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Answer:
For the pairs that satisfy that the sum is less than 7 we have:
(1,1) (2,1) (1,2) (3,1) (2,2) (1,3) (4,1) (3,2) (2,3) (1,4) (5,1) (4.2) (3,3) (2,4) (1,5)
So we have a total of 15 pairs where the sum is less than 7
And for an odd sum we have the following pairs
(2,1) (1,2) (4,1) (3,2) (2,3) (1,4) (6,2) (5,2) (4,3) (3,4) (2,5) (1,6) (6,3) (5,4) (4,5) (3,6) (6,5) (5,6)
A total of 18 pairs and we have 6 pairs who are in both cases for the sum less than 7 and the sum an odd number that represent the intersection and using the total probability rule we got:
Step-by-step explanation:
For this case when a pair of dice is rolled we have the following sample pace for the outcomes:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
We see 36 possible outcomes and we want to find how many of these pairs we got rolling an odd sum or a sum less than 7
For the pairs that satisfy that the sum is less than 7 we have:
(1,1) (2,1) (1,2) (3,1) (2,2) (1,3) (4,1) (3,2) (2,3) (1,4) (5,1) (4.2) (3,3) (2,4) (1,5)
So we have a total of 15 pairs where the sum is less than 7
And for an odd sum we have the following pairs
(2,1) (1,2) (4,1) (3,2) (2,3) (1,4) (6,2) (5,2) (4,3) (3,4) (2,5) (1,6) (6,3) (5,4) (4,5) (3,6) (6,5) (5,6)
A total of 18 pairs and we have 6 pairs who are in both cases for the sum less than 7 and the sum an odd number that represent the intersection and using the total probability rule we got:
To enable the completion of the proof that line <em>l</em> is parallel to line <em>m</em>, a
diagram showing the lines and their common transversal is attached.
The completed two column proof is presented as follows;
Statement Reason
1. ∠1 and ∠2 are supplementary angles 1. Given
2. m∠1 + m∠2 = 180° 2. <u>Definition of supplementary ∠s</u>
3. ∠1 and ∠3 are supplementary angles 3. Exterior sides in opposite rays
4. <u>m∠1 + m∠3 = 180° </u> 4. <u>Definition of supplementary ∠s</u>
5. m∠1 + m∠2 = m∠1 + m∠3 5. <u>Transitive property of equality</u>
6. <u>m∠2 = m∠3 </u> 6. <u>Subtraction property of equality</u>
7. l ║ m 7. <u>Converse of alternate interior </u>
<u>angles postulate</u>
Reasons:
- Reason for statement 2: Supplementary angles are defined as two angles that sum up to 180°
- Reason for statement 3: Two angles are supplementary if the exterior sides that form each angle are opposite rays (rays that are drawn out infinitely in opposite direction but have the same endpoint)
- Statement 4: Mathematical expression of the sum of ∠1 and ∠3; Reason for statement 4 is the definition of supplementary angles
- Reason for statement 5: Transitive property of equality describes the property that if a number <em>x</em> = <em>y</em>, and <em>z </em>= <em>y</em>, then <em>x</em> = <em>z</em>.
- Statement 6: Subtracting m∠1 from both sides of the equation in statement 5. gives; m∠1 + m∠2 - m∠1 = m∠1 + m∠3 - m∠1 ⇒ m∠2 = m∠3. Reason for statement 6 is the subtraction property of equality
- Reason for statement 7: The converse of the alternate interior angles postulate states that if the alternate interior angles formed between two lines and a common transversal are congruent, the two lines are parallel.
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