Question

lesson quizinequalities in one triangle1. select all the angle relationships in $\triangle abc$ that are correct.$square$ a. $mangle a < mangle cLXB0square$ c. $mangle a < mangle bLXB1square$ e. $mangle c < mangle a < mangle b$2. select all the side-angle relationships in $\triangle abc$ that are true.$square$ a. the largest angle is opposite the smallest side.$square$ b. the smallest angle is opposite the largest side.$square$ c. the smallest angle is opposite the smallest side.$square$ d. the largest angle is opposite the largest side.$square$ e. the side lengths and angle measures are not related.3. in $\triangle xyz$, suppose $xy < xz$. what inequality relates two angles in $\triangle xyz$?$circled{a}$ $mangle x < mangle yLXB2circled{c}$ $mangle y < mangle zLXB3circled{b}$ $overline{mn}, overline{mo}, overline{no}LXB4circled{d}$ $overline{no}, overline{mo}, overline{mn}$5. a triangle has two sides measuring 8.5 cm and 15 cm. what is the greatest whole number length possible for the third side? explain.$circled{a}$ 23; the third side length must be greater than $8.5 + 15 = 23.5$. the smallest whole number greater than 23.5 is 23.$circled{b}$ 24; the third side length must be less than $8.5 + 15 = 23.5$. the greatest whole number less than 23.5 is 24.$circled{c}$ 23; the third side length must be less than $8.5 + 15 = 23.5$. the greatest whole number less than 23.5 is 23.$circled{d}$ 24; the third side length must be greater than $8.5 + 15 = 23.5$. the smallest whole number greater than 23.5 is 24.

Explanation:

Step1: Analyze △ABC side lengths

In △ABC, side lengths: $BC=3$, $AC=2.5$, $AB=2.1$

Step2: Match angles to opposite sides

• $m\angle A$ opposes $BC$ (longest side, so $m\angle A$ largest)
• $m\angle B$ opposes $AC$ (middle side, so $m\angle B$ middle)
• $m\angle C$ opposes $AB$ (shortest side, so $m\angle C$ smallest)

Thus $m\angle C < m\angle B < m\angle A$, so correct options: C, E

Step3: Side-angle relationship rules

For any triangle, largest angle ↔ longest side; smallest angle ↔ shortest side. Correct options: C, D

Step4: Relate sides to angles in △XYZ

$XY < XZ$: $XY$ opposes $m\angle Z$, $XZ$ opposes $m\angle Y$. So $m\angle Z < m\angle Y$. Correct option: D

Step5: Find angles of △MNO

• $m\angle M = 61^\circ$, $m\angle N = 54^\circ$
• $m\angle O = 180^\circ - 61^\circ - 54^\circ = 65^\circ$

Order of angles: $m\angle N < m\angle M < m\angle O$
Opposite sides: $\overline{MO} < \overline{NO} < \overline{MN}$. Correct option: A

Step6: Triangle inequality for third side

Third side $x$: $15-8.5 < x < 15+8.5$ → $6.5 < x < 23.5$. Greatest whole number: 23. Correct option: C

Answer:

1. C. $m\angle A < m\angle B$, E. $m\angle C < m\angle A < m\angle B$
2. C. The smallest angle is opposite the smallest side., D. The largest angle is opposite the largest side.
3. D. $m\angle Z < m\angle Y$
4. A. $\overline{MO}, \overline{NO}, \overline{MN}$
5. C. 23; the third side length must be less than $8.5 + 15 = 23.5$. The greatest whole number less than 23.5 is 23.