Question

identify which conditions make a unique triangle, more than one triangle, or no triangle. move a response to the table for each set of conditions.

triangle conditions	unique, more than one, or no triangle

|side lengths of 5 in., 12 in., and 20 in.|
|angle measures of 42°, 53°, and 85°|
|side lengths of 8 cm, 17 cm, and 24 cm|
|angle measures of 36°, 36°, and 108°|
|angle measures of 39° and 74°, and a side length of 9 cm between those angles|
|side lengths of 14 in. and 16 in., and an angle measure of 27° between those sides|

unique triangle more than one triangle no triangle

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For angles, the sum of the interior angles of a triangle is 180°. Also, certain combinations of side - lengths and angles determine the uniqueness of a triangle.

Step2: Analyze side - lengths 5 in., 12 in., and 20 in.

$5 + 12=17<20$. By the triangle - inequality theorem, no triangle can be formed.

Step3: Analyze angle measures 42°, 53°, and 85°

$42 + 53+85 = 180$. But just knowing the angle measures gives more than one triangle (since we can have similar triangles of different sizes), so more than one triangle.

Step4: Analyze side - lengths 8 cm, 17 cm, and 24 cm

$8 + 17=25>24$, $8 + 24=32>17$, $17 + 24 = 41>8$. By the triangle - inequality theorem, a triangle can be formed. And given three side - lengths, a unique triangle is formed (SSS congruence criterion), so unique triangle.

Step5: Analyze angle measures 36°, 36°, and 108°

$36+36 + 108=180$. Just knowing the angle measures gives more than one triangle (similar triangles), so more than one triangle.

Step6: Analyze angle measures 39° and 74°, and side - length of 9 cm between those angles

By the ASA (angle - side - angle) congruence criterion, a unique triangle is formed.

Step7: Analyze side - lengths 14 in. and 16 in., and angle measure of 27° between those sides

By the SAS (side - angle - side) congruence criterion, a unique triangle is formed.

Answer:

Triangle Conditions	Unique, More Than One, or No Triangle
Angle measures of 42°, 53°, and 85°	More Than One Triangle
Side lengths of 8 cm, 17 cm, and 24 cm	Unique Triangle
Angle measures of 36°, 36°, and 108°	More Than One Triangle
Angle measures of 39° and 74°, and a side length of 9 cm between those angles	Unique Triangle
Side lengths of 14 in. and 16 in., and an angle measure of 27° between those sides	Unique Triangle