chapter 3: angle pairs

e. do the proof below.
given: $\overline{gd}$ bisects $\angle edf$; $\angle 1$ and $\angle 3$ are supplementary; $\angle 2$ and $\angle 4$ are supplementary.
prove: $m\angle 1 = m\angle 2$

problem set 16
tell whether each sentence below is true or false.
1. supplementary angles are angles with measures that add to equal 180.
2. the supplement of an acute angle is acute.

complete each sentence below with the best of the choices given.
3. the of a line segment is the point that divides the line segment into two congruent line segments.
a. bisector b. vertex c. end point d. midpoint e. divisor point
4. a is a set of points that forms a flat surface which extends forever in all directions and has length and width, but no depth.
a. line b. ray c. line segment d. plane e. bisector
5. two points determine a unique.
a. plane b. ray c. surface d. angle e. straight line

find the measure of the complement and supplement of each angle below.
6. $m\angle a=25$
7. $m\angle b = 88$
(a)8. $m\angle c=x$
a. complement: $90 + x$ supplement: $180 + x$
b. complement: $180 - x$ supplement: $90 - x$
c. complement: $90 - x$ supplement: $180 - x$
d. complement: $45 + x$ supplement: $90 + x$
e. complement: $x - 90$ supplement: $x - 180$

from each given statement below, select the definition, property, postulate, or theorem that justifies the prove statement.
9. given: $ab + de=gh$ prove: $gh = ab + de$
a. definition of a segment bisector b. symmetric property c. addition property d. transitive property e. definition of equivalent line segments
10. given: $m\angle pst+m\angle lqr = 90$ and $m\angle def=m\angle lqr$
prove: $m\angle pst + m\angle def=90$
a. definition of an angle bisector b. symmetric property c. betweenness of rays d. substitution property e. if two angles are complementary to the same angle or congruent angles, then they are congruent.

Question

chapter 3: angle pairs

e. do the proof below.
 given: $\overline{gd}$ bisects $\angle edf$; $\angle 1$ and $\angle 3$ are supplementary; $\angle 2$ and $\angle 4$ are supplementary.
 prove: $m\angle 1 = m\angle 2$

problem set 16
 tell whether each sentence below is true or false.
 1. supplementary angles are angles with measures that add to equal 180.
 2. the supplement of an acute angle is acute.

complete each sentence below with the best of the choices given.
 3. the of a line segment is the point that divides the line segment into two congruent line segments.
 a. bisector b. vertex c. end point d. midpoint e. divisor point
 4. a is a set of points that forms a flat surface which extends forever in all directions and has length and width, but no depth.
 a. line b. ray c. line segment d. plane e. bisector
 5. two points determine a unique.
 a. plane b. ray c. surface d. angle e. straight line

find the measure of the complement and supplement of each angle below.
 6. $m\angle a=25$
 7. $m\angle b = 88$
 (a)8. $m\angle c=x$
 a. complement: $90 + x$ supplement: $180 + x$
 b. complement: $180 - x$ supplement: $90 - x$
 c. complement: $90 - x$ supplement: $180 - x$
 d. complement: $45 + x$ supplement: $90 + x$
 e. complement: $x - 90$ supplement: $x - 180$

from each given statement below, select the definition, property, postulate, or theorem that justifies the prove statement.
 9. given: $ab + de=gh$ prove: $gh = ab + de$
 a. definition of a segment bisector b. symmetric property c. addition property d. transitive property e. definition of equivalent line segments
 10. given: $m\angle pst+m\angle lqr = 90$ and $m\angle def=m\angle lqr$
 prove: $m\angle pst + m\angle def=90$
 a. definition of an angle bisector b. symmetric property c. betweenness of rays d. substitution property e. if two angles are complementary to the same angle or congruent angles, then they are congruent.

Accepted Answer

# Explanation:
## Step1: Recall angle - bisector property
Since $\overline{GD}$ bisects $\angle EDF$, we have $\angle 3=\angle 4$ by the definition of an angle - bisector.
## Step2: Recall supplementary - angle property
We know that $\angle 1$ and $\angle 3$ are supplementary, so $m\angle 1 + m\angle 3=180^{\circ}$, which implies $m\angle 1=180^{\circ}-m\angle 3$. Also, $\angle 2$ and $\angle 4$ are supplementary, so $m\angle 2 + m\angle 4 = 180^{\circ}$, which implies $m\angle 2=180^{\circ}-m\angle 4$.
## Step3: Substitute equal angles
Since $\angle 3=\angle 4$, we can substitute $\angle 3$ for $\angle 4$ in the equation for $m\angle 2$. So $m\angle 2=180^{\circ}-m\angle 3$.
Since $m\angle 1=180^{\circ}-m\angle 3$ and $m\angle 2=180^{\circ}-m\angle 3$, we conclude that $m\angle 1 = m\angle 2$.

1. **True - False questions**:
   - 1. True. Supplementary angles are defined as angles whose measures add up to 180.
   - 2. False. The supplement of an acute angle (an angle with measure less than 90) is an obtuse angle (an angle with measure greater than 90 and less than 180).
2. **Fill - in - the - blank questions**:
   - 3. D. The mid - point of a line segment is the point that divides the line segment into two congruent line segments.
   - 4. D. A plane is a set of points that forms a flat surface which extends forever in all directions and has length and width, but no depth.
   - 5. E. Two points determine a unique straight line.
3. **Complementary and supplementary angles**:
   - 6. Complement of $\angle A$: $90 - 25=65$, Supplement of $\angle A$: $180 - 25 = 155$.
   - 7. Complement of $\angle B$: $90 - 88 = 2$, Supplement of $\angle B$: $180 - 88=92$.
4. **Justification questions**:
   - 9. B. Given $AB + DE=GH$, then $GH = AB + DE$ by the Symmetric Property (if $a = b$, then $b = a$).
   - 10. D. Given $m\angle PST+m\angle LQR = 90$ and $m\angle DEF=m\angle LQR$, we can substitute $\angle LQR$ with $\angle DEF$ in the first equation to get $m\angle PST + m\angle DEF=90$ by the Substitution Property.

# Answer:
e. Proven that $m\angle 1 = m\angle 2$ as shown above.
1. True
2. False
3. D. mid - point
4. D. plane
5. E. straight line
6. Complement: 65, Supplement: 155
7. Complement: 2, Supplement: 92
8. C. complement: $90 - x$, supplement: $180 - x$
9. B. Symmetric Property
10. D. Substitution Property

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall angle - bisector property

Step2: Recall supplementary - angle property

Step3: Substitute equal angles

Answer: