Middle School Math Quiz: Construct Geometric Shapes With Conditions

20 questions · exam conditions

0:00

Construct Geometric Shapes With ConditionsQuestion 1 of 20

A designer wants to create a triangular logo with sides measuring $5$ inches, $12$ inches, and $13$ inches. Before finalizing the design, the designer checks whether these measurements will form a valid triangle and what type it will be. What should the designer conclude?

A valid right triangle can be constructed since

5^2 + 12^2 = 13^2

, and this satisfies both triangle inequality and Pythagorean theorem.

A valid obtuse triangle can be constructed since the longest side exceeds the sum of the squares of the other two sides.

No triangle can be constructed because the ratio between the longest and shortest sides is too large for geometric stability.

A valid acute triangle can be constructed since all three sides satisfy the triangle inequality with comfortable margins for construction.

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Construct Geometric Shapes With Conditions

Practice Construct Geometric Shapes With Conditions in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Construct Geometric Shapes With Conditions, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A valid right triangle can be constructed since $5^2 + 12^2 = 13^2$ , and this satisfies both triangle inequality and Pythagorean theorem. (correct answer)
A valid obtuse triangle can be constructed since the longest side exceeds the sum of the squares of the other two sides.
No triangle can be constructed because the ratio between the longest and shortest sides is too large for geometric stability.
A valid acute triangle can be constructed since all three sides satisfy the triangle inequality with comfortable margins for construction.

Explanation: First, check triangle inequality:

5 + 12 = 17 > 13

5 + 13 = 18 > 12

, and

12 + 13 = 25 > 5

. The triangle inequality is satisfied. Then, check:

5^2 + 12^2 = 25 + 144 = 169 = 13^2

. Since the Pythagorean theorem holds exactly, this is a right triangle. Choice B is wrong because

13^2

equals (not exceeds)

5^2 + 12^2

. Choice C is wrong because the triangle inequality is satisfied. Choice D is wrong because it's a right triangle, not acute.

Question 2

A construction worker needs to build a triangular frame with side lengths of $8$ feet, $3$ feet, and $12$ feet. Before ordering materials, what should the worker determine about this triangular frame?

The frame can be constructed and will form a right triangle since $8^2 + 3^2 = 12^2$ is approximately true.
The frame cannot be constructed because the sum of the two shorter sides is less than the longest side. (correct answer)
The frame can be constructed but will be unstable due to the large difference between the shortest and longest sides.
Multiple different triangular frames are possible with these measurements depending on the construction angle approach used.

Explanation: By the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here,

8 + 3 = 11 < 12

, so no triangle can be formed. Choice A is wrong because

8^2 + 3^2 = 73 ≠ 144 = 12^2

, and the triangle inequality fails anyway. Choice C is wrong because the triangle cannot exist regardless of stability. Choice D is wrong because no triangle is possible with these side lengths.

Question 3

A student attempts to construct a triangle with angles measuring $30°$ , $70°$ , and $85°$ . After drawing the first two angles, the student realizes something about the third angle. What issue will the student encounter?

The triangle can be completed successfully, but it will be obtuse due to the $85°$ angle being close to $90°$ .
Multiple valid triangles could be constructed depending on which angle is drawn first in the construction sequence.
The triangle will be impossible to close properly because the $85°$ angle conflicts with the acute nature of the other angles.
The third angle cannot be drawn as $85°$ because the first two angles already determine it must be $80°$ . (correct answer)

Explanation: When you encounter triangle construction problems, remember that the angles in any triangle must always add up to exactly

180°

. This is a fundamental rule that cannot be violated. Let's check the given angles:

30° + 70° + 85° = 185°

. This sum exceeds

180°

5°

, which means these three angles cannot form a valid triangle. Once the student draws the first two angles (

30°

and

70°

), the third angle is automatically determined by the rule that all angles must sum to

