Question

1. abc is a triangle
2. $m\angle a + m\angle b + m\angle c = 180^{\circ}$
3. $m\angle a = 3x + 8$
4. $m\angle b = 3x$
5. $m\angle c = 3x + 10$
6. $3x + 8 + 3x + 3x + 10 = 180$
7. $9x + 18 = 180$
8. $9x = 162$
9. $x = 18$
10. $m\angle c = 3(18) + 10$
11. $m\angle c = 64^{\circ}$

what geometric reason justifies the transition from statement 5 to statement 10?
subtraction property of equality
division property of equality
substitution
symmetric property

Explanation:

In statement 5, \(m\angle C = 3x + 10\). In statement 10, \(x = 18\) (from statement 9) is substituted into the expression for \(m\angle C\) (i.e., \(3x+10\) becomes \(3(18)+10\)). The Substitution Property allows us to replace a variable with its known value. The Subtraction Property of Equality (used in step 8) and Division Property of Equality (used in step 9) are about operations on equations to solve for \(x\), not for substituting a value into an expression. The Symmetric Property (\(a = b\) implies \(b = a\)) is not relevant here. So the geometric reason is Substitution.

Answer:

C. Substitution