Answer:
1. m∠R > 90°
2. m∠S + m∠T < 90°
4. m∠R > m∠T
5. m∠R > m∠S
Step-by-step explanation:
<h3>General strategy</h3>
- prove the statement starting from known facts, or
- disprove the statement by finding a counterexample
Helpful fact: Recall that the Triangle Sum Theorem states that m∠R + m∠S + m∠T = 180°.
<u>Option 1. m∠R > 90°</u>
Start with m∠R > m∠S + m∠T.
Adding m∠R to both sides of the inequality...
m∠R + m∠R > m∠R + m∠S + m∠T
There are two things to note here:
- The left side of this inequality is 2*m∠R
- The right side of the inequality is exactly equal to the Triangle Sum Theorem expression
2* m∠R > 180°
Dividing both sides of the inequality by 2...
m∠R > 90°
So, the first option must be true.
<u>Option 2. m∠S + m∠T < 90°</u>
Start with m∠R > m∠S + m∠T.
Adding (m∠S + m∠T) to both sides of the inequality...
m∠R + (m∠S + m∠T) > m∠S + m∠T + (m∠S + m∠T)
There are two things to note here:
- The left side of this inequality is exactly equal to the Triangle Sum Theorem expression
- The right side of the inequality is 2*(m∠S+m∠T)
Substituting
180° > 2* (m∠S+m∠T)
Dividing both sides of the inequality by 2...
90° > m∠S+m∠T
So, the second option must be true.
<u>Option 3. m∠S = m∠T</u>
Not necessarily. While m∠S could equal m∠T, it doesn't have to.
Example 1: m∠S = m∠T = 10°; By the triangle sum Theorem, m∠R = 160°, and the angles satisfy the original inequality.
Example 2: m∠S = 15°, and m∠T = 10°; By the triangle sum Theorem, m∠R = 155°, and the angles still satisfy the original inequality.
So, option 3 does NOT have to be true.
<u>Option 4. m∠R > m∠T</u>
Start with the fact that ∠S is an angle of a triangle, so m∠S cannot be zero or negative, and thus m∠S > 0.
Add m∠T to both sides.
(m∠S) + m∠T > (0) + m∠T
m∠S + m∠T > m∠T
Recall that m∠R > m∠S + m∠T.
By the transitive property of inequalities, m∠R > m∠T.
So, option 4 must be true.
<u>Option 5. m∠R > m∠S</u>
Start with the fact that ∠T is an angle of a triangle, so m∠T cannot be zero or negative, and thus m∠T > 0.
Add m∠S to both sides.
m∠S + (m∠T) > m∠S + (0)
m∠S + m∠T > m∠S
Recall that m∠R > m∠S + m∠T.
By the transitive property of inequalities, m∠R > m∠S.
So, option 5 must be true.
<u>Option 6. m∠S > m∠T</u>
Not necessarily. While m∠S could be greater than m∠T, it doesn't have to be. (See examples 1 and 2 from option 3.)
So, option 6 does NOT have to be true.