180°

. The third angle must be

180° - 30° - 70° = 80°

, not

85°

. Option A is incorrect because the triangle cannot be completed at all—the issue isn't about the triangle being obtuse, but about the impossible angle sum. Option B misses the point entirely; the construction sequence doesn't matter when the fundamental angle sum rule is violated. Option C contains a misconception—there's nothing wrong with mixing an

85°

angle with acute angles in general, but these specific three angles simply don't add to

180°

. Option D correctly identifies that once two angles are drawn, the third angle is mathematically determined and must be

80°

to satisfy the angle sum requirement. Study tip: Whenever you see a triangle problem, immediately check if the angles add to

180°

. If they don't, the triangle is impossible. Remember that in any triangle, once you know two angles, the third is automatically determined.

Question 4

An architect needs to design a triangular support beam. She has determined that two sides must be $15$ feet and $8$ feet, and she can choose any angle between them. What constraint must she consider for the included angle to ensure a valid triangle construction?

The included angle must be greater than $0°$ and less than $180°$ to allow for any valid triangle construction. (correct answer)
The included angle must be less than $60°$ to ensure the triangle inequality is satisfied with the given side lengths.
The included angle must be exactly $90°$ to create the strongest triangular support structure possible.
The included angle must be at least $45°$ to prevent the third side from becoming longer than the sum of the given sides.

Explanation: When you encounter questions about triangle construction with two given sides and a variable angle between them, you're dealing with the fundamental requirements for forming any valid triangle. With two fixed sides of 15 feet and 8 feet, you can form a triangle using any included angle between

0°

and

180°

. As the angle approaches

0°

, the two sides nearly overlap, creating a very flat triangle. As it approaches

180°

, the sides point in nearly opposite directions, creating another very flat triangle. Any angle strictly between these extremes will produce a valid triangle, making choice A correct. Choice B incorrectly suggests a

60°

limit is needed to satisfy the triangle inequality. However, the triangle inequality (the sum of any two sides must exceed the third side) will be satisfied for any angle between

0°

and

180°

when you already have two fixed sides. Choice C claims the angle must be exactly

90°

. While a right triangle might be structurally strong, the question asks about the constraint for valid construction, not optimal strength. Many other angles would create perfectly valid triangles. Choice D incorrectly states the angle must be at least

45°

to prevent the third side from exceeding the sum of the given sides. This misunderstands how the triangle inequality works—with sides of 15 and 8 feet, the third side will always be less than their sum (23 feet) regardless of the included angle. Remember: when two sides of a triangle are fixed, any included angle between

0°

and

180°

(exclusive) will create a valid triangle.

Question 5

A student is told to construct $\triangle PQR$ where $PQ=4\text{ cm}$ , $PR=5\text{ cm}$ , and the included angle $\angle QPR=70^\circ$ . How many different triangles are possible with these conditions (up to flipping/rotation)?

No triangle is possible because the angle is greater than $60^\circ$ .
Exactly two triangles are possible because SSA is ambiguous.
Infinitely many triangles are possible because two sides are not enough information.
Exactly one triangle is possible because SAS determines a unique triangle. (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). In this case, the conditions are SAS (two sides 4 cm, 5 cm with included angle 70°), which determines exactly one unique triangle up to flipping or rotation. A common error is confusing SAS with SSA and claiming ambiguity, but SAS includes the angle between the sides, fixing the triangle rigidly. To determine: (1) identify conditions (two sides + included angle—SAS), (2) check feasibility (no inequality directly, but assumes possible), (3) determine uniqueness (SAS unique), (4) reason (SAS determines because sides and included angle lock the shape and size). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 6

A student is told to construct a triangle with angles $40^\circ$ and $60^\circ$ , and the side between those two angles is $7\text{ cm}$ . How many different triangles can be constructed with these conditions?

Infinitely many triangles, because angles do not determine the triangle.
Exactly two triangles, because two angles can be arranged in two ways.
Exactly one triangle, because ASA determines a unique triangle. (correct answer)
No triangle, because $40^\circ+60^\circ<180^\circ$ .

