Understanding Triangle Theorems

What does the Angle-Side Relationship Theorem state?

• In a triangle, the smaller angle is opposite the longer side.

• In a triangle, the longer side is opposite the smaller angle.

• In a triangle, the angles are always equal to the lengths of the sides.

• In a triangle, the larger angle is opposite the longer side.

If two angles in a triangle are not congruent, what can be said about the sides opposite those angles?

• The side opposite the smaller angle is longer than the side opposite the larger angle.

• The lengths of the sides do not relate to the angles at all.

• The side opposite the larger angle is longer than the side opposite the smaller angle.

• The sides opposite the angles are equal in length.

According to the Triangle Inequality Theorem, what must be true about the lengths of any two sides of a triangle?

• The difference of the lengths of any two sides must be greater than the length of the third side.

• The lengths of any two sides can be less than or equal to the length of the third side.

• The lengths of any two sides must be equal to the length of the third side.

• The sum of the lengths of any two sides must be greater than the length of the third side.

What is the Exterior Angle Inequality Theorem?

• The measure of an exterior angle is less than the sum of the two adjacent interior angles.

• An exterior angle is always equal to the adjacent interior angle.

• The measure of an exterior angle of a triangle is greater than the measures of the two non-adjacent interior angles.

• The sum of the interior angles of a triangle is equal to the exterior angle.

How does the Hinge Theorem relate the sides and angles of two triangles?

• The Hinge Theorem states that the smaller angle corresponds to the longer side.

• The Hinge Theorem indicates that two triangles with equal angles must have equal sides.

• The Hinge Theorem applies only to right triangles and their angles.

• The Hinge Theorem relates the sides and angles of two triangles by stating that the larger angle corresponds to the longer side when two sides are congruent.

What does the Converse of Hinge Theorem state about the sides of two triangles?

• If one triangle has a longer angle, then its sides must also be longer.

• If two triangles have equal angles, then their corresponding sides are equal.

• If two triangles have one side equal, then the other sides must also be equal.

• If one triangle has two sides longer than the corresponding sides of another triangle, then the angle opposite the longer side is greater.

If angle A is larger than angle B in triangle ABC, what can be said about the sides opposite those angles?

• The side opposite angle A is equal to the side opposite angle C.

• The side opposite angle A is shorter than the side opposite angle B.

• The side opposite angle A is longer than the side opposite angle B.

• The sides opposite angles A and B are equal in length.

In a triangle, if side a is longer than side b, what can be inferred about the angles opposite those sides?

• Angle opposite side a is equal to angle opposite side b.

• Angle opposite side a is larger than angle opposite side b.

• Angles opposite sides a and b are equal.

• Angle opposite side a is smaller than angle opposite side b.

What is the significance of the included angle in the Hinge Theorem?

• The included angle determines the length of the triangle's sides.

• The included angle determines the relative area of triangles sharing a side.

• The included angle affects the color of the triangles.

• The included angle is irrelevant to the triangle's area.

Can a triangle have two sides that are equal and still satisfy the Triangle Inequality Theorem?

• Yes, a triangle can have two sides that are equal and still satisfy the Triangle Inequality Theorem.

• No, a triangle cannot have two equal sides.

• Yes, but only if all sides are equal.

• No, it must have three unequal sides.