Testinar Grade 7 math

Constructing Triangles from Three Measurements

Students will build Grade 7 mastery of constructing triangles from three measurements using models, equations, and test-style reasoning.

CH06-04 Geometry and Scale

Name
Date
Score: ____ / 8

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

How many unique triangles can be constructed with sides of length \(6\) cm, \(8\) cm, and \(10\) cm?
1. \(0\)
2. \(1\)
3. \(2\)
4. More than \(2\)
Which set of side lengths will NOT form a triangle?
1. \(5\) cm, \(7\) cm, \(11\) cm
2. \(4\) cm, \(6\) cm, \(9\) cm
3. \(3\) cm, \(4\) cm, \(6\) cm
4. \(5\) cm, \(5\) cm, \(8\) cm
Triangle \(ABC\) with \(AB=5\) cm, \(AC=4\) cm, and a right angle at \(A\) satisfies which congruence condition?
1. SSS
2. SAS
3. ASA
4. RHS
Two triangles each have angles measuring 50°, 60°, and 70°. Are the triangles congruent?
1. Yes, always congruent
2. No, never congruent
3. Only if one side length is the same
4. Cannot be determined without side lengths
Triangle \(DEF\) has sides \(DE=7\) cm, \(EF=6\) cm, and \(FD=8\) cm. Is this triangle unique?
1. No, infinitely many triangles have these side lengths
2. Yes, SSS guarantees uniqueness
3. No, SAS allows variation
4. Yes, but only if we specify one angle
A triangle has sides of length \(5\) m and \(9\) m. Which length CANNOT be the third side?
1. \(6\) m
2. \(10\) m
3. \(13\) m
4. \(15\) m
If \(GH=7\) cm and angles \(\alpha\) and \(\beta\) at \(G\) and \(H\) are known, which congruence condition applies?
1. SAS
2. ASA
3. AAS
4. SSS
Triangle \(JKL\) has base \(JK=8\) cm and perpendicular height \(h=3\) cm. How many distinct triangles satisfy these conditions?
1. \(1\) (unique)
2. \(2\) (two possible positions for \(L\))
3. Infinitely many
4. \(0\) (impossible)

Teacher and parent reference

Answer key and explanations

B. \(1\)
Given three specific side lengths that satisfy the Triangle Inequality, exactly one unique triangle (up to congruence) can be formed. In fact, this is a right triangle because \(6^2+8^2=10^2\).
B. \(4\) cm, \(6\) cm, \(9\) cm
The Triangle Inequality states the sum of any two sides must exceed the third. Check: \(4+6=10\) is not greater than \(9\), so this fails.
B. SAS
Two sides and the included right angle are given: \(AC=4\) cm, \(AB=5\) cm, and angle \(A=90°\) is between them. This is SAS (Side\textendash{}Angle\textendash{}Side) congruence.
D. Cannot be determined without side lengths
AAA (Angle\textendash{}Angle\textendash{}Angle) guarantees similarity, not congruence. The triangles could be different sizes but the same shape. At least one side length is needed to prove congruence.
B. Yes, SSS guarantees uniqueness
When all three side lengths are specified (7 cm, 6 cm, 8 cm), exactly one unique triangle (up to congruence) can be constructed. This is the SSS (Side\textendash{}Side\textendash{}Side) condition.
D. \(15\) m
The Triangle Inequality requires the sum of any two sides to exceed the third. With sides 5 and 9: we need \(5+9>c\), so \(c<14\) m. Also \(5+c>9\), so \(c>4\) m. Thus \(4
B. ASA
We have angle \(\alpha\) at \(G\), the side \(GH=7\) cm between them, and angle \(\beta\) at \(H\). This is Angle\textendash{}Side\textendash{}Angle (ASA): two angles and the included side are given.
C. Infinitely many
The vertex \(L\) may lie anywhere on a line parallel to \(JK\) at distance 3 cm above it. Each position yields a different triangle with the same base and height but different side lengths and angles.