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that exactly one triangle is possible because ASA determines a unique triangle. A common error is claiming infinitely many thinking angles don't determine the triangle, or no triangle because the sum of two angles is less than 180° (but third angle makes it 180°). To determine this: (1) identify conditions (two angles + included side ASA), (2) check feasibility (angle sum with third=80°=180° holds), (3) determine uniqueness (ASA unique), (4) reason (ASA determines because angles and included side fix shape and size). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 7

For a design project, you are told to build a triangular frame with side lengths $2\text{ cm}$ , $3\text{ cm}$ , and $10\text{ cm}$ . How many triangles can be constructed with these side lengths?

Exactly one triangle is possible (SSS always works).
Exactly two triangles are possible because SSA is ambiguous.
Infinitely many triangles are possible because the frame could be scaled.
No triangle is possible because the triangle inequality fails: $2+3\le 10$ . (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that no triangle is possible because the triangle inequality fails: 2+3≤10. A common error is claiming exactly one triangle assuming SSS always works without checking inequality, or thinking it's ambiguous like SSA. To determine this: (1) identify conditions (three sides SSS), (2) check feasibility (triangle inequality all pairwise sums > third, here 2+3=5<10 fails), (3) determine uniqueness (violating rules give none), (4) reason (SSS determines because rigid triangle—sides lock angles, but inequality violation means sides don't close). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 8

A student is given only two side lengths, $6\text{ cm}$ and $9\text{ cm}$ , and is told to construct a triangle. How many triangles can be constructed with only this information?

Exactly one triangle, because two sides determine the third side.
No triangle is possible because you must always know three sides.
Infinitely many triangles are possible because the third side (and angles) can vary while still satisfying the triangle inequality. (correct answer)
Exactly two triangles are possible because SSA is ambiguous.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that infinitely many triangles are possible because the third side (and angles) can vary while still satisfying the triangle inequality (third side between |6-9|=3 and 6+9=15). A common error is claiming exactly one thinking two sides determine the third, or no triangle because three sides are needed (but third can vary). To determine this: (1) identify conditions (only two sides), (2) check feasibility (third side must satisfy inequality with the two), (3) determine uniqueness (infinite possibilities for third side), (4) reason (without fixing third side or angles, many triangles possible). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 9

A student tries to construct a triangle using sticks of lengths $2\text{ cm}$ , $3\text{ cm}$ , and $10\text{ cm}$ . Which statement correctly describes what happens?

Exactly one triangle is possible because three sides always make a triangle.
Exactly two triangles are possible because the longest side can tilt two ways.
Infinitely many triangles are possible because there are three side lengths.
No triangle is possible because $2+3\le 10$ , so the triangle inequality is violated. (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 2 cm, 3 cm, 10 cm, check inequality (2+3=5<10✗ fails—no triangle). The correct determination is no triangle is possible because 2+3≤10, so the triangle inequality is violated. A common error is claiming exactly one triangle because three sides always make a triangle, but inequality must be checked, or thinking infinitely many, but sides determine uniqueness if possible. To determine: (1) identify conditions (three sides SSS), (2) check feasibility (triangle inequality fails), (3) determine uniqueness (none possible), (4) reason (sides don't close into triangle). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 10

Maria is trying to construct a triangle using three given angle measures: $65°$ , $45°$ , and $80°$ . After checking her work, she realizes there's an issue with these measurements. What should Maria conclude about constructing a triangle with these angle measures?

The triangle can be constructed, but it will be obtuse due to the $80°$ angle being the largest.
No triangle can be constructed because the sum of the angles exceeds $180°$ , violating the triangle angle sum theorem. (correct answer)
Multiple different triangles can be constructed because only angles are given without any side length constraints.
A unique triangle can be constructed because three angles are sufficient to determine exactly one triangle shape.

