Triangle Inequality Theorem

Given a triangle with sides measuring 5 cm and 8 cm, which of the following could be the length of the third side, adhering to the Triangle Inequality Theorem?

If two sides of a triangle are 7 inches and 11 inches, which inequality represents all possible lengths, $x$, for the third side?

In triangle ABC, AB = 6 cm and BC = 8 cm. What is the range of possible lengths for side AC?

Which of the following sets of side lengths can form a triangle?

Can a triangle be formed with sides of the following lengths: 4, 5, and 10?

In $\triangle PQR$, if $PQ = 9$ and $QR = 12$, which of the following could be a possible length for PR?

Two sides of a triangle measure 6 cm and 11 cm. Which measurement could not be the length of the third side?

If the sides of a triangle are 3x, 4x, and 5x, what range of values must x have for the triangle to exist?

Given $\triangle ABC$ with sides $AB=5$, $BC=6$, and $CA=7$. Which angle has the smallest measure?

In $\triangle XYZ$, $XY = 8$, $YZ = 10$, and $XZ = 12$. Which angle has the largest measure?

In triangle $DEF$, if $DE = 15$, $EF = 13$, and $DF = 18$, which angle is the largest?

If in $\triangle ABC$, $AB < BC < AC$, then which of the following is true regarding the angles?

Given a triangle where $\angle A = 80^\circ$ and $\angle B = 60^\circ$, which side of the triangle is the longest?

In $\triangle PQR$, if $\angle P = 50^\circ$ and $\angle Q = 70^\circ$, which side is the shortest?

In $\triangle ABC$, if $m\angle A = 90^\circ$, $m\angle B = 60^\circ$, and $m\angle C = 30^\circ$, which side is the longest?

Given $\triangle XYZ$ with $\angle X = 45^\circ$, $\angle Y = 75^\circ$, and $\angle Z = 60^\circ$, arrange the sides in ascending order.

Two triangles, $\triangle ABC$ and $\triangle DEF$, have $AB = DE$ and $BC = EF$. If $m\angle B > m\angle E$, which statement is true according to the Hinge Theorem?

In $\triangle PQR$ and $\triangle STU$, $PQ = ST$ and $QR = TU$. If $PR < SU$, what can be concluded about the angles $\angle Q$ and $\angle T$?

Given $\triangle ABC$ and $\triangle DEF$ where $AB \cong DE$, $BC \cong EF$, and $AC > DF$. Which of the following statements is true?

Two triangles, $\triangle LMN$ and $\triangle PQR$, have sides such that $LM = PQ$ and $MN = QR$. If $\angle M$ measures $70^\circ$ and $\angle Q$ measures $55^\circ$, which of the following is always true?

In $\triangle ABC$, point D is on side AC such that BD bisects $\angle ABC$. If $AB=8$, $BC=10$, then which of the following must be true?

Given that in $\triangle ABC$, $\angle A$ is an obtuse angle, which side is the longest?

If $m\angle 1 = 130^\circ$ is an exterior angle of a triangle, what can you conclude about the measures of the two non-adjacent interior angles?

In a triangle, if one exterior angle measures $120^\circ$, what is the largest possible measure of one of the remote interior angles?

Considering the Exterior Angle Inequality Theorem, which statement is always true about an exterior angle of a triangle?

Given $\triangle ABC$, where D is a point on BC such that AD is not an altitude. If $\angle ADB$ is obtuse, then...

In quadrilateral ABCD, diagonals AC and BD intersect at point E. If $AB = BC$ and $\angle AEB > \angle BEC$, then:

In $\triangle ABC$, point $D$ lies on $BC$ such that $AD$ bisects $\angle BAC$. If $AB = 10$ and $AC = 15$, what can be said about the lengths of $BD$ and $DC$?

If in $\triangle ABC$, $D$ is a point on $BC$ such that $AD \perp BC$. Which of the following is always true?

Given the sides of a triangle are 5, 7 and 8, arrange the angles ($\angle A$, $\angle B$, $\angle C$) from smallest to largest, where $A$ is opposite the side of length 5, $B$ is opposite the side of length 7, and $C$ is opposite the side of length 8.

In triangles $\triangle ABC$ and $\triangle DEF$, $AB=DE$ and $BC=EF$. It is known that $AC > DF$. Which of the following can be concluded?

Which of the following statements correctly uses the Hinge Theorem?

If you were to apply triangle inequalities to estimate the distance between three cities, A, B, and C, what real-world assumption must you make to apply the theorem correctly?

In the real world, how might understanding triangle inequalities help ensure the structural integrity of a bridge design?

A surveyor needs to determine the possible distances between two points across a lake. He knows the distance from his position to each of the points. How can he use the Triangle Inequality Theorem?

How does the concept of triangle inequality apply to navigation and route planning?

Given a fixed length of fencing material, how does the Triangle Inequality Theorem inform the maximum area you can enclose in a triangular shape?

How could the principles behind the triangle inequality theorem be applied to optimize network routing in computer science?

Consider the design of a suspension bridge. How does understanding triangle inequalities contribute to the stability and safety of the structure?

Imagine you're placing two security cameras to cover a specific area. How might triangle inequalities guide your decision-making?

How can understanding triangle inequalities aid in architectural design, particularly in ensuring the stability of roof structures?

Explain how triangle inequalities could be applied to GPS technology to improve the accuracy of location tracking.