Question

1. m∠qlm = x + 70, m∠klm = 91°, and m∠klq = x + 31. find m∠klq.
2. m∠bef = x + 162, m∠deb = x + 38, and m∠def = 176°. find m∠deb.

classify each angle as acute, obtuse, right, or straight.
11)
12)
13)
14)
name the relationship: complementary, linear pair, vertical, or adjacent.
15)
16)
17)
18)

Explanation:

Step1: Set up equation for problem 9

Since $\angle QLM=\angle KLQ + \angle KLM$, we have $(x + 70)=(x + 31)+91$. Simplifying the right - hand side gives $x+70=x + 122$, which is incorrect. It should be $\angle KLM=\angle KLQ+\angle QLM$. So, $91=(x + 31)+(x + 70)$.

Step2: Solve the equation for problem 9

Combine like terms: $91 = 2x+101$. Subtract 101 from both sides: $91−101=2x$, so $- 10 = 2x$. Divide both sides by 2: $x=-5$.

Step3: Find $\angle KLQ$ for problem 9

Substitute $x = - 5$ into the expression for $\angle KLQ$. $\angle KLQ=x + 31=-5 + 31 = 26^{\circ}$.

Step4: Set up equation for problem 10

Since $\angle DEF=\angle BEF+\angle DEB$, we have $176=(x + 162)+(x + 38)$.

Step5: Solve the equation for problem 10

Combine like terms: $176=2x + 200$. Subtract 200 from both sides: $176−200=2x$, so $-24 = 2x$. Divide both sides by 2: $x=-12$.

Step6: Find $\angle DEB$ for problem 10

Substitute $x=-12$ into the expression for $\angle DEB$. $\angle DEB=x + 38=-12 + 38 = 26^{\circ}$.

Step7: Classify angles for 11 - 14

• For 11: Acute (less than $90^{\circ}$).
• For 12: Right (forms a $90^{\circ}$ angle).
• For 13: Acute (less than $90^{\circ}$).
• For 14: Obtuse (greater than $90^{\circ}$ and less than $180^{\circ}$).

Step8: Name relationships for 15 - 18

• For 15: Complementary (sum to $90^{\circ}$).
• For 16: Adjacent (share a common side and a common vertex).
• For 17: Adjacent (share a common side and a common vertex).
• For 18: Vertical (opposite angles formed by two intersecting lines).

Answer:

1. $m\angle KLQ = 26^{\circ}$
2. $m\angle DEB = 26^{\circ}$
3. Acute
4. Right
5. Acute
6. Obtuse
7. Complementary
8. Adjacent
9. Adjacent
10. Vertical