Explanation: The sum of the given angles is

65° + 45° + 80° = 190°

, which exceeds

180°

. Since the sum of angles in any triangle must equal exactly

180°

, no triangle can be constructed with these angle measures. Choice A is wrong because the issue isn't about the triangle being obtuse. Choice C is wrong because the angles are impossible regardless of side lengths. Choice D is wrong because these angles cannot form any triangle.

Question 11

A student is asked to construct a triangle with angles $60^\circ$ , $70^\circ$ , and $80^\circ$ . Before drawing, the student checks whether such a triangle is possible. What is the correct conclusion?

Infinitely many triangles are possible because the angles add to $180^\circ$ .
Exactly two triangles are possible because SSA is ambiguous.
Exactly one triangle is possible because three angles always determine a triangle.
No triangle is possible because $60^\circ+70^\circ+80^\circ\ne180^\circ$ . (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For angles 60°, 70°, 80° sum to 210°≠180° (impossible—no triangle). The correct determination is no triangle is possible because 60°+70°+80°≠180°. A common error is claiming infinitely many because angles add to 180° (but they add to 210°), or exactly one because three angles always determine a triangle, but sum must be exactly 180°. To determine: (1) identify conditions (three angles AAA), (2) check feasibility (angle sum≠180° fails), (3) determine uniqueness (none possible), (4) reason (angle sum must be 180° for plane triangle). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 12

A student is told: $\angle A=50^\circ$ , $\angle B=60^\circ$ , and side $AB=7\text{ cm}$ . They want to construct $\triangle ABC$ . How many different triangles are possible with these conditions (up to flipping/rotation)?

Exactly one triangle is possible because ASA determines a unique triangle. (correct answer)
No triangle is possible because two angles are not enough information.
Infinitely many triangles are possible because angles determine only shape.
Exactly two triangles are possible because SSA is ambiguous.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). Here, the conditions are ASA (angles 50° and 60° with included side AB=7 cm; third angle is 70° summing to 180°), determining exactly one unique triangle. A common error is thinking two angles without the included side lead to infinity, but here the side is included between them, fixing size. To determine: (1) identify conditions (two angles + included side—ASA), (2) check feasibility (angle sum=180° with third angle), (3) determine uniqueness (ASA unique), (4) reason (ASA fixes angles and the side between, determining the rest rigidly). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 13

A student tries to construct a triangle with side lengths $2\text{ cm}$ , $3\text{ cm}$ , and $10\text{ cm}$ . Which statement correctly describes what happens?

Infinitely many triangles are possible because the sides can be scaled.
No triangle is possible because $2+3\le 10$ violates the triangle inequality. (correct answer)
Exactly one triangle is possible because three sides always make a triangle.
Exactly two triangles are possible because SSA is ambiguous.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). In this case, the sides 2 cm, 3 cm, 10 cm violate the triangle inequality since 2+3=5<10 (and while 2+10>3 and 3+10>2 hold, one failure means no triangle). A common error is assuming three sides always form a triangle without checking all inequalities, but here the sides cannot close as the two shorter ones don't reach across the longest. To determine: (1) identify conditions (three sides—SSS attempt), (2) check feasibility (triangle inequality: not all pairwise sums > third, fails), (3) determine uniqueness (none possible), (4) reason (violation means sides don't form a closed shape). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 14

A student is given three angle measures for a triangle: $60^\circ$ , $70^\circ$ , and $80^\circ$ . They try to construct a triangle with these angles. What is true?

Exactly one triangle is possible because AAA determines a unique triangle.
Exactly two triangles are possible because the angles can be arranged in two different orders.
No triangle is possible because the angles sum to $210^\circ$ , not $180^\circ$ . (correct answer)
Infinitely many triangles are possible because any three angles always form a triangle.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). Here, angles 60°, 70°, 80° sum to 210° > 180°, so no triangle is possible as the angle sum must be exactly 180°. A common error is accepting angles that don't sum to 180° or claiming infinitely many despite the violation, but the sum rule is fundamental. To determine: (1) identify conditions (three angles—AAA attempt), (2) check feasibility (angle sum≠180° fails), (3) determine uniqueness (none possible), (4) reason (angles must sum to 180° for a plane triangle). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 15

A student wants to construct $\triangle PQR$ with $PQ=4\text{ cm}$ , $PR=5\text{ cm}$ , and the included angle $\angle QPR=70^\circ$ . How many different triangles are possible with these conditions (up to congruence)?

