Question

b a triangle has two sides with lengths 9 inches and 16 inches. describe the possible lengths of the third side of the triangle.

in exercises 2 and 3, determine which two statements contradict each other. explain your reasoning.
②
a $\triangle lmn$ is equilateral.
b $lm \
eq mn$
c $\angle l = \angle m$
③
a $\triangle abc$ is a right triangle.
b $\angle a$ is acute.
c $\angle c$ is obtuse.

c complete the phrase with the most logical conclusion.

1. if there is no right angle in a triangle, then...
2. if two lines do not have the same slope, then...
3. if a quadrilateral does not have four right angles, then...
4. if no two angles of a triangle are congruent, then...
5. if the sum of the measures of the interior angles of a polygon is not $180^\circ$, then...

Explanation:

Part B

Step1: Apply triangle inequality theorem

For a triangle with sides $a$, $b$, $c$, the inequalities are:
$a + b > c$, $a + c > b$, $b + c > a$
Let the known sides be $9$ and $16$, unknown side be $x$.

Step2: Set up first inequality

$9 + 16 > x$
$\implies 25 > x$

Step3: Set up second inequality

$9 + x > 16$
$\implies x > 16 - 9$
$\implies x > 7$

Step4: Set up third inequality

$16 + x > 9$
$\implies x > 9 - 16$
$\implies x > -7$ (irrelevant, side length positive)

For Exercise 2:

Step1: Recall equilateral triangle properties

An equilateral triangle has all sides equal ($LM = MN = NL$) and all angles equal ($\angle L = \angle M = \angle N$).

Step2: Contradict with given statements

Statement A says $\triangle LMN$ is equilateral, which requires $LM = MN$ and $\angle L = \angle M$. Statements B ($LM
eq MN$) and C ($\angle L = \angle M$) contradict A: B directly violates side equality, and A requires equal angles so C aligns with A, but B and A cannot both be true.

For Exercise 3:

Step1: Recall right triangle angle properties

A right triangle has one right angle ($90^\circ$), so the other two angles are acute (sum to $90^\circ$).

Step2: Contradict with given statements

Statement A says $\triangle ABC$ is a right triangle (has one right angle). Statement C says $\angle C$ is obtuse (greater than $90^\circ$). A triangle cannot have both a right angle and an obtuse angle (sum of angles would exceed $180^\circ$), so A and C contradict each other.

1. A triangle with no right angle cannot be a right triangle, so it is an oblique triangle (acute or obtuse).
2. Lines with different slopes are not parallel, so they intersect at exactly one point.
3. A quadrilateral without four right angles is not a rectangle or square, so it does not have all angles equal to $90^\circ$.
4. If no two angles of a triangle are congruent, no two sides opposite those angles are congruent, so it is a scalene triangle.
5. The sum of interior angles of a triangle is $180^\circ$, so if the sum is not $180^\circ$, the polygon is not a triangle.

Answer:

The length of the third side must be greater than 7 inches and less than 25 inches, or $7 < x < 25$.

---

Exercises 2 and 3