Order the steps below to complete the proof. Given: m<p+m<q=90,m<q=5m<p
Prove:
M<q=75
1) Use the given information that m<p+m<q = 90 and m<q = 5m<p to substitute in values and simplify the equation.
2) Substitute the equation m<q = 5m<p into the equation m<p+m<q = 90 to get 5m<p<p+m<q = 90.
3) Use the transitive property of inequality to combine the equations 5m<p<p+m<q = 90 and m<q=5m<p to get 5m<p<p+m<q.
4) Simplify the inequality 5m<p<p+m<q to 5m<2p.
5) Use the given information m<q = 5m<p to substitute in values and simplify the inequality 5m<2p to 5m<10m.
6) Divide both sides of the inequality 5m<10m by 5 to get m<2m.
7) Subtract m from both sides of the inequality m<2m to get 0<m.
8) Use the given information m<q = 5m<p to substitute in values and simplify the inequality 0<m to 0<5m.
9) Divide both sides of the inequality 0<5m by 5 to get 0<m.
10) Simplify the inequality 0<m to m>0.
11) Use the given information m<q = 5m<p to substitute in values and simplify the inequality m>0 to m<q>0.
12) Simplify the inequality m<q>0 to m<q>0.
13) Use the given information m<q=5m<p to substitute in values and conclude that m<q=75.
The order of the steps to complete the proof would be as follows:
1) m<p+m<q = 90
2) m<q = 5m<p
3) 5m<p<p+m<q = 90
4) 5m<p<p+m<q
5) 5m<2p
6) 5m<10m
7) m<2m
8) 0<m
9) 0<5m
10) m>0
11) m<q>0
12) m<q=75
To prove m<q=75, we need to follow these steps:
Step 1: Given: m<p+m<q=90 and m<q=5m<p.
Step 2: Combine the two given statements to get: m<q=5m<p+m<q=90.
Step 3: Set up an equation using the angle sum property of a triangle: m+q+p=180.
Step 4: Substitute the value of p from the given statements into the equation: m+q+5m=180.
Step 5: Simplify the equation: 6m+q=180.
Step 6: Substitute the expression for q from 5m<p into the equation: 6m+5m=180.
Step 7: Simplify the equation: 11m=180.
Step 8: Solve for m: m=180/11.
Step 9: Substitute the value of m into the equation for q from 5m<p: q=5(180/11).
Step 10: Simplify the equation: q=900/11.
Step 11: Substitute the values of m and q into the equation m<q=5m<p: 180/11<900/11=5(180/11).
Step 12: Simplify the equation: 180<900.
Step 13: Therefore, the statement m<q=75 is proven to be true, as we have shown that m<q=180/11<900/11=75.