No triangle, because $4+5$ is greater than $70$ .
Infinitely many triangles, because two sides are not enough information.
Exactly two triangles, because SSA is ambiguous.
Exactly one triangle, because SAS determines a unique triangle. (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For two sides 4 cm, 5 cm with included angle 70° (SAS gives unique triangle). The correct determination is exactly one triangle, because SAS determines a unique triangle. A common error is claiming exactly two triangles because SSA is ambiguous, but this is SAS (included angle), not SSA, or no triangle because 4+5>70 (but inequality is for sides, not angles). To determine: (1) identify conditions (two sides + included angle SAS), (2) check feasibility (no inequality violation for SAS, angles valid), (3) determine uniqueness (SAS unique), (4) reason (SAS determines because rigid—sides and included angle lock shape and size). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 16

A student wants to construct triangle $\triangle ABC$ with side lengths $AB=3\text{ cm}$ , $BC=4\text{ cm}$ , and $AC=5\text{ cm}$ . After constructing it with a ruler and compass, how many different triangles are possible with these measurements (up to congruence)?

Infinitely many triangles are possible because three sides do not determine a triangle.
No triangle is possible because $3+4=7$ is not greater than $5$ .
Exactly one triangle is possible because SSS determines a unique triangle and the triangle inequality is satisfied. (correct answer)
Exactly two triangles are possible because SSA is ambiguous.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3 cm, 4 cm, 5 cm, check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS). The correct determination is exactly one triangle is possible because SSS determines a unique triangle and the triangle inequality is satisfied. A common error is claiming no triangle because 3+4=7 is not greater than 5, but 7>5 is true, or thinking three sides give infinitely many, but SSS is unique. To determine: (1) identify conditions (three sides SSS), (2) check feasibility (triangle inequality all pass), (3) determine uniqueness (SSS unique), (4) reason (SSS determines because rigid triangle—sides lock angles). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 17

A student is given two side lengths $4\text{ cm}$ and $5\text{ cm}$ and the included angle between them is $70^\circ$ (SAS). The student constructs the triangle with a ruler and protractor. How many different triangles satisfy these conditions (up to rotation and reflection)?

Exactly one triangle, because SAS determines a unique triangle. (correct answer)
Infinitely many triangles, because the third side could vary.
Exactly two triangles, because SSA is ambiguous.
No triangle, because two sides and an angle are not enough information.

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that exactly one triangle is possible because SAS determines a unique triangle. A common error is confusing SAS with SSA and claiming two triangles due to ambiguity, or thinking it's infinite because the third side varies (but included angle fixes it). To determine this: (1) identify conditions (two sides + included angle SAS), (2) check feasibility (no specific inequality here, but assumes valid), (3) determine uniqueness (SAS unique), (4) reason (SAS determines because rigid—sides and included angle lock the shape and size). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 18

A student is given two side lengths $8\text{ cm}$ and $5\text{ cm}$ and a non-included angle of $30^\circ$ (SSA). The $30^\circ$ angle is opposite the $5\text{ cm}$ side. How many different triangles could be constructed from this information?

