NC.7.G.2

The right triangle has an angle measure of 45°. What is the measure of the other angle in the triangle?

Mathematical Equivalence

ON

A triangle has two sides that measure 7 cm and 12 cm and an included angle of 75°. Which statement must be true about this triangle?

• The third side is more than 15 cm.

• The third side is between 5 cm and 19 cm.

• The third side is between 10 cm and 14 cm.

• One of the other two angles must be less than 40°.

Answer explanation

Using the triangle inequality theorem, the third side must be less than the sum of the other two sides (19 cm) and greater than their difference (5 cm). Thus, the third side is between 5 cm and 19 cm.

If two sides of a triangle are 6 km and 14 km long, the third side of the triangle could measure _____.

• 7 km

• 8 km

• 19 km

• 20 km

Answer explanation

The third side of a triangle must be less than the sum and greater than the difference of the other two sides. Here, 6 + 14 = 20 and 14 - 6 = 8. Thus, the third side can be between 8 km and 20 km, making 19 km a valid option.

Sarah is sketching a triangle. One side has a length of 8 units, and another side has a length of 13 units. What could be the length of the third side of the triangle?

• 3 units

• 5 units

• 19 units

• 21 units

Answer explanation

To find the possible length of the third side of a triangle, we use the triangle inequality theorem. The sum of the lengths of any two sides must be greater than the length of the third side. Here, 8 + 13 > x, so x < 21. Also, 8 + x > 13 and 13 + x > 8. The only valid option is 19.

In triangle PQR, segment QR measures 8 ft, segment PR measures 12 ft, and segment PQ measures 9 ft. Place the angles of the triangle in order from smallest angle to largest angle.​ ​

• Angle R

• Angle P
• Angle Q

Which set of measurements forms a unique triangle?

• A triangle with angle measurements of 25°, 55°, and 100°.

• A triangle with a 7 in side, a 14 in side, and an included 360° angle.

• A triangle with a 4 in side, a 9 in side, and an included 120° angle.

• A triangle with angle measurements of 35°, 45°, and 100°.

Answer explanation

The triangle with a 4 in side, a 9 in side, and an included 120° angle can be uniquely determined using the Law of Cosines. The other options either violate triangle properties or do not provide unique solutions.

Sarah has 3 pieces of rope that measure 3 feet, 4 feet, and 5 feet. What type of triangle can Sarah create?

• no type of triangle

• a scalene triangle

• an equilateral triangle

• an obtuse triangle

Answer explanation

The lengths 3 feet, 4 feet, and 5 feet can form a triangle since they satisfy the triangle inequality. Since all sides are of different lengths, Sarah can create a scalene triangle.

One side of a triangle measures 15 meters. Another side of the same triangle measures 9 meters. Which inequalities represent all possible lengths of the third side, x?

• x > 6 and x < 24

• x ≥ 6 and x ≤ 24

• x > 9 and x < 15

• x ≥ 9 and x ≤ 15

Answer explanation

To find the possible lengths of the third side, x, we use the triangle inequality theorem. The sum of any two sides must be greater than the third. Thus, 15 + 9 > x and x + 9 > 15 lead to 6 < x < 24, making x > 6 and x < 24 the correct choice.

Mrs. Smith gave her class three strips of paper to make triangles. David has strips that measure 4 inches, 9 inches, and 13 inches. How many triangles can David create with his 3 strips?

• infinitely many

• one

• two

• none

Answer explanation

David's strips measure 4, 9, and 13 inches. To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Here, 4 + 9 = 13, which is not greater than 13. Thus, he can create none.

If two angles of a triangle are 45° and 45°, what kind of triangle is it?

• acute

• obtuse

• right

Answer explanation

In a triangle, the sum of angles is 180°. With two angles at 45°, the third angle must be 90°. A triangle with a 90° angle is classified as a right triangle.