This ONLY works if you are ONLY given three angles. If they add up to 180 degrees, then it will form a triangle, but the side lengths can be different. It will form more than one triangle, thus being ambiguously defined. (Yep, examples coming for that too!)

RULES:

Angle Sum Rule: The sum of the interior angles of a triangle MUST be 180 degrees.

Side Sum Rule: The sum of any 2 sides of a triangle MUST be larger than the 3rd side.

UNIQUELY DEFINED

Triangles

If you are given ONLY 3 sides, and they pass the Side Sum Rule, then forms a unique triangle (only one triangle).

THEN. . .you have to decide if the triangle will be right or acute. To do this, square each of the side lengths. If you can add any of these two squared numbers to equal the third, then it is a right triangle. Otherwise, it is an acute.

This is the Pythagorean Theorem using the easy button. (DON"T WORRY! I've got examples too!)

Nonexistent (means won't make any triangle)

If you check either the Angle Sum Rule or the Side Sum Rule, and neither one works, then the info you've been given will NOT make any triangle.

Examples: If you are given, angle measures of 60, 40, and 100. . .use the Angle Sum Rule. Will they add to 180? No? So the triangle is nonexistent.

Example: If you are given side measures 4, 7, and 12. . .use the Side Sum Rule. 7+12 is bigger than 4, 4 + 12 is bigger than 7, BUT 4+ 7 is NOT bigger than 12, so these won't make a triangle.

Examples

1. If you are given sides measuring 6, 8, and 9, will it make

a. ambiguously defined

b. nonexistent

c. unique, acute triangle

d. unique, right triangle

HERE's how to solve:

Use the Side Sum Rule first. It checks out because any 2 sides have a sum bigger than the 3rd side. So that means it is unique.

Now. . .square each side length. 6X6=36, 8x8=64, 9x9=81

Now, look at 36, 64, and 81. Can you add 36+64 to get 81? NO, so the triangle can't be right. Has to be acute. SO "C" is the correct answer.

Try another one like that. . .

If you are given side lengths of 5, 12, and 13, will it make

a. ambiguously defined

b. nonexistent

c. unique, acute triangle

d. unique, right triangle

SOLVE: Does it pass the side sum rule? Yes, so it must be a unique triangle.

Now, square each side. . .to get 25, 144, and 169. Can you add 25 + 144 to get 169? Yep! So it must be a right triangle.

CORRECT ANSWER: D

**Constructing Triangles**

M7G2 by Mrs. Cherveny

M7G2 by Mrs. Cherveny

Example:

If you are given angles measuring 34, 21, and 125. . .

a. ambiguously defined

b. nonexistent

c. unique, acute triangle

d. unique, right triangle

SOLVE: 34 + 21 + 125 = 180, so answer is A