7th Grade Math Quiz: Construct Geometric Shapes With Conditions

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 7th Grade 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.

Question 1

1. The triangle can be constructed, but it will be obtuse due to the $80°$ angle being the largest.
2. No triangle can be constructed because the sum of the angles exceeds $180°$, violating the triangle angle sum theorem. (correct answer)
3. Multiple different triangles can be constructed because only angles are given without any side length constraints.
4. 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 2

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)
2. A valid obtuse triangle can be constructed since the longest side exceeds the sum of the squares of the other two sides.
3. No triangle can be constructed because the ratio between the longest and shortest sides is too large for geometric stability.
4. 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 3

1. The frame can be constructed and will form a right triangle since $8^2 + 3^2 = 12^2$ is approximately true.
2. The frame cannot be constructed because the sum of the two shorter sides is less than the longest side. (correct answer)
3. The frame can be constructed but will be unstable due to the large difference between the shortest and longest sides.
4. 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 4

1. Exactly one unique triangle can be constructed since the three angles completely determine the triangle's shape and size.
2. No triangle can be constructed because one angle exceeds $75°$, making the triangle construction geometrically impossible.
3. Infinitely many similar triangles can be constructed since the angles determine shape but not size or scale. (correct answer)
4. Three different triangles are possible, each emphasizing a different angle as the largest angle in the construction process.

Explanation: When only three angles are given (and they sum to 180°), infinitely many similar triangles can be constructed. The angles determine the shape but not the size. All such triangles would be similar to each other but could have different side lengths. Choice A is wrong because angles alone don't determine size. Choice B is wrong because angles up to (but not including) 180° are valid, and the sum here is exactly 180°. Choice D is wrong because the number of triangles isn't three—it's infinite.

Question 5

1. The triangle can be completed successfully, but it will be obtuse due to the $85°$ angle being close to $90°$.
2. Multiple valid triangles could be constructed depending on which angle is drawn first in the construction sequence.
3. The triangle will be impossible to close properly because the $85°$ angle conflicts with the acute nature of the other angles.
4. 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 $180°$ by $5°$, 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 6

1. The included angle must be greater than $0°$ and less than $180°$ to allow for any valid triangle construction. (correct answer)
2. The included angle must be less than $60°$ to ensure the triangle inequality is satisfied with the given side lengths.
3. The included angle must be exactly $90°$ to create the strongest triangular support structure possible.
4. 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 7

1. No triangle is possible because the angle is greater than $60^\circ$.
2. Exactly two triangles are possible because SSA is ambiguous.
3. Infinitely many triangles are possible because two sides are not enough information.
4. 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 8

1. Infinitely many triangles, because angles do not determine the triangle.
2. Exactly two triangles, because two angles can be arranged in two ways.
3. Exactly one triangle, because ASA determines a unique triangle. (correct answer)
4. 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 9

1. Exactly two triangles are possible because SSA is ambiguous.
2. Infinitely many triangles are possible because any three angles work.
3. No triangle is possible because the angles sum to $210^\circ$, not $180^\circ$. (correct answer)
4. Exactly one triangle is possible because three angles always determine a unique 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). The correct determination is that no triangle is possible because the angles sum to 210°, not 180°. A common error is accepting angles summing to more than 180° and claiming infinitely many or one triangle, violating the angle sum requirement. To determine this: (1) identify conditions (three angles AAA), (2) check feasibility (angle sum=180° fails), (3) determine uniqueness (violating rules give none), (4) reason (angles must sum to exactly 180° for a 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

1. Exactly one triangle is possible (SSS always works).
2. Exactly two triangles are possible because SSA is ambiguous.
3. Infinitely many triangles are possible because the frame could be scaled.
4. 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 11

1. Infinitely many triangles, because AAA determines shape but not size. (correct answer)
2. Exactly two triangles, because there are two ways to place the third angle.
3. Exactly one triangle, because three angles determine a unique triangle.
4. No triangle, because the angles are all the same.

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°-60°-60° sum to 180° (AAA gives infinitely many equilateral triangles all same angles, different sizes—not unique). The correct determination is infinitely many triangles, because AAA determines shape but not size. A common error is claiming exactly one triangle because three angles determine a unique triangle, but AAA gives infinite similar triangles, or no triangle because angles are the same, but equilateral is possible. To determine: (1) identify conditions (three angles AAA), (2) check feasibility (angle sum=180°), (3) determine uniqueness (AAA infinite), (4) reason (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 12

1. Exactly one triangle is possible because AAA determines a unique triangle.
2. No triangle is possible because the angles are too large to fit in a triangle.
3. Infinitely many triangles are possible because AAA fixes the shape but not the size. (correct answer)
4. Exactly two triangles are possible because the angles can be arranged in two ways.

