Question

complete the proof below by filling in the blanks. (c) 25. given: m∠dac = m∠dbc; ab bisects ∠dac; ab bisects ∠dbc. prove: m∠1 = m∠2 statements reasons 1. ab bisects ∠dac. 1. given 2. 2. given 3. m∠1 = 1/2 m∠dac 3. 4. 4. definition of angle bisector 5. m∠dac = m∠dbc 5. 6. 1/2 m∠dac = 1/2 m∠dbc 6. 7. 7. substitution property

Explanation:

Step1: Recall angle - bisector property

Since $\overline{AB}$ bisects $\angle DAC$, by the definition of an angle - bisector, the measure of $\angle1$ is half of the measure of $\angle DAC$, so $m\angle1=\frac{1}{2}m\angle DAC$.

Step2: Use given equality of angles

We are given that $m\angle DAC = m\angle DBC$.

Step3: Apply property of equality

If $m\angle DAC = m\angle DBC$, then multiplying both sides by $\frac{1}{2}$, we get $\frac{1}{2}m\angle DAC=\frac{1}{2}m\angle DBC$ by the multiplication property of equality.

Step4: Substitute

Since $m\angle1=\frac{1}{2}m\angle DAC$ and $\frac{1}{2}m\angle DAC=\frac{1}{2}m\angle DBC$, and also since $\overline{AB}$ bisects $\angle DBC$ so $m\angle2=\frac{1}{2}m\angle DBC$, by substitution, $m\angle1 = m\angle2$.

The completed proof with reasons:

Statements	Reasons
2. $\overline{AB}$ bisects $\angle DBC$	2. Given
3. $m\angle1=\frac{1}{2}m\angle DAC$	3. Definition of angle bisector
4. $m\angle2=\frac{1}{2}m\angle DBC$	4. Definition of angle bisector
5. $m\angle DAC = m\angle DBC$	5. Given
6. $\frac{1}{2}m\angle DAC=\frac{1}{2}m\angle DBC$	6. Multiplication property of equality
7. $m\angle1 = m\angle2$	7. Substitution property

Answer:

Statements	Reasons
2. $\overline{AB}$ bisects $\angle DBC$	2. Given
3. $m\angle1=\frac{1}{2}m\angle DAC$	3. Definition of angle bisector
4. $m\angle2=\frac{1}{2}m\angle DBC$	4. Definition of angle bisector
5. $m\angle DAC = m\angle DBC$	5. Given
6. $\frac{1}{2}m\angle DAC=\frac{1}{2}m\angle DBC$	6. Multiplication property of equality
7. $m\angle1 = m\angle2$	7. Substitution property