Question

specification cards
card set 1
a. angles 40°, 50°, 90° no side lengths given.
c. sides 4 cm, 5 cm, 10 cm
e. angles 30°, 60°, 90° side opposite 90° = 10 cm
g. angle x = 90°, angle y = 90° side z = 12 cm
card set 2
b. angles 100°, 100°,? side length 5 cm
d. angle a = 60°, angle b = 60° included side = 5 inches
f. angles 50°, 50°, 80° no side lengths given.
h. angles 70°, 70°, 40° side between 70° angles = 6 cm
sort the cards
place the letters from the cards above into the correct box.
❌ impossible triangle (can’t build it)
✅ one unique triangle (exactly one shape)
☑️ many triangles (same shape, different size)
justify your choice
pick one card from the ‘impossible’ category and explain specifically why it cannot form a triangle.

Explanation:

Step1: Check Card C validity

Triangle inequality: $4 + 5 = 9 < 10$

Step2: Check Card B angle sum

Sum: $100^\circ + 100^\circ + 9^\circ = 209^\circ
eq 180^\circ$

Step3: Check Card G angle sum

Sum: $90^\circ + 90^\circ + \text{Angle } Z = 180^\circ \implies \text{Angle } Z=0^\circ$ (invalid)

Step4: Sort valid cards

• Impossible: B, C, G
• One Unique Triangle: E, H, D
• Many Triangles: A, F

Step5: Justify one impossible card

Card C: $4+5<10$ violates triangle inequality.

Sort the Cards:

Impossible Triangle (Can't build it)

B, C, G

One Unique Triangle (Exactly one shape)

D, E, H

Many Triangles (Same shape, different sizes)

A, F

Justify Your Choice:

Card C has side lengths 4 cm, 5 cm, 10 cm. For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Here, the sum of the two shorter sides is $4 + 5 = 9$ cm, which is less than the longest side (10 cm). This violates the triangle inequality theorem, so no triangle can be formed.

Answer:

Card C cannot form a triangle because $4 + 5 < 10$, violating the triangle inequality theorem.