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 AAA with 60°, 60°, 60° summing to 180°, so infinitely many similar equilateral triangles of different sizes are possible. A common error is claiming AAA determines a unique triangle, but it only fixes the shape, not the size, allowing scaling. To determine: (1) identify conditions (three angles—AAA), (2) check feasibility (angle sum=180° yes), (3) determine uniqueness (infinite similar triangles), (4) reason (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 13

1. Two angles and the included side (ASA) (correct answer)
2. Two sides only (SS)
3. Two sides and a non-included angle (SSA)
4. Three angles (AAA)

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 option D, two angles and included side (ASA guarantees unique triangle when possible). The correct option is two angles and the included side (ASA) because it guarantees a unique triangle every time it is possible. A common error is choosing AAA, which gives infinitely many, or SSA, which can be ambiguous (0-2), or SS, which doesn't determine any unique triangle. To determine: (1) identify conditions (various), (2) check feasibility (varies), (3) determine uniqueness (ASA always unique if possible), (4) reason (ASA locks shape with angles and size with included side). 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

1. Exactly one triangle, because two sides determine the third side.
2. No triangle is possible because you must always know three sides.
3. Infinitely many triangles are possible because the third side (and angles) can vary while still satisfying the triangle inequality. (correct answer)
4. 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 15

1. Exactly one triangle is possible because three sides always make a triangle.
2. Exactly two triangles are possible because the longest side can tilt two ways.
3. Infinitely many triangles are possible because there are three side lengths.
4. 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 16

1. Infinitely many triangles are possible, because SSA always gives infinitely many.
2. No triangle is possible, because $8+5$ is too large.
3. Exactly two triangles are possible, because SSA can be ambiguous for these values. (correct answer)
4. Exactly one triangle, because two sides and an angle always determine a unique 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 8 cm, 5 cm with non-included angle 30° at B (SSA ambiguous case, can give two triangles depending on values). The correct determination is exactly two triangles are possible, because SSA can be ambiguous for these values. A common error is claiming exactly one because two sides and an angle always determine unique, but SSA is not always unique, or infinitely many, but SSA gives 0-2 specific triangles. To determine: (1) identify conditions (two sides + non-included angle SSA), (2) check feasibility (depends on values, here possible), (3) determine uniqueness (SSA ambiguous 0-2 triangles), (4) reason (SSA can have two configurations where the side opposite the angle intersects the arc in two points). 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

1. Exactly one triangle is possible because ASA determines a unique triangle. (correct answer)
2. No triangle is possible because two angles are not enough information.
3. Infinitely many triangles are possible because angles determine only shape.
4. 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 18

1. Yes, infinitely many triangles can be constructed because the sides can be rearranged.
2. No, a triangle cannot be constructed because $13$ is an odd number.
3. Yes, exactly one triangle can be constructed because all three sides are given.
4. No, a triangle cannot be constructed because $6+7\le 13$ violates the triangle inequality. (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). For sides 6 cm, 7 cm, 13 cm, the inequality fails since 6+7=13≯13 (equality means degenerate, not a strict triangle), so no triangle can be constructed. A common error is checking only some inequalities or assuming equality allows a triangle, but strict inequality is required for a non-degenerate triangle. To determine: (1) identify conditions (three sides—SSS attempt), (2) check feasibility (triangle inequality: 6+7=13 not >13, fails), (3) determine uniqueness (none), (4) reason (equality in inequality results in a straight line, not a 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

1. SSS (three sides) (correct answer)
2. Two sides only (no angle)
3. SSA (two sides and a non-included angle)
4. AAA (three angles)

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 set of conditions that guarantees a unique triangle (assuming possible) is SSS (three sides), as it fixes both shape and size when inequalities hold. A common error is selecting AAA, which only gives infinitely many similar triangles, or SSA, which can be ambiguous. To determine: (1) identify conditions, (2) check feasibility (inequalities or sum), (3) determine uniqueness (SSS/SAS/ASA unique if feasible), (4) reason (SSS locks all aspects 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 20

1. Infinitely many triangles are possible because the sides can be scaled.
2. No triangle is possible because $2+3\le 10$ violates the triangle inequality. (correct answer)
3. Exactly one triangle is possible because three sides always make a triangle.
4. 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).