Construct Geometric Shapes With Conditions
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7th Grade Math › Construct Geometric Shapes With Conditions
Which set of conditions would allow a student to construct infinitely many different triangles (all with the same shape but different sizes)?
$\angle A=50^\circ$, $\angle B=60^\circ$, $\angle C=70^\circ$
$AB=5\text{ cm}$, $BC=6\text{ cm}$, $AC=7\text{ cm}$
$AB=4\text{ cm}$, $AC=9\text{ cm}$, $\angle A=30^\circ$
$\angle A=35^\circ$, $\angle B=65^\circ$, and $AB=8\text{ cm}$
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 B, angles 50°, 60°, 70° sum to 180° (AAA gives infinitely many similar triangles). The correct set is ∠A=50°, ∠B=60°, ∠C=70° because AAA allows infinitely many different triangles with same shape but different sizes. A common error is choosing SSS like A, which gives unique if inequality holds, or SSA like C, which can give 0-2, or ASA like D, which gives unique. To determine: (1) identify conditions (three angles AAA for B), (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).
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)?
Infinitely many triangles are possible because two sides are not enough information.
Exactly one triangle is possible because SAS determines a unique triangle.
Exactly two triangles are possible because SSA is ambiguous.
No triangle is possible because the angle is greater than $60^\circ$.
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). 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).
A student checks whether side lengths $6\text{ cm}$, $7\text{ cm}$, and $13\text{ cm}$ can form a triangle. What is the correct conclusion?
Exactly one triangle is possible because $6+7=13$.
Infinitely many triangles are possible because the sides are different.
No triangle is possible because $6+7\le 13$, so the triangle inequality fails.
Exactly two triangles are possible because the longest side can be placed in two directions.
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 6 cm, 7 cm, 13 cm, check inequality (6+7=13=13 not >13✗ fails—no triangle). The correct determination is no triangle is possible because 6+7≤13, so the triangle inequality fails. A common error is claiming exactly one because 6+7=13 (but must be strictly greater), or infinitely many because sides are different, but SSS would be unique 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, degenerate case). 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).
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?
Exactly one triangle, because ASA determines a unique triangle.
Exactly two triangles, because two angles can be arranged in two ways.
Infinitely many triangles, because angles do not determine the triangle.
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).
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?
Infinitely many triangles are possible because any three angles work.
Exactly two triangles are possible because SSA is ambiguous.
Exactly one triangle is possible because three angles always determine a unique triangle.
No triangle is possible because the angles sum to $210^\circ$, not $180^\circ$.
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 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).
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).
No triangle is possible because the triangle inequality fails: $2+3\le 10$.
Exactly two triangles are possible because SSA is ambiguous.
Infinitely many triangles are possible because the frame could be scaled.
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).
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?
Infinitely many triangles, because AAA determines shape but not size.
No triangle, because the angles are all the same.
Exactly two triangles, because there are two ways to place the third angle.
Exactly one triangle, because three angles 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 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).
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?
Exactly one triangle is possible because AAA determines a unique triangle.
Infinitely many triangles are possible because AAA fixes the shape but not the size.
No triangle is possible because the angles are too large to fit in a triangle.
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).
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?
Two angles and the included side (ASA)
Two sides only (SS)
Two sides and a non-included angle (SSA)
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).
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?
Infinitely many triangles are possible because the third side (and angles) can vary while still satisfying the triangle inequality.
No triangle is possible because you must always know three sides.
Exactly two triangles are possible because SSA is ambiguous.
Exactly one triangle, because two sides determine the third side.
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).