Construct Geometric Shapes With Conditions - 7th Grade Math

1. A student is told to draw a triangle with angle measures $60^\circ$, $60^\circ$, and $60^\circ$, but no side lengths are given. How many different triangles satisfy these conditions?
   Answer: Infinitely many triangles, because AAA determines shape but not size. 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).
2. 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)?
   Answer: Exactly one triangle is possible because SAS determines a unique triangle. 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).
3. A student is asked to draw a triangle with angles $60^\circ$, $60^\circ$, and $60^\circ$ (no side length is given). How many different triangles satisfy these conditions?
   Answer: Infinitely many triangles, because AAA fixes the shape but not the size. 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 AAA fixes the shape but not the size. A common error is claiming exactly one triangle thinking three angles determine a unique triangle, or no triangle because the sum is miscalculated as not 180°. To determine this: (1) identify conditions (three angles AAA), (2) check feasibility (angle sum=180° holds), (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).
4. 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)?
   Answer: Exactly one triangle is possible because SSS determines a unique triangle and the triangle inequality is satisfied. 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).
5. 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)?
   Answer: Exactly one triangle, because SAS determines a unique triangle. 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).
6. Which set of conditions guarantees a unique triangle can be constructed (assuming the measurements are possible)?
   Answer: SSS (three sides) 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).
7. 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?
   Answer: The included angle must be greater than $0°$ and less than $180°$ to allow for any valid triangle construction. 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.
8. A student claims the side lengths $6\text{ cm}$, $7\text{ cm}$, and $13\text{ cm}$ form a triangle. Which choice correctly describes whether a triangle can be constructed?
   Answer: No, a triangle cannot be constructed because $6+7\le 13$ violates the triangle inequality. 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).
9. 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?
   Answer: A valid right triangle can be constructed since $5^2 + 12^2 = 13^2$, and this satisfies both triangle inequality and Pythagorean theorem. 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.
10. 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)?
    Answer: Exactly one triangle is possible because ASA determines a unique triangle. 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).
11. Refer to the construction attempt shown. A student is trying to construct a triangle using the given measurements. What can be concluded about this construction?
    Answer: The construction will fail because the sum of the two shorter sides equals the longest side, creating a degenerate case. The three sides shown are 4 cm, 6 cm, and 10 cm. Checking the triangle inequality: $4 + 6 = 10$. When the sum of two sides equals the third side, the three points are collinear, forming a degenerate triangle (essentially a line segment). No actual triangle can be constructed. Choice A is wrong because the triangle inequality fails. Choice C is wrong because no triangle exists. Choice D is wrong because the side lengths are fixed and no triangle is possible.
12. 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)?
    Answer: Exactly one triangle is possible because SSS determines a unique triangle and $3+4>5$. 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).
13. A student checks whether side lengths 6 cm, 7 cm, and 13 cm can form a triangle. What is the correct conclusion?
    Answer: No triangle is possible because 6+7≤13, so the triangle inequality fails. The triangle inequality says that the sum of any two sides must be greater than the third side. Here, 6 + 7 = 13, which is equal to the third side, not greater than it, so the triangle inequality fails and no triangle can be formed. Choice C is incorrect because it treats 6 + 7 = 13 as if it still allows a triangle, when equality actually means the sides would lie flat instead of forming a shape. Choice A and Choice B describe outcomes for when a triangle can be built, which does not apply here since no triangle exists.
14. A student is given $\angle A=40^\circ$, $\angle B=60^\circ$, and the included side $AB=6\text{ cm}$. The student constructs $\triangle ABC$ with a protractor and ruler. How many different triangles are possible with these conditions (up to congruence)?
    Answer: Exactly one triangle, because ASA determines a unique triangle. 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 40°, 60° with included side 6 cm (ASA gives unique triangle, third angle 80° sums to 180°). The correct determination is exactly one triangle, because ASA determines a unique triangle. A common error is claiming no triangle because 40°+60°=100°≠180° (but third angle makes 180°), or infinitely many because angles do not determine a triangle (but ASA includes side for size). To determine: (1) identify conditions (two angles + included side ASA), (2) check feasibility (angle sum with third=180°), (3) determine uniqueness (ASA unique), (4) reason (ASA determines because angles and included side 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).
15. 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?
    Answer: No triangle is possible because the triangle inequality fails: $2+3\le 10$. 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).
16. A student tries to construct a triangle with angle measures $60^\circ$, $70^\circ$, and $80^\circ$. How many triangles can be constructed with these angles?
    Answer: No triangle is possible because the angles sum to $210^\circ$, not $180^\circ$. 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).
17. Which set of conditions would allow a student to construct infinitely many different triangles (all with the same shape but different sizes)?
    Answer: angle A=50 degrees, angle B=60 degrees, angle C=70 degrees When only the three angles of a triangle are known and they sum to 180 degrees, the triangle's shape is fixed but its size is not, so infinitely many similar triangles can be drawn with those same angles. Choice D gives angle A = 50 degrees, angle B = 60 degrees, and angle C = 70 degrees, which sum to 180 degrees and represent this situation, so it is correct. Choice A gives three side lengths, which produces exactly one triangle if the triangle inequality holds. Choice B gives two sides and a non-included angle, which can produce zero, one, or two triangles depending on the values. Choice C gives two angles and the included side, which produces exactly one triangle.
18. A student wants to know which information is enough to construct a unique triangle (up to congruence). Which option guarantees a unique triangle every time it is possible?
    Answer: Two angles and the included side (ASA) 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).
19. 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?
    Answer: The third angle cannot be drawn as $85°$ because the first two angles already determine it must be $80°$. 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.
20. A student is only told the angle measures of a triangle: $60^\circ$, $60^\circ$, and $60^\circ$. They want to draw a triangle that matches these angles. How many different triangles are possible?
    Answer: Infinitely many triangles are possible because AAA fixes the shape but not the size. 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).

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