Question

5 given: $overline{db}$ bisects $angle abc$, $angle1$ and $angle2$ are complementary; $angle1$ and $angle4$ are complementary; $angle3$ and $angle4$ are complementary; prove: $angle1congangle4$ statements reasons 1. $overline{db}$ bisects $angle abc$; $angle1$ and $angle2$ are complementary; $angle3$ and $angle4$ are complementary; 1. given 2. 2. defn. of angle bisector 3. $mangle1 + mangle2=90^{circ}$; $mangle3 + mangle4 = 90^{circ}$ 3. 4. $mangle1 + mangle2=mangle3 + mangle4$ 4. 5. $mangle1 + mangle2=mangle2 + mangle4$ 5. 6. 6. subtraction property 7. $angle1congangle4$ 7.

Explanation:

Step1: Use angle - bisector definition

$\angle2=\angle3$

Step2: Recall complementary - angle property

Definition of complementary angles (if two angles are complementary, their sum is 90°)

Step3: Transitive property of equality

Since $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle4 = 90^{\circ}$, we can say $m\angle1 + m\angle2=m\angle3 + m\angle4$ by the transitive property of equality.

Step4: Substitute $\angle2$ for $\angle3$

Because $\angle2=\angle3$, we substitute $\angle3$ with $\angle2$ in $m\angle1 + m\angle2=m\angle3 + m\angle4$ to get $m\angle1 + m\angle2=m\angle2 + m\angle4$.

Step5: Subtract $m\angle2$ from both sides

$m\angle1=m\angle4$

Step6: Definition of congruent angles

If two angles have equal measures, they are congruent. So $\angle1\cong\angle4$.

Answer:

1. $\angle2 = \angle3$; 3. Definition of complementary angles; 4. Transitive property of equality; 5. Substitution property (since $\angle2=\angle3$); 6. $m\angle1=m\angle4$; 7. Definition of congruent angles