Question

complete each sentence below by selecting always, sometimes, or never. 14. complementary angles are congruent. a. always b. sometimes c. never 15. an angle bisector divides the angle into two congruent angles. a. always b. sometimes c. never find the measure of each angle below. (b)16. ∠lno (18) 17. ∠kln (18) answer each question below. 18. ∠ads and ∠bct are complementary. if m∠ads = 3z - 1 and m∠bct = z + 7, then find m∠ads. (c)19. ∠r and ∠j are a linear pair. if m∠r = 3x - 5 and m∠j = 5x + 1, find m∠j. 20. if g - h - i, and if gh = 2x - 9, hi = 14, and gi = 3x - 8, then find gi. (d)21. if (overline{pb}) bisects ∠apc, and if m∠apb = 2x + 9 and m∠apc = 3x + 35 then find m∠apc.

Explanation:

Step1: Define complementary angles

Complementary angles add up to 90°. They are congruent when each is 45°, so sometimes.

Step2: Recall angle - bisector definition

By definition, an angle bisector always divides an angle into two congruent angles.

Step3: Find vertical - angle relationship

∠JLM and ∠KLN are vertical angles, so ∠KLN = 100°. Also, ∠LNO and $\frac{2}{7}x^{\circ}$ are vertical angles. And since ∠JLM and ∠KLN are vertical angles, ∠KLN = 100°.
For ∠LNO, we know that vertical angles are equal.
For 18:
Since ∠ADS and ∠BCT are complementary, $m\angle ADS + m\angle BCT=90^{\circ}$. So, $(3z - 1)+(z + 7)=90$.

Step4: Solve the equation for z

Combine like terms: $4z+6 = 90$. Subtract 6 from both sides: $4z=84$, then $z = 21$. So, $m\angle ADS=3z - 1=3\times21-1=62^{\circ}$.
For 19:
Since ∠R and ∠J are a linear - pair, $m\angle R+m\angle J = 180^{\circ}$. So, $(3x - 5)+(5x + 1)=180$.

Step5: Solve the equation for x

Combine like terms: $8x-4 = 180$. Add 4 to both sides: $8x=184$, then $x = 23$. So, $m\angle J=5x + 1=5\times23+1 = 116^{\circ}$.
For 20:
Since $G - H - I$, $GH+HI=GI$. So, $(2x - 9)+14=3x - 8$.

Step6: Solve the equation for x

Combine like terms: $2x + 5=3x - 8$. Subtract 2x from both sides: $5=x - 8$. Add 8 to both sides: $x = 13$. So, $GI=3x - 8=3\times13-8=31$.
For 21:
Since $\overline{PB}$ bisects ∠APC, $m\angle APC = 2m\angle APB$. So, $3x + 35=2(2x + 9)$.

Step7: Solve the equation for x

Expand the right - hand side: $3x + 35=4x + 18$. Subtract 3x from both sides: $35=x + 18$. Subtract 18 from both sides: $x = 17$. So, $m\angle APC=3x + 35=3\times17+35=86^{\circ}$.

Answer:

1. B. sometimes
2. A. always
3. $\frac{2}{7}x^{\circ}$
4. $100^{\circ}$
5. $m\angle ADS = 62^{\circ}$
6. $m\angle J = 116^{\circ}$
7. $GI = 31$
8. $m\angle APC = 86^{\circ}$