Exactly one triangle, because two sides and an angle always determine a unique triangle.
No triangle, because SSA can never form a triangle.
Infinitely many triangles, because the third side is not given.
Possibly two different triangles (or fewer), because SSA is the ambiguous case. (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that possibly two different triangles (or fewer) are possible because SSA is the ambiguous case. A common error is claiming exactly one triangle thinking two sides and an angle always determine unique, or infinite because third side not given (but SSA can yield 0,1, or 2). To determine this: (1) identify conditions (two sides + non-included angle SSA), (2) check feasibility (depends on specific values, but ambiguous), (3) determine uniqueness (SSA ambiguous 0-2 triangles), (4) reason (SSA can lead to two possible configurations for the triangle). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 19

A student wants to construct a triangle using a ruler. The side lengths are $3\text{ cm}$ , $4\text{ cm}$ , and $5\text{ cm}$ . After constructing it, how many different triangles are possible with these measurements (up to rotation and reflection)?

Infinitely many triangles are possible because the angles could change.
Exactly one triangle is possible because SSS determines a unique triangle and $3+4>5$ , $3+5>4$ , $4+5>3$ . (correct answer)
Exactly two triangles are possible because two sides and a third side can make an ambiguous case.
No triangle is possible because $3+4=5$ .

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is that exactly one triangle is possible because SSS determines a unique triangle and the inequalities hold. A common error is claiming no triangle because 3+4=7>5 is miscalculated as equal, or thinking SSS is ambiguous like SSA. To determine this: (1) identify conditions (three sides SSS), (2) check feasibility (triangle inequality all pairwise sums > third), (3) determine uniqueness (SSS unique), (4) reason (SSS determines because rigid triangle—sides lock angles, AAA doesn't determine size—similar triangles all match). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).

Question 20

A student wants to construct $\triangle ABC$ using a ruler and compass. They are told that $AB=3\text{ cm}$ , $BC=4\text{ cm}$ , and $AC=5\text{ cm}$ . After constructing it, how many different triangles are possible with these measurements (up to flipping/rotation)?

No triangle is possible because the triangle inequality is not satisfied.
Infinitely many triangles are possible because three side lengths do not determine a unique triangle.
Exactly two triangles are possible because SSA is ambiguous.
Exactly one triangle is possible because SSS determines a unique triangle and $3+4>5$ . (correct answer)

Explanation: This question tests constructing triangles from conditions (sides/angles) and determining uniqueness: SSS/SAS/ASA give unique triangle, AAA gives infinitely many similar triangles, inequality violations or angle sum≠180° give no triangle, SSA ambiguous. Triangle uniqueness: SSS (three sides) gives unique if triangle inequality satisfied (sum any two sides > third: check 3+4>5✓, 4+5>3✓, 3+5>4✓ all true for 3-4-5 triangle), SAS (two sides, included angle) and ASA (two angles, included side) give unique. AAA (three angles) gives infinitely many similar triangles (same angles, different sizes—angles determine shape not size). Triangle inequality: a+b>c, b+c>a, a+c>b all required (if 2+3=5≤10, cannot form triangle—sides don't reach). Angle sum: must equal 180° (if 60°+70°+80°=210°, impossible). For sides 3,4,5 check inequality (3+4=7>5✓, 4+5=9>3✓, 3+5=8>4✓, all pass—forms unique triangle SSS), or sides 2,3,10 check (2+3=5<10✗ fails—no triangle), or angles 60°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). In this case, the conditions are SSS with sides 3 cm, 4 cm, 5 cm, and since all triangle inequalities hold (3+4>5, 3+5>4, 4+5>3), exactly one unique triangle is possible up to flipping or rotation. A common error is claiming infinitely many for SSS, but SSS fixes both shape and size uniquely if inequalities are satisfied, unlike AAA which only fixes shape. To determine: (1) identify conditions (three sides—SSS), (2) check feasibility (triangle inequality: all pairwise sums > third, here yes), (3) determine uniqueness (SSS unique), (4) reason (SSS determines because rigid triangle—sides lock angles and size). Triangle inequality: must check ALL THREE pairwise (a+b>c AND b+c>a AND a+c>b), one violation means impossible (sides don't close into triangle). Common mistakes: assuming all conditions give unique (AAA doesn't), not checking inequality (accepts impossible side combinations), checking one inequality only (missing violations in other pairs).