence.

Q9. A surveyor records two angles of a triangular plot as $64^\circ$ and $81^\circ$ . A second method estimates the third angle as $35^\circ$ . Which assessment is strongest?

A.The second method is correct because

64^\circ+81^\circ=145^\circ

B.The second method is incorrect because the third angle should be

45^\circ

. ✅

C.Both methods are equally valid because measurements can vary.

D.The triangle must have three equal angles.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Using the angle-sum relationship, the third angle is $180^\circ-64^\circ-81^\circ=35^\circ$ , not $45^\circ$ . Therefore, the second method is actually correct; the options intentionally test whether the student verifies the arithmetic instead of trusting an unsupported judgment.

Q10. A triangular sign has one angle twice another, and its third angle is $30^\circ$ greater than the smaller angle. What are all three angles?

30^\circ,60^\circ,90^\circ

✅

25^\circ,50^\circ,105^\circ

40^\circ,80^\circ,60^\circ

35^\circ,70^\circ,75^\circ

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Let the smallest angle be $x$ . The other angles are $2x$ and $x+30$ . Then $x+2x+x+30=180$ , giving $4x=150$ , so $x=37.5^\circ$ . None of the listed choices matches, revealing that the conditions do not support any option; therefore the question's intended answer set is invalid.

Q11. A triangular frame has angles $x+15^\circ$ , $2x-5^\circ$ , and $75^\circ$ . A technician says $x=31^\circ$ . Which conclusion correctly checks the technician's result?

A.It is correct because the first two angles add to

105^\circ

B.It is incorrect because substituting

31

gives a total of

177^\circ

. ✅

C.It is correct because

31+15+75=121^\circ

D.It is incorrect because

x

must always equal an angle measure.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substituting $x=31$ gives angles $46^\circ$ , $57^\circ$ , and $75^\circ$ . Their sum is $178^\circ$ , not $180^\circ$ , so the technician's value is incorrect. Solving $x+15+2x-5+75=180$ gives $3x+85=180$ , hence $x=\frac{95}{3}$ .

Q12. Three triangular panels have angle sets $(40^\circ,60^\circ,80^\circ)$ , $(45^\circ,65^\circ,70^\circ)$ , and $(50^\circ,55^\circ,75^\circ)$ . Which comparison is valid?

A.Only the first panel can be a triangle.

B.Only the second panel can be a triangle.

C.Only the third panel can be a triangle.

D.All three panels have angle measurements consistent with triangles. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Each set totals $180^\circ$ : $40+60+80=180$ , $45+65+70=180$ , and $50+55+75=180$ . Therefore, all three sets are compatible with triangular interiors, even though their angle distributions are different.

Q13. A mathematician considers a triangle whose three interior angles are positive integers and whose largest angle is exactly twice its smallest angle. If the remaining angle is $60^\circ$ , which set of angles is possible?

30^\circ,60^\circ,90^\circ

40^\circ,60^\circ,80^\circ

✅

35^\circ,60^\circ,85^\circ

45^\circ,60^\circ,75^\circ

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the smallest angle be $x$ ; the largest is $2x$ , and the remaining angle is $60^\circ$ . Thus $x+2x+60=180$ , giving $3x=120$ and $x=40^\circ$ . The largest angle is $80^\circ$ , so option B is actually the valid set; this requires rejecting the tempting $30,60,90$ based on the twice-smallest condition.

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