Question

guided practice
vocabulary apply the vocabulary from this lesson to answer each question.

1. an angle measures 1. what is the measure of its complement? what is the measure of its supplement?
2. ∠aoc and ∠cob are adjacent angles. which side do the angles have in common?
3. tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.
4. ∠1 and ∠2
5. ∠1 and ∠3
6. ∠2 and ∠4
7. ∠2 and ∠3
8. find the measure of each of the following:
9. supplement of ∠1
10. complement of ∠1
11. supplement of ∠8
12. complement of ∠8
13. multi - step an angles measure is 6 degrees more than 3 times the measure of its complement. find the measure of the angle.
14. landscaping a sprinkler swings back and forth between a and b in such a way that ∠1≅∠2. ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary. if m∠1 = 47.5°, find m∠2, m∠3, and m∠4.
15. name each pair of vertical angles.

Explanation:

1. For question 1:

• # Explanation:
• ## Step1: Recall the definition of complementary and supplementary angles
• Complementary angles add up to $90^{\circ}$, and supplementary angles add up to $180^{\circ}$.
• ## Step2: Calculate the complement
• If an angle measures $x = 7^{\circ}$, the measure of its complement is $90 - 7=83^{\circ}$.
• ## Step3: Calculate the supplement
• The measure of its supplement is $180 - 7 = 173^{\circ}$.
• # Answer:
• Complement: $83^{\circ}$, Supplement: $173^{\circ}$

1. For question 11:

• # Explanation:
• ## Step1: Let the measure of the angle be $x$ and its complement be $y$.
• We know that $x + y=90^{\circ}$ (by the definition of complementary - angles), and $x = 3y+6$.
• ## Step2: Substitute $x$ in the first - equation
• Substitute $x = 3y + 6$ into $x + y=90$. We get $(3y + 6)+y=90$.
• Combine like terms: $4y+6 = 90$.
• ## Step3: Solve for $y$
• Subtract 6 from both sides: $4y=90 - 6=84$.
• Divide both sides by 4: $y = 21^{\circ}$.
• ## Step4: Solve for $x$
• Since $x=3y + 6$, substitute $y = 21$ into this equation. $x=3\times21+6=63 + 6=69^{\circ}$.
• # Answer:
• The measure of the angle is $69^{\circ}$

1. For question 12:

• # Explanation:
• ## Step1: Use the given equal - angle relationship
• Given $\angle1=\angle2$ and $m\angle1 = 42.5^{\circ}$, so $m\angle2=42.5^{\circ}$.
• ## Step2: Use the complementary - angle relationship for $\angle1$ and $\angle3$
• Since $\angle1$ and $\angle3$ are complementary, and the sum of complementary angles is $90^{\circ}$. Let $m\angle3=x$. Then $m\angle1+x = 90^{\circ}$.
• Substitute $m\angle1 = 42.5^{\circ}$ into the equation: $42.5+x = 90$.
• Solve for $x$: $x=m\angle3=90 - 42.5 = 47.5^{\circ}$.
• ## Step3: Use the complementary - angle relationship for $\angle2$ and $\angle4$
• Since $\angle2$ and $\angle4$ are complementary, and $m\angle2 = 42.5^{\circ}$. Let $m\angle4 = y$. Then $m\angle2+y = 90^{\circ}$.
• Substitute $m\angle2 = 42.5^{\circ}$ into the equation: $42.5+y = 90$.
• Solve for $y$: $y=m\angle4=90 - 42.5 = 47.5^{\circ}$.
• # Answer:
• $m\angle2 = 42.5^{\circ}$, $m\angle3 = 47.5^{\circ}$, $m\angle4 = 47.5^{\circ}$

Answer:

1. For question 1:

• # Explanation:
• ## Step1: Recall the definition of complementary and supplementary angles
• Complementary angles add up to $90^{\circ}$, and supplementary angles add up to $180^{\circ}$.
• ## Step2: Calculate the complement
• If an angle measures $x = 7^{\circ}$, the measure of its complement is $90 - 7=83^{\circ}$.
• ## Step3: Calculate the supplement
• The measure of its supplement is $180 - 7 = 173^{\circ}$.
• # Answer:
• Complement: $83^{\circ}$, Supplement: $173^{\circ}$

1. For question 11:

• # Explanation:
• ## Step1: Let the measure of the angle be $x$ and its complement be $y$.
• We know that $x + y=90^{\circ}$ (by the definition of complementary - angles), and $x = 3y+6$.
• ## Step2: Substitute $x$ in the first - equation
• Substitute $x = 3y + 6$ into $x + y=90$. We get $(3y + 6)+y=90$.
• Combine like terms: $4y+6 = 90$.
• ## Step3: Solve for $y$
• Subtract 6 from both sides: $4y=90 - 6=84$.
• Divide both sides by 4: $y = 21^{\circ}$.
• ## Step4: Solve for $x$
• Since $x=3y + 6$, substitute $y = 21$ into this equation. $x=3\times21+6=63 + 6=69^{\circ}$.
• # Answer:
• The measure of the angle is $69^{\circ}$

1. For question 12:

• # Explanation:
• ## Step1: Use the given equal - angle relationship
• Given $\angle1=\angle2$ and $m\angle1 = 42.5^{\circ}$, so $m\angle2=42.5^{\circ}$.
• ## Step2: Use the complementary - angle relationship for $\angle1$ and $\angle3$
• Since $\angle1$ and $\angle3$ are complementary, and the sum of complementary angles is $90^{\circ}$. Let $m\angle3=x$. Then $m\angle1+x = 90^{\circ}$.
• Substitute $m\angle1 = 42.5^{\circ}$ into the equation: $42.5+x = 90$.
• Solve for $x$: $x=m\angle3=90 - 42.5 = 47.5^{\circ}$.
• ## Step3: Use the complementary - angle relationship for $\angle2$ and $\angle4$
• Since $\angle2$ and $\angle4$ are complementary, and $m\angle2 = 42.5^{\circ}$. Let $m\angle4 = y$. Then $m\angle2+y = 90^{\circ}$.
• Substitute $m\angle2 = 42.5^{\circ}$ into the equation: $42.5+y = 90$.
• Solve for $y$: $y=m\angle4=90 - 42.5 = 47.5^{\circ}$.
• # Answer:
• $m\angle2 = 42.5^{\circ}$, $m\angle3 = 47.5^{\circ}$, $m\angle4 = 47.5^{\circ}$