Question

1. given: be bisects ∠abd, bd bisects ∠ebc. prove: ∠abe = ∠dbc. statements: 1. be bisects ∠abd. 2. ∠abe = ∠ebd. 3. bd bisects ∠ebc. 4. ∠ebd = ∠dbc. 5. ∠abe = ∠dbc. reasons: 1. 2. 3. 4. 5. 5. given: ∠rsu ≅ ∠vst. prove: ∠rsv ≅ ∠ust. statements: 1. ∠rsu ≅ ∠vst. 2. m∠rsu = m∠vst. 3. m∠rsu + m∠usv = m∠rsv. 4. m∠vst + m∠usv = m∠ust. 5. m∠rsu + m∠usv = m∠ust. 6. m∠rsv = m∠ust. 7. ∠rsv ≅ ∠ust. reasons: 1. 2. 3. 4. 5. 6. 7. 6. given: ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary. prove: ∠1 ≅ ∠4. statements: 1. ∠1 and ∠2 are complementary. 2. ∠3 and ∠4 are complementary. 3. m∠1 + m∠2 = 90°. 4. m∠3 + m∠4 = 90°. 5. ∠2 ≅ ∠3. 6. m∠2 = m∠3. 7. m∠1 + m∠2 = m∠3 + m∠4. 8. m∠1 + m∠3 = m∠3 + m∠4. 9. m∠1 = m∠4. 10. ∠1 ≅ ∠4. reasons: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Explanation:

Question 4

Step1: Given information

Given that $BE$ bisects $\angle ABD$. By the definition of an angle - bisector, if a ray bisects an angle, it divides the angle into two congruent angles. So the reason for "1. $BE$ bisects $\angle ABD$" is "Given" and the reason for "2. $\angle ABE=\angle EBD$" is "Definition of angle - bisector".

Step2: Given information

Given that $BD$ bisects $\angle EBC$. So the reason for "3. $BD$ bisects $\angle EBC$" is "Given". And by the definition of an angle - bisector, $\angle EBD=\angle DBC$, so the reason for "4. $\angle EBD = \angle DBC$" is "Definition of angle - bisector".

Step3: Transitive property

Since $\angle ABE=\angle EBD$ and $\angle EBD=\angle DBC$, by the transitive property of equality (if $a = b$ and $b = c$, then $a = c$), we have $\angle ABE=\angle DBC$. So the reason for "5. $\angle ABE=\angle DBC$" is "Transitive property of equality".

Question 5

Step1: Given information

The reason for "1. $\angle RSU\cong\angle VST$" is "Given".

Step2: Congruent angles have equal measures

If two angles are congruent, then their measures are equal. So the reason for "2. $m\angle RSU = m\angle VST$" is "Definition of congruent angles".

Step3: Angle - addition postulate

The measure of the whole angle is the sum of the measures of its non - overlapping parts. So for $\angle RSV$, $m\angle RSU+m\angle USV=m\angle RSV$, and the reason is "Angle - addition postulate". Similarly, for $\angle UST$, $m\angle VST+m\angle USV=m\angle UST$ (reason: Angle - addition postulate).

Step4: Substitution

Since $m\angle RSU = m\angle VST$, we can substitute $m\angle RSU$ for $m\angle VST$ in the equation $m\angle VST+m\angle USV=m\angle UST$. So $m\angle RSU+m\angle USV=m\angle UST$. The reason for "6. $m\angle RSV=m\angle UST$" is substitution. And if two angles have equal measures, they are congruent. So the reason for "7. $\angle RSV\cong\angle UST$" is "Definition of congruent angles".

Question 6

Step1: Given information

The reasons for "1. $\angle1$ and $\angle2$ are complementary" and "2. $\angle3$ and $\angle4$ are complementary" are "Given".

Step2: Definition of complementary angles

If two angles are complementary, the sum of their measures is $90^{\circ}$. So the reason for "3. $m\angle1 + m\angle2=90^{\circ}$" and "4. $m\angle3 + m\angle4 = 90^{\circ}$" is "Definition of complementary angles".

Step3: Given (assumed from the problem context, missing given information about $\angle2\cong\angle3$)

Let's assume $\angle2\cong\angle3$ is given. Then the reason for "5. $\angle2\cong\angle3$" is "Given" and for "6. $m\angle2=m\angle3$" is "Definition of congruent angles".

Step4: Substitution

Substitute $m\angle2$ with $m\angle3$ in the equation $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle4 = 90^{\circ}$. We get $m\angle1 + m\angle2=m\angle3 + m\angle4$ (reason: Substitution). Then, since $m\angle2=m\angle3$, we can subtract $m\angle3$ from both sides of the equation $m\angle1 + m\angle3=m\angle3 + m\angle4$ to get $m\angle1=m\angle4$ (reason: Subtraction property of equality). And if two angles have equal measures, they are congruent. So the reason for "10. $\angle1\cong\angle4$" is "Definition of congruent angles".

Answer:

Question 4

1. Given
2. Definition of angle - bisector
3. Given
4. Definition of angle - bisector
5. Transitive property of equality

Question 5

1. Given
2. Definition of congruent angles
3. Angle - addition postulate
4. Angle - addition postulate
5. Substitution
6. Substitution
7. Definition of congruent angles

Question 6

1. Given
2. Given
3. Definition of complementary angles
4. Definition of complementary angles
5. Given (assumed)
6. Definition of congruent angles
7. Substitution
8. Subtraction property of equality
9. Definition of congruent angles
10. Definition of congruent angles