Test: Inequalities - Grade 11 MCQ

In a triangle, the side opposite the largest angle is the longest. Therefore, if AB is the longest side, angle B must be greater than angle C, confirming the relationship between angles and sides.

The side opposite the largest angle in a triangle is always the longest. Therefore, if angle A is the largest angle, side AC must be the longest side.

When an angle is bisected, it splits the angle into two equal parts. Hence, if \( AC = AD \) and angle A is bisected, then angle ACD must be equal to angle ADC.

The triangle inequality theorem states that the difference between two sides must be less than the length of the third side for a triangle to exist. Hence, if this condition is met, triangle ABC is valid.

If angle A is greater than angle B, then according to the properties of triangles, the side opposite the larger angle (AE) must be longer than the side opposite the smaller angle (BE). This principle helps in understanding triangle comparisons.

If side AB is longer than side AC, then angle A must be greater than angle C, according to the properties of triangles where the longer side is opposite the larger angle.

The theorem regarding medians states that the sum of the lengths of the two sides of a triangle is greater than twice the length of the median to the third side. Therefore, \( AB + AC > 2AD \).

In a triangle, the side opposite the larger angle must be longer. Therefore, if angle A is greater than angle B, side AB must be shorter than side AC.

The triangle inequality states that the sum of the lengths of any two sides must always be greater than the length of the third side. This principle is critical for determining the feasibility of triangle construction.

Since \( BP \) is a median, it divides the triangle into two smaller triangles. The triangle inequality theorem asserts that the sum of the lengths of any two sides must be greater than the length of the third side, leading to \( PB + PA < bc="" +="" ac="">

The symbol ">" indicates that one value is greater than another. For example, if \( a > b \), it means that \( a \) is larger than \( b \). This concept is fundamental in understanding inequalities in geometry, particularly when comparing lengths and angles.

If two sides of a triangle are unequal, the theorem states that the greater side has the greater angle opposite to it. This principle helps in understanding the relationship between side lengths and angles in triangles.

In any triangle, if two angles are equal, the sides opposite those angles must also be equal. This is a fundamental property of isosceles triangles.

When two sides of a triangle are equal, the angles opposite those sides are also equal. Therefore, if \( AB = AC \), then angle \( A \) equals angle \( C \).

Since \( AC \) and \( AD \) are equal and angles opposite to equal sides are equal, it follows that triangle ACD has two sides equal, thus \( AC = DC \).

According to the converse of the triangle inequality, if angle CAB is the largest angle, then side AC, which is opposite this angle, must be longer than side BC. This illustrates how angles and sides relate in triangles.

When a segment bisects an angle, it implies that the sides opposite the resulting equal angles can be compared. Here, since angle \( BAD = angle CAD \), it follows that side \( AB \) is greater than side \( BD \).

The corollary states that the sum of any two sides of a triangle is always greater than the length of the third side. This property is essential in triangle construction and proves that triangles exist under specific conditions.

If \( AB \) is the shortest side, then angle C, which is opposite this side, must be the smallest angle. This follows the rule that shorter sides correspond to smaller angles.

The theorem states that among all lines drawn from a point outside a given straight line to that line, the perpendicular line is the shortest. This is derived from the properties of right triangles where the angle opposite the longest side is the largest.