11. supply the missing statements and reasons.
given: pr bisects $ \angle qps ; \angle qps $ is a right angle.
prove: $ m \angle r p s = 45 ^ { \circ } $
| statements | reasons |
| :---: | :---: |
| 1. $ p r $ bisects $ \angle q p s $. | 1. |
| 2. $ m \angle q p r = m \angle r p s $ | 2. definition of |
| 3. $ \angle q p s $ is a right angle. | 3. |
| 4. $ m \angle q p s = $ | 4. definition of |
| 5. $ m \angle q p r + m \angle r p s = $ $ m \angle q p s $ | 5. postulate |
| 6. $ m \angle r p s + m \angle r p s = $ $ 90 ^ { \circ } $ | 6. property |
| 7. $ 2 ( m \angle r p s ) = 90 ^ { \circ } $ | 7. distributive property |
| 8. $ m \angle r p s = $ | 8. |

Question

Accepted Answer

# Explanation:
## Step1: Identify Given
Given that $ \overline{PR} $ bisects $ \angle QPS $, so this is the given information for reason 1.
## Step2: Definition of Angle Bisector
By definition of angle bisector, $ m\angle QPR = m\angle RPS $, so reason 2 is "angle bisector".
## Step3: Given (Right Angle)
$ \angle QPS $ is a right angle is given, so reason 3 is "Given".
## Step4: Measure of Right Angle
A right angle measures $ 90^\circ $, so $ m\angle QPS = 90^\circ $ by definition of right angle, reason 4 is "right angle".
## Step5: Angle Addition Postulate
$ m\angle QPR + m\angle RPS = m\angle QPS $ by Angle Addition Postulate, so reason 5 is "Angle Addition".
## Step6: Substitution Property
Since $ m\angle QPR = m\angle RPS $, substitute $ m\angle QPR $ with $ m\angle RPS $, so $ m\angle RPS + m\angle RPS = 90^\circ $ by Substitution Property, reason 6 is "Substitution".
## Step7: Simplify
$ 2(m\angle RPS)=90^\circ $ (already given as step 7 with Distributive Property, but actually it's combining like terms).
## Step8: Solve for $ m\angle RPS $
Divide both sides by 2: $ m\angle RPS=\frac{90^\circ}{2} = 45^\circ $ by Division Property of Equality, reason 8 is "Division Property of Equality".

# Answer:
1. Given
2. angle bisector
3. Given
4. $ 90^\circ $; right angle
5. Angle Addition
6. Substitution
7. (As is, combining like terms from step 6)
8. $ 45^\circ $; Division Property of Equality

Filling the table:

| Statements | Reasons |
| --- | --- |
| 1. $ \overline{PR} $ bisects $ \angle QPS $. | 1. Given |
| 2. $ m\angle QPR = m\angle RPS $ | 2. Definition of angle bisector |
| 3. $ \angle QPS $ is a right angle. | 3. Given |
| 4. $ m\angle QPS = \boldsymbol{90^\circ} $ | 4. Definition of right angle |
| 5. $ m\angle QPR + m\angle RPS = m\angle QPS $ | 5. Angle Addition Postulate |
| 6. $ m\angle RPS + m\angle RPS = 90^\circ $ | 6. Substitution Property |
| 7. $ 2(m\angle RPS) = 90^\circ $ | 7. Distributive Property (or Combining Like Terms) |
| 8. $ m\angle RPS = \boldsymbol{45^\circ} $ | 8. Division Property of Equality |

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QUESTION IMAGE

Question

Explanation:

Step1: Identify Given

Step2: Definition of Angle Bisector

Step3: Given (Right Angle)

Step4: Measure of Right Angle

Step5: Angle Addition Postulate

Step6: Substitution Property

Step7: Simplify

Step8: Solve for \( m\angle RPS \)

Answer:

statements	reasons
2. \\( m \angle q p r = m \angle r p s \\)	2. definition of
3. \\( \angle q p s \\) is a right angle.	3.
4. \\( m \angle q p s = \\)	4. definition of
5. \\( m \angle q p r + m \angle r p s = \\) \\( m \angle q p s \\)	5. postulate
6. \\( m \angle r p s + m \angle r p s = \\) \\( 90 ^ { \circ } \\)	6. property
7. \\( 2 ( m \angle r p s ) = 90 ^ { \circ } \\)	7. distributive property
8. \\( m \angle r p s = \\)	8.

Statements	Reasons
2. \( m\angle QPR = m\angle RPS \)	2. Definition of angle bisector
3. \( \angle QPS \) is a right angle.	3. Given
4. \( m\angle QPS = \boldsymbol{90^\circ} \)	4. Definition of right angle
5. \( m\angle QPR + m\angle RPS = m\angle QPS \)	5. Angle Addition Postulate
6. \( m\angle RPS + m\angle RPS = 90^\circ \)	6. Substitution Property
7. \( 2(m\angle RPS) = 90^\circ \)	7. Distributive Property (or Combining Like Terms)
8. \( m\angle RPS = \boldsymbol{45^\circ} \)	8. Division Property of Equality